Transmission line

<h2 id="overview">Overview</h2>
<p>Ordinary electrical cables suffice to carry low frequency <a href="/facts/Alternating_current/qxICS3xa">alternating current</a> (AC), such as <a href="/facts/Mains_power/rjE90Hg0">mains power</a>, which reverses direction 100 to 120 times per second, and <a href="/facts/Audio_signal/rFSTI9IF">audio signals</a>.  However, they are not generally used to carry currents in the <a href="/facts/Radio_frequency/aWku7h4R">radio frequency</a> range,<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> above about 30 kHz, because the energy tends to radiate off the cable as <a href="/facts/Radio_wave/XmB6WPpF">radio waves</a>, causing power losses.  Radio frequency currents also tend to reflect from discontinuities in the cable such as <a href="/facts/Electrical_connector/rJh0cHs6">connectors</a> and joints, and travel back down the cable toward the source.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>  These reflections act as bottlenecks, preventing the signal power from reaching the destination.  Transmission lines use specialized construction, and <a href="/facts/Impedance_matching/0mdy9TAp">impedance matching</a>, to carry electromagnetic signals with minimal reflections and power losses.  The distinguishing feature of most transmission lines is that they have uniform cross sectional dimensions along their length, giving them a uniform <a href="/facts/Electrical_impedance/I06NSXtN">impedance</a>, called the <i><a href="/facts/Characteristic_impedance/oUg0mhah">characteristic impedance</a></i>,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> to prevent reflections.  Types of transmission line include parallel line (<a href="/facts/Ladder_line/BfRCmX8A">ladder line</a>, <a href="/facts/Twisted_pair/L426Jk2C">twisted pair</a>), <a href="/facts/Coaxial_cable/aG9FfCpL">coaxial cable</a>, and <a href="/facts/Planar_transmission_line/PtQSrIpB">planar transmission lines</a> such as <a href="/facts/Stripline/RNdJWRQJ">stripline</a> and <a href="/facts/Microstrip/QkKwyAN7">microstrip</a>.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The higher the frequency of electromagnetic waves moving through a given cable or medium, the shorter the <a href="/facts/Wavelength/kcJU0ymz">wavelength</a> of the waves.  Transmission lines become necessary when the transmitted frequency's wavelength is sufficiently short that the length of the cable becomes a significant part of a wavelength.
</p><p>At frequencies of <a href="/facts/Microwave/HmVSN31Y">microwave</a> and higher, power losses in transmission lines become excessive, and <a href="/facts/Waveguide/tJS9Y0GH">waveguides</a> are used instead,<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> which function as "pipes" to confine and guide the electromagnetic waves.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>  Some sources define waveguides as a type of transmission line;<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> however, this article will not include them.
</p>
<h2 id="history">History</h2>
<p>Mathematical analysis of the behaviour of electrical transmission lines grew out of the work of <a href="/facts/James_Clerk_Maxwell/jtYyze61">James Clerk Maxwell</a>, <a href="/facts/Lord_Kelvin/LTIxG1jV">Lord Kelvin</a>, and <a href="/facts/Oliver_Heaviside/ktYBpAt7">Oliver Heaviside</a>. In 1855, Lord Kelvin formulated a diffusion model of the current in a submarine cable. The model correctly predicted the poor performance of the 1858 trans-Atlantic <a href="/facts/Submarine_communications_cable/WA4elMsQ">submarine telegraph cable</a>. In 1885, Heaviside published the first papers that described his analysis of propagation in cables and the modern form of the <a href="/facts/Telegrapher%27s_equations/U6ky4CFx">telegrapher's equations</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
<h2 id="the-four-terminal-model">The four terminal model</h2>

<p>For the purposes of analysis, an electrical transmission line can be modelled as a <a href="/facts/Two-port_network/cqUrBTYq">two-port network</a> (also called a quadripole), as follows:
</p><p>In the simplest case, the network is assumed to be linear (i.e. the <a href="/facts/Complex_number/2w7mqI7e">complex</a> voltage across either port is proportional to the complex current flowing into it when there are no reflections), and the two ports are assumed to be interchangeable.  If the transmission line is uniform along its length, then its behaviour is largely described by two parameters called <i><a href="/facts/Characteristic_impedance/oUg0mhah">characteristic impedance</a></i>, symbol Z0 and <i><a href="/facts/Signal_propagation_delay/tUcrAwwe">propagation delay</a></i>, symbol 
  
    
      
        
          τ
          
            p
          
        
      
    
    {\displaystyle \tau _{p}}
  
.  Z0 is the ratio of the complex voltage of a given wave to the complex current of the same wave at any point on the line. Typical values of Z0 are 50 or 75 <a href="/facts/Ohm_(unit)/QNVrlGjw">ohms</a> for a <a href="/facts/Coaxial_cable/aG9FfCpL">coaxial cable</a>, about 100 ohms for a twisted pair of wires, and about 300 ohms for a common type of untwisted pair used in radio transmission. Propagation delay is proportional to the length of the transmission line and is never less than the length divided by the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a>.  Typical delays for modern communication transmission lines vary from 3.33 ns/m to 5 ns/m.
</p><p>When sending power down a transmission line, it is usually desirable that as much power as possible will be absorbed by the load and as little as possible will be reflected back to the source.  This can be ensured by making the load impedance equal to Z0, in which case the transmission line is said to be <i><a href="/facts/Impedance_matching/0mdy9TAp">matched</a></i>.
</p>

<p>Some of the power that is fed into a transmission line is lost because of its resistance.  This effect is called <i>ohmic</i> or <i>resistive</i> loss (see <a href="/facts/Ohmic_heating/eANEHVzP">ohmic heating</a>).  At high frequencies, another effect called <i><a href="/facts/Dielectric_absorption/6J1NXsxD">dielectric loss</a></i> becomes significant, adding to the losses caused by resistance.  Dielectric loss is caused when the insulating material inside the transmission line absorbs energy from the alternating electric field and converts it to <a href="/facts/Heat/FjDGssCs">heat</a> (see <a href="/facts/Dielectric_heating/LvxNcysf">dielectric heating</a>). The transmission line is modelled with a resistance (R) and inductance (L) in series with a capacitance (C) and conductance (G) in parallel. The resistance and conductance contribute to the loss in a transmission line.
</p><p>The total loss of power in a transmission line is often specified in <a href="/facts/Decibels/0LyxAPXh">decibels</a> per <a href="/facts/Metre/hPhzc1CV">metre</a> (dB/m), and usually depends on the frequency of the signal.  The manufacturer often supplies a chart showing the loss in dB/m at a range of frequencies.  A loss of 3 dB corresponds approximately to a halving of the power.
</p><p>Propagation delay is often specified in units of <a href="/facts/Nanoseconds/gARLTp4c">nanoseconds</a> per metre.  While propagation delay usually depends on the frequency of the signal, transmission lines are typically operated over frequency ranges where the propagation delay is approximately constant.
</p>

<h2 id="telegraphers-equations">Telegrapher's equations</h2>
<p class="note">Main article: <a href="/facts/Telegrapher%27s_equations/U6ky4CFx">Telegrapher's equations</a></p>
<p class="note">See also: <a href="/facts/Reflections_on_copper_lines/49IPJpWz">Reflections on copper lines</a></p>
<p>The telegrapher's equations (or just telegraph equations) are a pair of linear differential equations which describe the <a href="/facts/Voltage/EN2po6N2">voltage</a> (
  
    
      
        V
      
    
    {\displaystyle V}
  
) and <a href="/facts/Electric_current/Dc1xdnUX">current</a> (
  
    
      
        I
      
    
    {\displaystyle I}
  
) on an electrical transmission line with distance and time. They were developed by <a href="/facts/Oliver_Heaviside/ktYBpAt7">Oliver Heaviside</a> who created the <i>transmission line model</i>, and are based on <a href="/facts/Maxwell%27s_equations/LfJ1drHS">Maxwell's equations</a>.
</p>

<p>The transmission line model is an example of the <a href="/facts/Distributed-element_model/UxArsYdy">distributed-element model</a>. It represents the transmission line as an infinite series of two-port elementary components, each representing an infinitesimally short segment of the transmission line:
</p>
<ul><li>The distributed resistance 
  
    
      
        R
      
    
    {\displaystyle R}
  
 of the conductors is represented by a series resistor (expressed in <a href="/facts/Ohm/QNVrlGjw">ohms</a> per unit length).</li>
<li>The distributed inductance 
  
    
      
        L
      
    
    {\displaystyle L}
  
 (due to the <a href="/facts/Magnetic_field/Ta8Ev588">magnetic field</a> around the wires, <a href="/facts/Self-inductance/oIxj7JxX">self-inductance</a>, etc.) is represented by a series <a href="/facts/Inductor/YnfAXB3x">inductor</a> (in <a href="/facts/Henry_(unit)/uwVv9zue">henries</a> per unit length).</li>
<li>The capacitance 
  
    
      
        C
      
    
    {\displaystyle C}
  
 between the two conductors is represented by a <a href="/facts/Shunt_(electrical)/EEBIeEdo">shunt</a> <a href="/facts/Capacitor/hyowqv7n">capacitor</a> (in <a href="/facts/Farad/gPAYcVoE">farads</a> per unit length).</li>
<li>The <a href="/facts/Electric_conductance/hxSt0UAj">conductance</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 of the dielectric material separating the two conductors is represented by a shunt resistor between the signal wire and the return wire (in <a href="/facts/Siemens_(unit)/kLUHhcCj">siemens</a> per unit length).</li></ul>
<p>The model consists of an <i>infinite series</i> of the elements shown in the figure, and the values of the components are specified <i>per unit length</i> so the picture of the component can be misleading. 
  
    
      
        R
      
    
    {\displaystyle R}
  
, 
  
    
      
        L
      
    
    {\displaystyle L}
  
, 
  
    
      
        C
      
    
    {\displaystyle C}
  
, and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 may also be functions of frequency. An alternative notation is to use 
  
    
      
        
          R
          ′
        
      
    
    {\displaystyle R'}
  
, 
  
    
      
        
          L
          ′
        
      
    
    {\displaystyle L'}
  
, 
  
    
      
        
          C
          ′
        
      
    
    {\displaystyle C'}
  
 and 
  
    
      
        
          G
          ′
        
      
    
    {\displaystyle G'}
  
 to emphasize that the values are derivatives with respect to length.  These quantities can also be known as the <a href="/facts/Primary_line_constants/AfXhVc9o">primary line constants</a> to distinguish from the secondary line constants derived from them, these being the <a href="/facts/Propagation_constant/3XpFb0q1">propagation constant</a>, <a href="/facts/Attenuation_constant/3XpFb0q1">attenuation constant</a> and <a href="/facts/Phase_constant/3XpFb0q1">phase constant</a>.
</p><p>The line voltage 
  
    
      
        V
        (
        x
        )
      
    
    {\displaystyle V(x)}
  
 and the current 
  
    
      
        I
        (
        x
        )
      
    
    {\displaystyle I(x)}
  
 can be expressed in the frequency domain as
</p>

∂
              V
              (
              x
              )
            
            
              ∂
              x
            
          
        
        =
        −
        (
        R
        +
        j
        
        ω
        
        L
        )
        
        I
        (
        x
        )
      
    
    {\displaystyle {\frac {\partial V(x)}{\partial x}}=-(R+j\,\omega \,L)\,I(x)}

∂
              I
              (
              x
              )
            
            
              ∂
              x
            
          
        
        =
        −
        (
        G
        +
        j
        
        ω
        
        C
        )
        
        V
        (
        x
        )
         
        
        .
      
    
    {\displaystyle {\frac {\partial I(x)}{\partial x}}=-(G+j\,\omega \,C)\,V(x)~\,.}

(see <a href="/facts/Differential_equation/9MGbE3Ca">differential equation</a>, angular frequency <a href="/facts/Angular_frequency/VqLm3L3R">ω</a> and imaginary unit <a href="/facts/Imaginary_unit/1SCllQlU">j</a>)
<h3>Special case of a lossless line</h3>
<p>When the elements 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
 are negligibly small the transmission line is considered as a lossless structure. In this hypothetical case, the model depends only on the 
  
    
      
        L
      
    
    {\displaystyle L}
  
 and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 elements which greatly simplifies the analysis. For a lossless transmission line, the second order steady-state Telegrapher's equations are:
</p>

∂
                
                  2
                
              
              V
              (
              x
              )
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        +
        
          ω
          
            2
          
        
        L
        
        C
        
        V
        (
        x
        )
        =
        0
      
    
    {\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,V(x)=0}

∂
                
                  2
                
              
              I
              (
              x
              )
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        +
        
          ω
          
            2
          
        
        L
        
        C
        
        I
        (
        x
        )
        =
        0
         
        
        .
      
    
    {\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}+\omega ^{2}L\,C\,I(x)=0~\,.}

<p>These are <a href="/facts/Wave_equation/J2ksD2Az">wave equations</a> which have <a href="/facts/Plane_wave/0aDUbNhQ">plane waves</a> with equal propagation speed in the forward and reverse directions as solutions. The physical significance of this is that electromagnetic waves propagate down transmission lines and in general, there is a reflected component that interferes with the original signal. These equations are fundamental to transmission line theory.
</p>
<h3>General case of a line with losses</h3>
<p>In the general case the loss terms, 
  
    
      
        R
      
    
    {\displaystyle R}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
, are both included, and the full form of the Telegrapher's equations become:
</p>

∂
                
                  2
                
              
              V
              (
              x
              )
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        =
        
          γ
          
            2
          
        
        V
        (
        x
        )
        
      
    
    {\displaystyle {\frac {\partial ^{2}V(x)}{\partial x^{2}}}=\gamma ^{2}V(x)\,}

∂
                
                  2
                
              
              I
              (
              x
              )
            
            
              ∂
              
                x
                
                  2
                
              
            
          
        
        =
        
          γ
          
            2
          
        
        I
        (
        x
        )
        
      
    
    {\displaystyle {\frac {\partial ^{2}I(x)}{\partial x^{2}}}=\gamma ^{2}I(x)\,}

<p>where 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is the (<a href="/facts/Complex_number/2w7mqI7e">complex</a>) <a href="/facts/Propagation_constant/3XpFb0q1">propagation constant</a>. These equations are fundamental to transmission line theory. They are also <a href="/facts/Wave_equation/J2ksD2Az">wave equations</a>, and have solutions similar to the special case, but which are a mixture of sines and cosines with exponential decay factors. Solving for the propagation constant 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 in terms of the primary parameters 
  
    
      
        R
      
    
    {\displaystyle R}
  
, 
  
    
      
        L
      
    
    {\displaystyle L}
  
, 
  
    
      
        G
      
    
    {\displaystyle G}
  
, and 
  
    
      
        C
      
    
    {\displaystyle C}
  
 gives:
</p>

γ
        =
        
          
            (
            R
            +
            j
            
            ω
            
            L
            )
            (
            G
            +
            j
            
            ω
            
            C
            )
            
          
        
      
    
    {\displaystyle \gamma ={\sqrt {(R+j\,\omega \,L)(G+j\,\omega \,C)\,}}}

<p>and the characteristic impedance can be expressed as
</p>

Z
          
            0
          
        
        =
        
          
            
              
                
                  R
                  +
                  j
                  
                  ω
                  
                  L
                
                
                  G
                  +
                  j
                  
                  ω
                  
                  C
                
              
            
            
          
        
         
        
        .
      
    
    {\displaystyle Z_{0}={\sqrt {{\frac {R+j\,\omega \,L}{G+j\,\omega \,C}}\,}}~\,.}

<p>The solutions for 
  
    
      
        V
        (
        x
        )
      
    
    {\displaystyle V(x)}
  
 and 
  
    
      
        I
        (
        x
        )
      
    
    {\displaystyle I(x)}
  
 are:
</p>

V
        (
        x
        )
        =
        
          V
          
            (
            +
            )
          
        
        
          e
          
            −
            γ
            
            x
          
        
        +
        
          V
          
            (
            −
            )
          
        
        
          e
          
            +
            γ
            
            x
          
        
        
      
    
    {\displaystyle V(x)=V_{(+)}e^{-\gamma \,x}+V_{(-)}e^{+\gamma \,x}\,}

I
        (
        x
        )
        =
        
          
            1
            
              Z
              
                0
              
            
          
        
        
        
          (
          
            
              V
              
                (
                +
                )
              
            
            
              e
              
                −
                γ
                
                x
              
            
            −
            
              V
              
                (
                −
                )
              
            
            
              e
              
                +
                γ
                
                x
              
            
          
          )
        
         
        
        .
      
    
    {\displaystyle I(x)={\frac {1}{Z_{0}}}\,\left(V_{(+)}e^{-\gamma \,x}-V_{(-)}e^{+\gamma \,x}\right)~\,.}

<p>The constants 
  
    
      
        
          V
          
            (
            ±
            )
          
        
      
    
    {\displaystyle V_{(\pm )}}
  
 must be determined from boundary conditions. For a voltage pulse 
  
    
      
        
          V
          
            
              i
              n
            
          
        
        (
        t
        )
        
      
    
    {\displaystyle V_{\mathrm {in} }(t)\,}
  
, starting at 
  
    
      
        x
        =
        0
      
    
    {\displaystyle x=0}
  
 and moving in the positive 
  
    
      
        x
      
    
    {\displaystyle x}
  
 direction, then the transmitted pulse 
  
    
      
        
          V
          
            
              o
              u
              t
            
          
        
        (
        x
        ,
        t
        )
        
      
    
    {\displaystyle V_{\mathrm {out} }(x,t)\,}
  
 at position 
  
    
      
        x
      
    
    {\displaystyle x}
  
 can be obtained by computing the Fourier Transform, 
  
    
      
        
          
            
              V
              ~
            
          
        
        (
        ω
        )
      
    
    {\displaystyle {\tilde {V}}(\omega )}
  
, of 
  
    
      
        
          V
          
            
              i
              n
            
          
        
        (
        t
        )
        
      
    
    {\displaystyle V_{\mathrm {in} }(t)\,}
  
, attenuating each frequency component by 
  
    
      
        
          e
          
            −
            Re
            ⁡
            (
            γ
            )
            
            x
          
        
        
      
    
    {\displaystyle e^{-\operatorname {Re} (\gamma )\,x}\,}
  
, advancing its phase by 
  
    
      
        −
        Im
        ⁡
        (
        γ
        )
        
        x
        
      
    
    {\displaystyle -\operatorname {Im} (\gamma )\,x\,}
  
, and taking the <a href="/facts/Fourier_inversion_theorem/aM1aMkf0">inverse Fourier Transform</a>. The real and imaginary parts of 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 can be computed as
</p>

Re
        ⁡
        (
        γ
        )
        =
        α
        =
        (
        
          a
          
            2
          
        
        +
        
          b
          
            2
          
        
        
          )
          
            1
            
              /
            
            4
          
        
        cos
        ⁡
        (
        ψ
        )
        
      
    
    {\displaystyle \operatorname {Re} (\gamma )=\alpha =(a^{2}+b^{2})^{1/4}\cos(\psi )\,}

Im
        ⁡
        (
        γ
        )
        =
        β
        =
        (
        
          a
          
            2
          
        
        +
        
          b
          
            2
          
        
        
          )
          
            1
            
              /
            
            4
          
        
        sin
        ⁡
        (
        ψ
        )
        
      
    
    {\displaystyle \operatorname {Im} (\gamma )=\beta =(a^{2}+b^{2})^{1/4}\sin(\psi )\,}

a
         
        ≡
         
        R
        
        G
        
        −
        
          ω
          
            2
          
        
        L
        
        C
         
         
        =
         
        
          ω
          
            2
          
        
        L
        
        C
        
        
          [
          
            
              (
              
                
                  R
                  
                    ω
                    L
                  
                
              
              )
            
            
              (
              
                
                  G
                  
                    ω
                    C
                  
                
              
              )
            
            −
            1
          
          ]
        
      
    
    {\displaystyle a~\equiv ~R\,G\,-\omega ^{2}L\,C\ ~=~\omega ^{2}L\,C\,\left[\left({\frac {R}{\omega L}}\right)\left({\frac {G}{\omega C}}\right)-1\right]}

b
         
        ≡
         
        ω
        
        C
        
        R
        +
        ω
        
        L
        
        G
         
        =
         
        
          ω
          
            2
          
        
        L
        
        C
        
        
          (
          
            
              
                R
                
                  ω
                  
                  L
                
              
            
            +
            
              
                G
                
                  ω
                  
                  C
                
              
            
          
          )
        
      
    
    {\displaystyle b~\equiv ~\omega \,C\,R+\omega \,L\,G~=~\omega ^{2}L\,C\,\left({\frac {R}{\omega \,L}}+{\frac {G}{\omega \,C}}\right)}

<p>the right-hand expressions holding when neither 
  
    
      
        L
      
    
    {\displaystyle L}
  
, nor  
  
    
      
        C
      
    
    {\displaystyle C}
  
, nor 
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
 is zero, and with
</p>

ψ
         
        ≡
         
        
          
            
              1
              2
            
          
        
        atan2
        ⁡
        (
        b
        ,
        a
        )
        
      
    
    {\displaystyle \psi ~\equiv ~{\tfrac {1}{2}}\operatorname {atan2} (b,a)\,}

<p>where <a href="/facts/Atan2/0JDnUYfP">atan2</a> is the everywhere-defined form of two-parameter arctangent function, with arbitrary value zero when both arguments are zero.
</p><p>Alternatively, the complex square root can be evaluated algebraically, to yield:
</p>

α
        =
        
          
            
              ±
              b
            
            
              2
              
                (
                
                  −
                  a
                  +
                  
                    
                      
                        a
                        
                          2
                        
                      
                      +
                      
                        b
                        
                          2
                        
                      
                    
                  
                
                )
              
               
            
          
        
        ,
      
    
    {\displaystyle \alpha ={\frac {\pm b}{\sqrt {2\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}}},}

β
        =
        ±
        
          
            
              
                
                  1
                  2
                
              
            
            
              (
              
                −
                a
                +
                
                  
                    
                      a
                      
                        2
                      
                    
                    +
                    
                      b
                      
                        2
                      
                    
                  
                
              
              )
            
             
          
        
        ,
      
    
    {\displaystyle \beta =\pm {\sqrt {{\tfrac {1}{2}}\left(-a+{\sqrt {a^{2}+b^{2}}}\right)~}},}

<p>with the plus or minus signs chosen opposite to the direction of the wave's motion through the conducting medium. (a  is usually negative, since 
  
    
      
        G
      
    
    {\displaystyle G}
  
 and 
  
    
      
        R
      
    
    {\displaystyle R}
  
 are typically much smaller than 
  
    
      
        ω
        C
      
    
    {\displaystyle \omega C}
  
 and 
  
    
      
        ω
        L
      
    
    {\displaystyle \omega L}
  
, respectively, so  −a  is usually positive. b is always positive.)
</p>
<h3>Special, low loss case</h3>
<p>For small losses and high frequencies, the general equations can be simplified: If 
  
    
      
        
          
            
              R
              
                ω
                
                L
              
            
          
        
        ≪
        1
      
    
    {\displaystyle {\tfrac {R}{\omega \,L}}\ll 1}
  
 and 
  
    
      
        
          
            
              G
              
                ω
                
                C
              
            
          
        
        ≪
        1
      
    
    {\displaystyle {\tfrac {G}{\omega \,C}}\ll 1}
  
 then
</p>

Re
        ⁡
        (
        γ
        )
        =
        α
        ≈
        
          
            
              1
              2
            
          
        
        
          
            L
            
            C
            
          
        
        
        
          (
          
            
              
                R
                L
              
            
            +
            
              
                G
                C
              
            
          
          )
        
        
      
    
    {\displaystyle \operatorname {Re} (\gamma )=\alpha \approx {\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,}

Im
        ⁡
        (
        γ
        )
        =
        β
        ≈
        ω
        
        
          
            L
            
            C
            
          
        
         
        .
        
      
    
    {\displaystyle \operatorname {Im} (\gamma )=\beta \approx \omega \,{\sqrt {L\,C\,}}~.\,}

<p>Since an advance in phase by 
  
    
      
        −
        ω
        
        δ
      
    
    {\displaystyle -\omega \,\delta }
  
 is equivalent to a time delay by 
  
    
      
        δ
      
    
    {\displaystyle \delta }
  
, 
  
    
      
        
          V
          
            o
            u
            t
          
        
        (
        t
        )
      
    
    {\displaystyle V_{out}(t)}
  
 can be simply computed as
</p>

V
          
            
              o
              u
              t
            
          
        
        (
        x
        ,
        t
        )
        ≈
        
          V
          
            
              i
              n
            
          
        
        (
        t
        −
        
          
            L
            
            C
            
          
        
        
        x
        )
        
        
          e
          
            −
            
              
                
                  1
                  2
                
              
            
            
              
                L
                
                C
                
              
            
            
            
              (
              
                
                  
                    R
                    L
                  
                
                +
                
                  
                    G
                    C
                  
                
              
              )
            
            
            x
          
        
        .
        
      
    
    {\displaystyle V_{\mathrm {out} }(x,t)\approx V_{\mathrm {in} }(t-{\sqrt {L\,C\,}}\,x)\,e^{-{\tfrac {1}{2}}{\sqrt {L\,C\,}}\,\left({\frac {R}{L}}+{\frac {G}{C}}\right)\,x}.\,}

<h3>Heaviside condition</h3>
<p class="note">Main article: <a href="/facts/Heaviside_condition/J8ejKVYT">Heaviside condition</a></p>
<p>The <a href="/facts/Heaviside_condition/J8ejKVYT">Heaviside condition</a> is 
  
    
      
        
          
            G
            C
          
        
        =
        
          
            R
            L
          
        
      
    
    {\displaystyle {\frac {G}{C}}={\frac {R}{L}}}
  
.  
</p><p>If R, G, L, and C are constants that are <i>not</i> frequency dependent and the Heaviside condition is met, 
then waves travel down the transmission line without <a href="/facts/Dispersion_(optics)/cUdHnYXz">dispersion</a> distortion.
</p>
<h2 id="input-impedance-of-transmission-line">Input impedance of transmission line</h2>

<p>The <a href="/facts/Characteristic_impedance/oUg0mhah">characteristic impedance</a> 
  
    
      
        
          Z
          
            0
          
        
      
    
    {\displaystyle Z_{0}}
  
 of a transmission line is the ratio of the amplitude of a <i>single</i> voltage wave to its current wave. Since most transmission lines also have a reflected wave, the characteristic impedance is generally not the impedance that is measured on the line.
</p><p>The impedance measured at a given distance 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 from the load impedance 
  
    
      
        
          Z
          
            
              L
            
          
        
      
    
    {\displaystyle Z_{\mathrm {L} }}
  
 may be expressed as
</p>

Z
          
            
              i
              n
            
          
        
        
          (
          ℓ
          )
        
        =
        
          
            
              V
              (
              ℓ
              )
            
            
              I
              (
              ℓ
              )
            
          
        
        =
        
          Z
          
            0
          
        
        
          
            
              1
              +
              
                
                  
                    Γ
                  
                
                
                  
                    L
                  
                
              
              
                e
                
                  −
                  2
                  γ
                  ℓ
                
              
            
            
              1
              −
              
                
                  
                    Γ
                  
                
                
                  
                    L
                  
                
              
              
                e
                
                  −
                  2
                  γ
                  ℓ
                
              
            
          
        
      
    
    {\displaystyle Z_{\mathrm {in} }\left(\ell \right)={\frac {V(\ell )}{I(\ell )}}=Z_{0}{\frac {1+{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}{1-{\mathit {\Gamma }}_{\mathrm {L} }e^{-2\gamma \ell }}}}
  
,
<p>where 
  
    
      
        γ
      
    
    {\displaystyle \gamma }
  
 is the propagation constant and 
  
    
      
        
          
            
              Γ
            
          
          
            
              L
            
          
        
        =
        
          
            
              
              
                Z
                
                  
                    L
                  
                
              
              −
              
                Z
                
                  0
                
              
              
            
            
              
                Z
                
                  
                    L
                  
                
              
              +
              
                Z
                
                  0
                
              
            
          
        
      
    
    {\displaystyle {\mathit {\Gamma }}_{\mathrm {L} }={\frac {\,Z_{\mathrm {L} }-Z_{0}\,}{Z_{\mathrm {L} }+Z_{0}}}}
  
 is the voltage <a href="/facts/Reflection_coefficient/se689p3C">reflection coefficient</a> measured at the load end of the transmission line. Alternatively, the above formula can be rearranged to express the input impedance in terms of the load impedance rather than the load voltage reflection coefficient:
</p>

Z
          
            
              i
              n
            
          
        
        (
        ℓ
        )
        =
        
          Z
          
            0
          
        
        
        
          
            
              
                Z
                
                  
                    L
                  
                
              
              +
              
                Z
                
                  0
                
              
              tanh
              ⁡
              
                (
                
                  γ
                  ℓ
                
                )
              
            
            
              
                Z
                
                  0
                
              
              +
              
                Z
                
                  
                    L
                  
                
              
              
              tanh
              ⁡
              
                (
                
                  γ
                  ℓ
                
                )
              
            
          
        
      
    
    {\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}\,{\frac {Z_{\mathrm {L} }+Z_{0}\tanh \left(\gamma \ell \right)}{Z_{0}+Z_{\mathrm {L} }\,\tanh \left(\gamma \ell \right)}}}
  
.
<h3>Input impedance of lossless transmission line</h3>
<p>For a lossless transmission line, the propagation constant is purely imaginary, 
  
    
      
        γ
        =
        j
        
        β
      
    
    {\displaystyle \gamma =j\,\beta }
  
, so the above formulas can be rewritten as
</p>

Z
          
            
              i
              n
            
          
        
        (
        ℓ
        )
        =
        
          Z
          
            0
          
        
        
          
            
              
                Z
                
                  
                    L
                  
                
              
              +
              j
              
              
                Z
                
                  0
                
              
              
              tan
              ⁡
              (
              β
              ℓ
              )
            
            
              
                Z
                
                  0
                
              
              +
              j
              
              
                Z
                
                  
                    L
                  
                
              
              tan
              ⁡
              (
              β
              ℓ
              )
            
          
        
      
    
    {\displaystyle Z_{\mathrm {in} }(\ell )=Z_{0}{\frac {Z_{\mathrm {L} }+j\,Z_{0}\,\tan(\beta \ell )}{Z_{0}+j\,Z_{\mathrm {L} }\tan(\beta \ell )}}}

<p>where 
  
    
      
        β
        =
        
          
            
              
              2
              π
              
            
            λ
          
        
      
    
    {\displaystyle \beta ={\frac {\,2\pi \,}{\lambda }}}
  
 is the <a href="/facts/Wavenumber/cLaLAeW5">wavenumber</a>.
</p><p>In calculating 
  
    
      
        β
        ,
      
    
    {\displaystyle \beta ,}
  
 the wavelength is generally different <i>inside</i> the transmission line to what it would be in free-space. Consequently, the velocity factor of the material the transmission line is made of needs to be taken into account when doing such a calculation.
</p>
<h3>Special cases of lossless transmission lines</h3>
<h4>Half wave length</h4>
<p>For the special case where 
  
    
      
        β
        
        ℓ
        =
        n
        
        π
      
    
    {\displaystyle \beta \,\ell =n\,\pi }
  
 where n is an integer (meaning that the length of the line is a multiple of half a wavelength), the expression reduces to the load impedance so that
</p>

Z
          
            
              i
              n
            
          
        
        =
        
          Z
          
            
              L
            
          
        
        
      
    
    {\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }\,}

<p>for all 
  
    
      
        n
        
        .
      
    
    {\displaystyle n\,.}
  
 This includes the case when 
  
    
      
        n
        =
        0
      
    
    {\displaystyle n=0}
  
, meaning that the length of the transmission line is negligibly small compared to the wavelength. The physical significance of this is that the transmission line can be ignored (i.e. treated as a wire) in either case.
</p>
<h4>Quarter wave length</h4>
<p class="note">Main article: <a href="/facts/Quarter-wave_impedance_transformer/3NqfTQkh">quarter-wave impedance transformer</a></p>
<p>For the case where the length of the line is one quarter wavelength long, or an odd multiple of a quarter wavelength long, the input impedance becomes
</p>

Z
          
            
              i
              n
            
          
        
        =
        
          
            
              Z
              
                0
              
              
                2
              
            
            
              Z
              
                
                  L
                
              
            
          
        
         
        
        .
      
    
    {\displaystyle Z_{\mathrm {in} }={\frac {Z_{0}^{2}}{Z_{\mathrm {L} }}}~\,.}

<h4>Matched load</h4>
<p>Another special case is when the load impedance is equal to the characteristic impedance of the line (i.e. the line is <i>matched</i>), in which case the impedance reduces to the characteristic impedance of the line so that
</p>

Z
          
            
              i
              n
            
          
        
        =
        
          Z
          
            
              L
            
          
        
        =
        
          Z
          
            0
          
        
        
      
    
    {\displaystyle Z_{\mathrm {in} }=Z_{\mathrm {L} }=Z_{0}\,}

<p>for all 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 and all 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.
</p>
<h4>Short</h4>

<p class="note">Main article: <a href="/facts/Stub_(electronics)/urTnUgYY">stub</a></p>
<p>For the case of a shorted load (i.e. 
  
    
      
        
          Z
          
            
              L
            
          
        
        =
        0
      
    
    {\displaystyle Z_{\mathrm {L} }=0}
  
), the input impedance is purely imaginary and a periodic function of position and wavelength (frequency)
</p>

Z
          
            
              i
              n
            
          
        
        (
        ℓ
        )
        =
        j
        
        
          Z
          
            0
          
        
        
        tan
        ⁡
        (
        β
        ℓ
        )
        .
        
      
    
    {\displaystyle Z_{\mathrm {in} }(\ell )=j\,Z_{0}\,\tan(\beta \ell ).\,}

<h4>Open</h4>
<p class="note">Main article: <a href="/facts/Stub_(electronics)/urTnUgYY">stub</a></p>
<p>For the case of an open load (i.e. 
  
    
      
        
          Z
          
            
              L
            
          
        
        =
        ∞
      
    
    {\displaystyle Z_{\mathrm {L} }=\infty }
  
), the input impedance is once again imaginary and periodic
</p>

Z
          
            
              i
              n
            
          
        
        (
        ℓ
        )
        =
        −
        j
        
        
          Z
          
            0
          
        
        cot
        ⁡
        (
        β
        ℓ
        )
        .
        
      
    
    {\displaystyle Z_{\mathrm {in} }(\ell )=-j\,Z_{0}\cot(\beta \ell ).\,}

<h2 id="matrix-parameters">Matrix parameters</h2>
<p>The <a href="/facts/Electronic_circuit_simulation/HH9KUc0K">simulation</a> of transmission lines embedded into larger systems generally utilize <a href="/facts/Admittance_parameters/0PJ6u6IH">admittance parameters</a> (Y matrix), <a href="/facts/Impedance_parameters/4ocIpsWI">impedance parameters</a> (Z matrix), and/or <a href="/facts/Scattering_parameters/z6ffeKh6">scattering parameters</a> (S matrix) that embodies the full transmission line model needed to support the simulation.
</p>
<h3>Admittance parameters</h3>
<p>Admittance (Y) parameters may be defined by applying a fixed voltage to one port (V1) of a transmission line with the other end shorted to ground and measuring the resulting current running into each port (I1, I2)<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> and computing the admittance on each port as a ratio of I/V  The admittance parameter Y11 is I1/V1, and the admittance parameter Y12 is I2/V1.  Since transmission lines are electrically passive and symmetric devices, Y12 = Y21, and Y11 = Y22.
</p><p>For lossless and lossy transmission lines respectively, the Y parameter matrix is as follows:<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a>
</p><p>
  
    
      
        
          Y
          
            Lossless
          
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        −
                        j
                        c
                        o
                        t
                        (
                        β
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
                
                  
                    
                      
                        j
                        c
                        s
                        c
                        (
                        β
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        j
                        c
                        s
                        c
                        (
                        β
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
                
                  
                    
                      
                        −
                        j
                        c
                        o
                        t
                        (
                        β
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
              
            
            ]
          
        
        
           
        
        
          Y
          
            Lossy
          
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        c
                        o
                        t
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
                
                  
                    
                      
                        −
                        c
                        s
                        c
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
              
              
                
                  
                    
                      
                        −
                        c
                        s
                        c
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
                
                  
                    
                      
                        c
                        o
                        t
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        Z
                        
                          o
                        
                      
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle Y_{\text{Lossless}}={\begin{bmatrix}{\frac {-jcot(\beta l)}{Z_{o}}}&{\frac {jcsc(\beta l)}{Z_{o}}}\\{\frac {jcsc(\beta l)}{Z_{o}}}&{\frac {-jcot(\beta l)}{Z_{o}}}\end{bmatrix}}{\text{    }}Y_{\text{Lossy}}={\begin{bmatrix}{\frac {coth(\gamma l)}{Z_{o}}}&{\frac {-csch(\gamma l)}{Z_{o}}}\\{\frac {-csch(\gamma l)}{Z_{o}}}&{\frac {coth(\gamma l)}{Z_{o}}}\end{bmatrix}}}

</p>
<h3>Impedance parameters</h3>
<p>Impedance (Z) parameter may defines by applying a fixed current into one port (I1) of a transmission line with the other port open and measuring the resulting voltage on each port (V1, V2)<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> and computing the impedance parameter Z11 is V1/I1, and the impedance parameter Z12 is V2/I1.  Since transmission lines are electrically passive and symmetric devices, V12 = V21, and V11 = V22.
</p><p>In the Y and Z matrix definitions, 
  
    
      
        Y
        =
        
          Z
          
            −
            1
          
        
      
    
    {\displaystyle Y=Z^{-1}}
  
 and 
  
    
      
        Z
        =
        
          Y
          
            −
            1
          
        
      
    
    {\displaystyle Z=Y^{-1}}
  
.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a>  Unlike ideal <a href="/facts/Lumped-element_model/Nqtt18v5">lumped 2 port elements</a> (<a href="/facts/Resistor/JrsGReM6">resistors</a>, <a href="/facts/Capacitor/hyowqv7n">capacitors</a>, <a href="/facts/Inductor/YnfAXB3x">inductors</a>, etc.) which do not have defined Z parameters, transmission lines have an internal path to ground, which permits the definition of Z parameters.
</p><p>For lossless and lossy transmission lines respectively, the Z parameter matrix is as follows:<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a><a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a>
</p><p>
  
    
      
        
          Z
          
            Lossless
          
        
        =
        
          
            [
            
              
                
                  −
                  j
                  
                    Z
                    
                      o
                    
                  
                  c
                  o
                  t
                  (
                  β
                  l
                  )
                
                
                  −
                  j
                  
                    Z
                    
                      o
                    
                  
                  c
                  s
                  c
                  (
                  β
                  l
                  )
                
              
              
                
                  −
                  j
                  
                    Z
                    
                      o
                    
                  
                  c
                  s
                  c
                  (
                  β
                  l
                  )
                
                
                  −
                  j
                  
                    Z
                    
                      o
                    
                  
                  c
                  o
                  t
                  (
                  β
                  l
                  )
                
              
            
            ]
          
        
        
           
        
        
          Z
          
            Lossy
          
        
        =
        
          
            [
            
              
                
                  
                    Z
                    
                      o
                    
                  
                  c
                  o
                  t
                  h
                  (
                  γ
                  l
                  )
                
                
                  
                    Z
                    
                      o
                    
                  
                  c
                  s
                  c
                  h
                  (
                  γ
                  l
                  )
                
              
              
                
                  
                    Z
                    
                      o
                    
                  
                  c
                  s
                  c
                  h
                  (
                  γ
                  l
                  )
                
                
                  
                    Z
                    
                      o
                    
                  
                  c
                  o
                  t
                  h
                  (
                  γ
                  l
                  )
                
              
            
            ]
          
        
      
    
    {\displaystyle Z_{\text{Lossless}}={\begin{bmatrix}-jZ_{o}cot(\beta l)&-jZ_{o}csc(\beta l)\\-jZ_{o}csc(\beta l)&-jZ_{o}cot(\beta l)\end{bmatrix}}{\text{     }}Z_{\text{Lossy}}={\begin{bmatrix}Z_{o}coth(\gamma l)&Z_{o}csch(\gamma l)\\Z_{o}csch(\gamma l)&Z_{o}coth(\gamma l)\end{bmatrix}}}

</p>
<h3>Scattering parameters</h3>
<p>Scattering (S) matrix parameters model the electrical behavior of the transmission line with <a href="/facts/Impedance_matching/0mdy9TAp">matched loads</a> at each <a href="/facts/Electrical_termination/MatLrePo">termination</a>.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a>
</p><p>For lossless and lossy transmission lines respectively, the S parameter matrix is as follows,<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a><a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> using standard <a href="/facts/Hyperbolic_functions/Y4Cb5S35">hyperbolic to circular complex translations</a>.
</p><p>
  
    
      
        
          S
          
            Lossless
          
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        −
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                        −
                        j
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        (
                        β
                        l
                        )
                      
                    
                  
                
                
                  
                    
                      
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                      
                      
                        j
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        (
                        β
                        l
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                      
                      
                        j
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        (
                        β
                        l
                        )
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        −
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        (
                        β
                        l
                        )
                        −
                        j
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        (
                        β
                        l
                        )
                      
                    
                  
                
              
            
            ]
          
        
        
           
        
        
          S
          
            Lossy
          
        
        =
        
          
            [
            
              
                
                  
                    
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        −
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        h
                        (
                        γ
                        l
                        )
                      
                    
                  
                
                
                  
                    
                      
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        h
                        (
                        γ
                        l
                        )
                      
                    
                  
                
              
              
                
                  
                    
                      
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        h
                        (
                        γ
                        l
                        )
                      
                    
                  
                
                
                  
                    
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        −
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                      
                      
                        (
                        
                          Z
                          
                            o
                          
                          
                            2
                          
                        
                        +
                        
                          Z
                          
                            p
                          
                          
                            2
                          
                        
                        )
                        s
                        i
                        n
                        h
                        (
                        γ
                        l
                        )
                        +
                        2
                        
                          Z
                          
                            o
                          
                        
                        
                          Z
                          
                            p
                          
                        
                        c
                        o
                        s
                        h
                        (
                        γ
                        l
                        )
                      
                    
                  
                
              
            
            ]
          
        
      
    
    {\displaystyle S_{\text{Lossless}}={\begin{bmatrix}{\frac {(Z_{o}^{2}-Z_{p}^{2})sin(\beta l)}{(Z_{o}^{2}+Z_{p}^{2})sin(\beta l)-j2Z_{o}Z_{p}cos(\beta l)}}&{\frac {2Z_{o}Z_{p}}{j(Z_{o}^{2}+Z_{p}^{2})sin(\beta l)+2Z_{o}Z_{p}cos(\beta l)}}\\{\frac {2Z_{o}Z_{p}}{j(Z_{o}^{2}+Z_{p}^{2})sin(\beta l)+2Z_{o}Z_{p}cos(\beta l)}}&{\frac {(Z_{o}^{2}-Z_{p}^{2})sin(\beta l)}{(Z_{o}^{2}+Z_{p}^{2})sin(\beta l)-j2Z_{o}Z_{p}cos(\beta l)}}\end{bmatrix}}{\text{    }}S_{\text{Lossy}}={\begin{bmatrix}{\frac {(Z_{o}^{2}-Z_{p}^{2})sinh(\gamma l)}{(Z_{o}^{2}+Z_{p}^{2})sinh(\gamma l)+2Z_{o}Z_{p}cosh(\gamma l)}}&{\frac {2Z_{o}Z_{p}}{(Z_{o}^{2}+Z_{p}^{2})sinh(\gamma l)+2Z_{o}Z_{p}cosh(\gamma l)}}\\{\frac {2Z_{o}Z_{p}}{(Z_{o}^{2}+Z_{p}^{2})sinh(\gamma l)+2Z_{o}Z_{p}cosh(\gamma l)}}&{\frac {(Z_{o}^{2}-Z_{p}^{2})sinh(\gamma l)}{(Z_{o}^{2}+Z_{p}^{2})sinh(\gamma l)+2Z_{o}Z_{p}cosh(\gamma l)}}\end{bmatrix}}}

</p>
<h3>Variable definitions</h3>
<p>In all matrix parameters above, the following variable definitions apply:
</p><p>
  
    
      
        
          Z
          
            o
          
        
      
    
    {\displaystyle Z_{o}}
  
 = <a href="/facts/Characteristic_impedance/oUg0mhah">characteristic impedance</a>
</p><p>Zp = port impedance, or <a href="/facts/Electrical_termination/MatLrePo">termination impedance</a>
</p><p>
  
    
      
        γ
        =
        α
        +
        j
        β
      
    
    {\displaystyle \gamma =\alpha +j\beta }
  
 = the <a href="/facts/Propagation_constant/3XpFb0q1">propagation constant</a> per unit length
</p><p>
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 = <a href="/facts/Attenuation_constant/3XpFb0q1">attenuation constant</a> in <a href="/facts/Neper/nzHnA6ev">nepers</a> per unit length
</p><p>
  
    
      
        β
        =
        
          
            
              2
              π
            
            λ
          
        
        =
        
          
            ω
            V
          
        
      
    
    {\displaystyle \beta ={\frac {2\pi }{\lambda }}={\frac {\omega }{V}}}
  
 = <a href="/facts/Wavenumber/cLaLAeW5">wave number</a> or <a href="/facts/Phase_constant/3XpFb0q1">phase constant</a>  radians per unit length
</p><p>
  
    
      
        ω
      
    
    {\displaystyle \omega }
  
 = <a href="/facts/Frequency/QUSbPaW8">frequency</a> radians / second
</p><p>
  
    
      
        V
        =
        
          
            1
            
              L
              C
            
          
        
        =
        
          
            
              V
              
                C
              
            
            
              
                E
                
                  r
                  e
                
              
            
          
        
      
    
    {\displaystyle V={\frac {1}{\sqrt {LC}}}={\frac {V_{C}}{\sqrt {E_{re}}}}}
  
 = <a href="/facts/Signal_velocity/3xpnmM4j">Speed of propagation</a>
</p><p>
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 = <a href="/facts/Wavelength/kcJU0ymz">wave length</a> in unit length
</p><p>L = <a href="/facts/Inductance/oIxj7JxX">inductance</a> per unit length
</p><p>C = <a href="/facts/Capacitance/0pH046TL">capacitance</a> per unit length
</p><p>
  
    
      
        
          E
          
            r
            e
          
        
      
    
    {\displaystyle E_{re}}
  
 = <a href="/facts/Relative_permittivity/vnKbPTn1">effective dielectric constant</a>
</p><p>
  
    
      
        
          V
          
            C
          
        
      
    
    {\displaystyle V_{C}}
  
 = 299,792,458 meters / second = <a href="/facts/Speed_of_light/sKSQwdNx">Speed of light</a> in a vacuum
</p>
<h2 id="coupled-transmission-lines">Coupled transmission lines</h2>
<p>Transmission lines may be placed in proximity to each other such that they electrically interact, such as two <a href="/facts/Microstrip/QkKwyAN7">microstrip</a> lines in close proximity.  Such transmission lines are said to be coupled transmission lines.  Coupled transmission lines are characterized by an even and odd mode analysis.  The even mode is characterized by excitation of the two conductors with a signal of equal amplitude and phase.  The odd mode is characterized by excitation with signals of equal and opposite magnitude.  The even and odd modes each have their own characteristic impedances (Zoe, Zoo) and phase constants (
  
    
      
        
          β
          
            e
          
        
        
          , 
        
        
          β
          
            o
          
        
      
    
    {\displaystyle \beta _{e}{\text{, }}\beta _{o}}
  
).  Lossy coupled transmission lines have their own even and odd mode attenuation constants (
  
    
      
        
          α
          
            e
          
        
        
          , 
        
        
          α
          
            o
          
        
      
    
    {\displaystyle \alpha _{e}{\text{, }}\alpha _{o}}
  
), which in turn leads to even and odd mode propagation constants (
  
    
      
        
          γ
          
            e
          
        
        
          , 
        
        
          γ
          
            o
          
        
      
    
    {\displaystyle \gamma _{e}{\text{, }}\gamma _{o}}
  
).<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a><a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a><a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a><a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a><a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a>
</p>
<h3>Coupled matrix parameters</h3>
<p>Coupled transmission lines may be modeled using even and odd mode transmission line parameters defined in the prior paragraph as shown with ports 1 and 2 on the input and ports 3 and 4 on the output,<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a>
</p><p>
  
    
      
        
          
            
              
                Y
              
              
                
                =
                
                  
                    [
                    
                      
                        
                          y
                          11
                        
                        
                          y
                          12
                        
                        
                          y
                          13
                        
                        
                          y
                          14
                        
                      
                      
                        
                          y
                          21
                        
                        
                          y
                          22
                        
                        
                          y
                          23
                        
                        
                          y
                          24
                        
                      
                      
                        
                          y
                          31
                        
                        
                          y
                          32
                        
                        
                          y
                          33
                        
                        
                          y
                          34
                        
                      
                      
                        
                          y
                          41
                        
                        
                          y
                          42
                        
                        
                          y
                          43
                        
                        
                          y
                          44
                        
                      
                    
                    ]
                  
                
              
            
            
              
                Z
              
              
                
                =
                [
                Y
                
                  ]
                  
                    −
                    1
                  
                
              
            
            
              
              
            
            
              
                
                  Where:
                
              
              
            
            
              
                
                  For lossless coupled lines:
                
              
              
            
            
              
                y
                11
              
              
                
                =
                y
                22
                =
                y
                33
                =
                y
                44
                =
                
                  
                    
                      −
                      j
                    
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      o
                      t
                      (
                      
                        β
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                +
                
                  
                    
                      c
                      o
                      t
                      (
                      
                        β
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                12
              
              
                
                =
                y
                22
                =
                y
                34
                =
                y
                43
                =
                
                  
                    
                      −
                      j
                    
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      o
                      t
                      (
                      
                        β
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                −
                
                  
                    
                      c
                      o
                      t
                      (
                      
                        β
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                13
              
              
                
                =
                y
                31
                =
                y
                24
                =
                y
                42
                =
                
                  
                    j
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      s
                      c
                      (
                      
                        β
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                +
                
                  
                    
                      c
                      s
                      c
                      (
                      
                        β
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                14
              
              
                
                =
                y
                41
                =
                y
                23
                =
                y
                32
                =
                
                  
                    j
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      s
                      c
                      (
                      
                        β
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                −
                
                  
                    
                      c
                      s
                      c
                      (
                      
                        β
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                
                  For lossy coupled lines:
                
              
              
            
            
              
                y
                11
              
              
                
                =
                y
                22
                =
                y
                33
                =
                y
                44
                =
                
                  
                    1
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      o
                      t
                      h
                      (
                      
                        γ
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                +
                
                  
                    
                      c
                      o
                      t
                      h
                      (
                      
                        γ
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                12
              
              
                
                =
                y
                22
                =
                y
                34
                =
                y
                43
                =
                
                  
                    1
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      o
                      t
                      h
                      (
                      
                        γ
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                −
                
                  
                    
                      c
                      o
                      t
                      h
                      (
                      
                        γ
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                13
              
              
                
                =
                y
                31
                =
                y
                24
                =
                y
                42
                =
                
                  
                    
                      −
                      1
                    
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      s
                      c
                      h
                      (
                      
                        γ
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                +
                
                  
                    
                      c
                      s
                      c
                      h
                      (
                      
                        γ
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
            
              
                y
                14
              
              
                
                =
                y
                41
                =
                y
                23
                =
                y
                32
                =
                
                  
                    
                      −
                      1
                    
                    2
                  
                
                
                  
                    (
                  
                
                
                  
                    
                      c
                      s
                      c
                      h
                      (
                      
                        γ
                        
                          e
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        e
                      
                    
                  
                
                −
                
                  
                    
                      c
                      s
                      c
                      h
                      (
                      
                        γ
                        
                          o
                        
                      
                      l
                      )
                    
                    
                      Z
                      
                        o
                        o
                      
                    
                  
                
                
                  
                    )
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}Y&={\begin{bmatrix}y11&y12&y13&y14\\y21&y22&y23&y24\\y31&y32&y33&y34\\y41&y42&y43&y44\\\end{bmatrix}}\\Z&=[Y]^{-1}\\&\\{\text{Where:}}&\\{\text{For lossless coupled lines:}}&\\y11&=y22=y33=y44={\frac {-j}{2}}{\bigg (}{\frac {cot(\beta _{e}l)}{Z_{oe}}}+{\frac {cot(\beta _{o}l)}{Z_{oo}}}{\bigg )}\\y12&=y22=y34=y43={\frac {-j}{2}}{\bigg (}{\frac {cot(\beta _{e}l)}{Z_{oe}}}-{\frac {cot(\beta _{o}l)}{Z_{oo}}}{\bigg )}\\y13&=y31=y24=y42={\frac {j}{2}}{\bigg (}{\frac {csc(\beta _{e}l)}{Z_{oe}}}+{\frac {csc(\beta _{o}l)}{Z_{oo}}}{\bigg )}\\y14&=y41=y23=y32={\frac {j}{2}}{\bigg (}{\frac {csc(\beta _{e}l)}{Z_{oe}}}-{\frac {csc(\beta _{o}l)}{Z_{oo}}}{\bigg )}\\{\text{For lossy coupled lines:}}&\\y11&=y22=y33=y44={\frac {1}{2}}{\bigg (}{\frac {coth(\gamma _{e}l)}{Z_{oe}}}+{\frac {coth(\gamma _{o}l)}{Z_{oo}}}{\bigg )}\\y12&=y22=y34=y43={\frac {1}{2}}{\bigg (}{\frac {coth(\gamma _{e}l)}{Z_{oe}}}-{\frac {coth(\gamma _{o}l)}{Z_{oo}}}{\bigg )}\\y13&=y31=y24=y42={\frac {-1}{2}}{\bigg (}{\frac {csch(\gamma _{e}l)}{Z_{oe}}}+{\frac {csch(\gamma _{o}l)}{Z_{oo}}}{\bigg )}\\y14&=y41=y23=y32={\frac {-1}{2}}{\bigg (}{\frac {csch(\gamma _{e}l)}{Z_{oe}}}-{\frac {csch(\gamma _{o}l)}{Z_{oo}}}{\bigg )}\\\end{aligned}}}
  
..
</p>
<h2 id="practical-types">Practical types</h2>
<h3>Coaxial cable</h3>
<p class="note">Main article: <a href="/facts/Coaxial_cable/aG9FfCpL">coaxial cable</a></p>
<p>Coaxial lines confine virtually all of the electromagnetic wave to the area inside the cable. Coaxial lines can therefore be bent and twisted (subject to limits) without negative effects, and they can be strapped to conductive supports without inducing unwanted currents in them.
In radio-frequency applications up to a few gigahertz, the wave propagates in the <a href="/facts/Transverse_wave/WVLe5XKX">transverse</a> electric and magnetic mode (TEM) only, which means that the electric and magnetic fields are both perpendicular to the direction of propagation (the electric field is radial, and the magnetic field is circumferential). However, at frequencies for which the wavelength (in the dielectric) is significantly shorter than the circumference of the cable other <a href="/facts/Transverse_mode/nCIl5D6a">transverse modes</a> can propagate. These modes are classified into two groups, transverse electric (TE) and transverse magnetic (TM) <a href="/facts/Waveguide/tJS9Y0GH">waveguide</a> modes. When more than one mode can exist, bends and other irregularities in the cable geometry can cause power to be transferred from one mode to another.
</p><p>The most common use for coaxial cables is for television and other signals with bandwidth of multiple megahertz. In the middle 20th century they carried <a href="/facts/Long_distance_telephone/gR0mtIeD">long distance telephone</a> connections.
</p>
<h3>Planar lines</h3>
<p class="note">Main article: <a href="/facts/Planar_transmission_line/PtQSrIpB">Planar transmission line</a></p>
<p>Planar transmission lines are transmission lines with <a href="/facts/Electrical_conductor/BSuyupLP">conductors</a>, or in some cases dielectric strips, that are flat, ribbon-shaped lines.  They are used to interconnect components on <a href="/facts/Printed_circuit/muSR2p43">printed circuits</a> and <a href="/facts/Integrated_circuit/fjn6vg7C">integrated circuits</a> working at microwave frequencies because the planar type fits in well with the manufacturing methods for these components. Several forms of planar transmission lines exist.
</p>
<h4>Microstrip</h4>

<p class="note">Main article: <a href="/facts/Microstrip/QkKwyAN7">microstrip</a></p>
<p>A microstrip circuit uses a thin flat conductor which is <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> to a <a href="/facts/Ground_plane/quwvjQl0">ground plane</a>. Microstrip can be made by having a strip of copper on one side of a <a href="/facts/Printed_circuit_board/muSR2p43">printed circuit board</a> (PCB) or ceramic substrate while the other side is a continuous ground plane. The width of the strip, the thickness of the insulating layer (PCB or ceramic) and the <a href="/facts/Dielectric_constant/vnKbPTn1">dielectric constant</a> of the insulating layer determine the characteristic impedance. Microstrip is an open structure whereas coaxial cable is a closed structure.
</p>
<h4>Stripline</h4>
<p class="note">Main article: <a href="/facts/Stripline/RNdJWRQJ">Stripline</a></p>
<p>A stripline circuit uses a flat strip of metal which is sandwiched between two parallel ground planes. The insulating material of the substrate forms a dielectric. The width of the strip, the thickness of the substrate and the relative permittivity of the substrate determine the characteristic impedance of the strip which is a transmission line.
</p>
<h4>Coplanar waveguide</h4>
<p class="note">Main article: <a href="/facts/Coplanar_waveguide/3G3P5uGk">Coplanar waveguide</a></p>
<p>A coplanar waveguide consists of a center strip and two adjacent outer conductors, all three of them flat structures that are deposited onto the same insulating substrate and thus are located in the same plane ("coplanar"). The width of the center conductor, the distance between inner and outer conductors, and the relative permittivity of the substrate determine the characteristic impedance of the coplanar transmission line.
</p>
<h3>Balanced lines</h3>
<p class="note">Main article: <a href="/facts/Balanced_line/XubfIRdc">Balanced line</a></p>
<p>A balanced line is a transmission line consisting of two conductors of the same type, and equal impedance to ground and other circuits.  There are many formats of balanced lines, amongst the most common are twisted pair, star quad and twin-lead.
</p>
<h4>Twisted pair</h4>
<p class="note">Main article: <a href="/facts/Twisted_pair/L426Jk2C">Twisted pair</a></p>
<p>Twisted pairs are commonly used for terrestrial <a href="/facts/Telephone/P7uEgWQX">telephone</a> communications.  In such cables, many pairs are grouped together in a single cable, from two to several thousand.<a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a>  The format is also used for data network distribution inside buildings, but the cable is more expensive because the transmission line parameters are tightly controlled.
</p>
<h4>Star quad</h4>
<p class="note">Main article: <a href="/facts/Star_quad_cable/jI9j0Ipz">Star quad cable</a></p>
<p>Star quad is a four-conductor cable in which all four conductors are twisted together around the cable axis.  It is sometimes used for two circuits, such as <a href="/facts/4-wire/C3InvEMe">4-wire</a> telephony and other telecommunications applications.  In this configuration each pair uses two non-adjacent conductors.  Other times it is used for a single, <a href="/facts/Balanced_line/XubfIRdc">balanced line</a>, such as audio applications and <a href="/facts/2-wire/j0hbjBtD">2-wire</a> telephony.  In this configuration two non-adjacent conductors are terminated together at both ends of the cable, and the other two conductors are also terminated together.
</p><p>When used for two circuits, crosstalk is reduced relative to cables with two separate twisted pairs.
</p><p>When used for a single, <a href="/facts/Balanced_line/XubfIRdc">balanced line</a>, magnetic interference picked up by the cable arrives as a virtually perfect common mode signal, which is easily removed by coupling transformers.
</p><p>The combined benefits of twisting, balanced signalling, and quadrupole pattern give outstanding noise immunity, especially advantageous for low signal level applications such as microphone cables, even when installed very close to a power cable.<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a><a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> The disadvantage is that star quad, in combining two conductors, typically has double the capacitance of similar two-conductor twisted and shielded audio cable. High capacitance causes increasing distortion and greater loss of high frequencies as distance increases.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a><a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a>
</p>
<h4>Twin-lead</h4>
<p class="note">Main article: <a href="/facts/Twin-lead/BfRCmX8A">Twin-lead</a></p>
<p>Twin-lead consists of a pair of conductors held apart by a continuous insulator.  By holding the conductors a known distance apart, the geometry is fixed and the line characteristics are reliably consistent.  It is lower loss than coaxial cable because the characteristic impedance of twin-lead is generally higher than coaxial cable, leading to lower resistive losses due to the reduced current.  However, it is more susceptible to interference.
</p>
<h4>Lecher lines</h4>
<p class="note">Main article: <a href="/facts/Lecher_lines/94lQXAnF">Lecher lines</a></p>
<p>Lecher lines are a form of parallel conductor that can be used at <a href="/facts/Ultra_high_frequency/S6XDUaDY">UHF</a> for creating resonant circuits.  They are a convenient practical format that fills the gap between <a href="/facts/Lumped-element_model/Nqtt18v5">lumped</a> components (used at <a href="/facts/High_frequency/JmDJLR5I">HF</a>/<a href="/facts/VHF/8V90M1Sl">VHF</a>) and <a href="/facts/Resonant_cavity/NjC413Gn">resonant cavities</a> (used at <a href="/facts/Ultra_high_frequency/S6XDUaDY">UHF</a>/<a href="/facts/Super_high_frequency/Bqrg4fcy">SHF</a>).
</p>
<h3>Single-wire line</h3>
<p><a href="/facts/Unbalanced_line/BcTHryVX">Unbalanced lines</a> were formerly much used for telegraph transmission, but this form of communication has now fallen into disuse.  Cables are similar to twisted pair in that many cores are bundled into the same cable but only one conductor is provided per circuit and there is no twisting.  All the circuits on the same route use a common path for the return current (earth return).  There is a <a href="/facts/Electric_power_transmission/I9w3lRmM">power transmission</a> version of <a href="/facts/Single-wire_earth_return/5j61avwv">single-wire earth return</a> in use in many locations.
</p>
<h2 id="general-applications">General applications</h2>
<h3>Signal transfer</h3>
<p>Electrical transmission lines are very widely used to transmit high frequency signals over long or short distances with minimum power loss.  One familiar example is the down lead from a TV or radio <a href="/facts/Antenna_(radio)/pGlcXBvc">aerial</a> to the receiver.
</p>
<h3>Transmission line circuits</h3>
<p class="note">Main article: <a href="/facts/Distributed-element_circuit/gRHcdUTX">Distributed-element circuit</a></p>
<p>A large variety of circuits can also be constructed with transmission lines including <a href="/facts/Impedance_matching/0mdy9TAp">impedance matching</a> circuits, <a href="/facts/Distributed-element_filter/Pkt1DPAj">filters</a>, <a href="/facts/Power_dividers_and_directional_couplers/4LNPkdnN">power dividers and directional couplers</a>.
</p>
<h4>Stepped transmission line</h4>
<p class="note">See also: <a href="/facts/Waveguide_filter/ACjb5SjG">Waveguide filter § Impedance matching</a></p>

<p>A stepped transmission line is used for broad range impedance matching. It can be considered as multiple transmission line segments connected in series, with the characteristic impedance of each individual element to be 
  
    
      
        
          Z
          
            
              0
              ,
              i
            
          
        
      
    
    {\displaystyle Z_{\mathrm {0,i} }}
  
.<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> The input impedance can be obtained from the successive application of the chain relation
</p>

Z
          
            
              i
              +
              1
            
          
        
        =
        
          Z
          
            
              0
              ,
              i
            
          
        
        
        
          
            
              
              
                Z
                
                  
                    i
                  
                
              
              +
              j
              
              
                Z
                
                  
                    0
                    ,
                    i
                  
                
              
              
              tan
              ⁡
              (
              
                β
                
                  
                    i
                  
                
              
              
                ℓ
                
                  
                    i
                  
                
              
              )
              
            
            
              
                Z
                
                  
                    0
                    ,
                    i
                  
                
              
              +
              j
              
              
                Z
                
                  
                    i
                  
                
              
              
              tan
              ⁡
              (
              
                β
                
                  
                    i
                  
                
              
              
                ℓ
                
                  
                    i
                  
                
              
              )
            
          
        
        
      
    
    {\displaystyle Z_{\mathrm {i+1} }=Z_{\mathrm {0,i} }\,{\frac {\,Z_{\mathrm {i} }+j\,Z_{\mathrm {0,i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })\,}{Z_{\mathrm {0,i} }+j\,Z_{\mathrm {i} }\,\tan(\beta _{\mathrm {i} }\ell _{\mathrm {i} })}}\,}

<p>where 
  
    
      
        
          β
          
            
              i
            
          
        
      
    
    {\displaystyle \beta _{\mathrm {i} }}
  
 is the wave number of the 
  
    
      
        
          i
        
      
    
    {\displaystyle \mathrm {i} }
  
-th transmission line segment and 
  
    
      
        
          ℓ
          
            
              i
            
          
        
      
    
    {\displaystyle \ell _{\mathrm {i} }}
  
 is the length of this segment, and 
  
    
      
        
          Z
          
            
              i
            
          
        
      
    
    {\displaystyle Z_{\mathrm {i} }}
  
 is the front-end impedance that loads the 
  
    
      
        
          i
        
      
    
    {\displaystyle \mathrm {i} }
  
-th segment.
</p>

<p>Because the characteristic impedance of each transmission line segment 
  
    
      
        
          Z
          
            
              0
              ,
              i
            
          
        
      
    
    {\displaystyle Z_{\mathrm {0,i} }}
  
 is often different from the impedance 
  
    
      
        
          Z
          
            0
          
        
      
    
    {\displaystyle Z_{0}}
  
 of the fourth, input cable (only shown as an arrow marked 
  
    
      
        
          Z
          
            0
          
        
      
    
    {\displaystyle Z_{0}}
  
 on the left side of the diagram above), the impedance transformation circle is off-centred along the 
  
    
      
        x
      
    
    {\displaystyle x}
  
 axis of the <a href="/facts/Smith_Chart/jdkHh9Ju">Smith Chart</a> whose impedance representation is usually normalized against 
  
    
      
        
          Z
          
            0
          
        
      
    
    {\displaystyle Z_{0}}
  
.
</p>
<h4>Approximating lumped elements</h4>
<p>At higher frequencies, the reactive parasitic effects of real world <a href="/facts/Lumped-element_model/Nqtt18v5">lumped elements</a>, including <a href="/facts/Inductor/YnfAXB3x">inductors</a> and <a href="/facts/Capacitor/hyowqv7n">capacitors</a>, limits their usefulness.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a>  Therefore, it is sometimes useful to approximate the electrical characteristics of inductors and capacitors with transmission lines at the higher frequencies using <a href="/facts/Richards%27_transformation/sAXoGIPQ">Richards' Transformations</a> and then substitute the transmission lines for the lumped elements.<a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a>
</p><p>More accurate forms of <a href="/facts/Multimode/nCIl5D6a">multimode</a> high frequency inductor modeling with transmission lines exist for advanced designers.<a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a>
</p>
<h3>Stub filters</h3>
<p class="note">See also: <a href="/facts/Distributed-element_filter/Pkt1DPAj">Distributed-element filter § Stub band-pass filters</a></p>
<p>If a short-circuited or open-circuited transmission line is wired in parallel with a line used to transfer signals from point A to point B, then it will function as a filter. The method for making stubs is similar to the method for using Lecher lines for crude frequency measurement, but it is 'working backwards'. One method recommended in the <a href="/facts/Radio_Society_of_Great_Britain/AQ9GdBq6">RSGB</a>'s radiocommunication handbook is to take an open-circuited length of transmission line wired in parallel with the <a href="/facts/Feed_line/y9qa2CWa">feeder</a> delivering signals from an aerial. By cutting the free end of the transmission line, a minimum in the strength of the signal observed at a receiver can be found. At this stage the stub filter will reject this frequency and the odd harmonics, but if the free end of the stub is shorted then the stub will become a filter rejecting the even harmonics.
</p><p>Wideband filters can be achieved using multiple stubs.  However, this is a somewhat dated technique.  Much more compact filters can be made with other methods such as parallel-line resonators.
</p>
<h3>Pulse generation</h3>
<p>Transmission lines are used as pulse generators. By charging the transmission line and then discharging it into a <a href="/facts/Resistive/hxSt0UAj">resistive</a> load, a rectangular pulse equal in length to twice the <a href="/facts/Electrical_length/TMWhkDft">electrical length</a> of the line can be obtained, although with half the voltage. A <a href="/facts/Blumlein_transmission_line/xWACoUAD">Blumlein transmission line</a> is a related pulse forming device that overcomes this limitation. These are sometimes used as the <a href="/facts/Pulsed_power/BTJN8xg5">pulsed power</a> sources for <a href="/facts/Radar/jk9BTbFA">radar</a> <a href="/facts/Transmitters/nFYghtKv">transmitters</a> and other devices.
</p>
<h2 id="sound">Sound</h2>
<p>The theory of <a href="/facts/Sound_wave/DlKXlPuF">sound wave</a> propagation is very similar mathematically to that of electromagnetic waves, so techniques from transmission line theory are also used to build structures to conduct acoustic waves; and these are called <a href="/facts/Acoustic_transmission_line/bAVStIE1">acoustic transmission lines</a>.
</p>
<h2 id="see-also">See also</h2>

Wikimedia Commons has media related to Transmission lines.

<ul><li><a href="/facts/Artificial_transmission_line/hC4kfQyd">Artificial transmission line</a></li>
<li><a href="/facts/Longitudinal_wave/KoysWuIN">Longitudinal electromagnetic wave</a></li>
<li><a href="/facts/Propagation_velocity/m43q3X33">Propagation velocity</a></li>
<li><a href="/facts/Radio_frequency_power_transmission/IaCR27nq">Radio frequency power transmission</a></li>
<li><a href="/facts/Time_domain_reflectometer/md9VY2JY">Time domain reflectometer</a></li></ul>

<p><i>Part of this article was derived from <a href="/facts/Federal_Standard_1037C/CA4jqghV">Federal Standard 1037C</a>.</i>
</p>

<ul><li><a href="/facts/Charles_Proteus_Steinmetz/j4TplnwJ">Steinmetz, Charles Proteus</a> (27 August 1898). "The natural period of a transmission line and the frequency of lightning discharge therefrom". <i>The Electrical World</i>: 203–205.</li>
<li>Grant, I.S.; Phillips, W.R. (1991-08-26). <a href="https://archive.org/details/electromagnetism0000gran"><i>Electromagnetism</i></a> (2nd ed.). John Wiley. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-471-92712-9.</li>
<li>Ulaby, F.T. (2004). <i>Fundamentals of Applied Electromagnetics</i> (2004 media ed.). Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-185089-7.</li>
<li>"Chapter 17". <i>Radio communication handbook</i>. <a href="/facts/Radio_Society_of_Great_Britain/AQ9GdBq6">Radio Society of Great Britain</a>. 1982. p. 20. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-900612-58-9.</li>
<li>Naredo, J.L.; Soudack, A.C.; Marti, J.R. (Jan 1995). "Simulation of transients on transmission lines with corona via the method of characteristics". <i>IEE Proceedings - Generation, Transmission and Distribution</i>. 142 (1): 81. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1049%2Fip-gtd%3A19951488">10.1049/ip-gtd:19951488</a> (inactive 7 December 2024). <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1350-2360">1350-2360</a>.{{cite journal}}:  CS1 maint: DOI inactive as of December 2024 (link)</li></ul>
<h2 id="further-reading">Further reading</h2>
<ul><li><a href="https://babel.hathitrust.org/cgi/pt?id=coo.31924066336946;view=1up;seq=115"><i>Honoring of Guglielmo Marconi</i></a>. Annual Dinner of the Institute at the Waldorf-Astoria. New York: American Institute of Electrical Engineers. 13 January 1902.</li>
<li><a href="https://web.archive.org/web/20050925041320/http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-124.html">"Using Transmission Line Equations and Parameters"</a>. Star-Hspice Manual. Avant! Software. June 2001. Archived from <a href="http://www.ece.cmu.edu/~ee762/hspice-docs/html/hspice_and_qrg/hspice_2001_2-124.html">the original</a> on 25 September 2005.</li>
<li>Cornille, P. (1990). "On the propagation of inhomogeneous waves". <i>Journal of Physics D: Applied Physics</i>. 23 (2): 129–135. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1990JPhD...23..129C">1990JPhD...23..129C</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1088%2F0022-3727%2F23%2F2%2F001">10.1088/0022-3727/23/2/001</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:250788839">250788839</a>.</li>
<li><a href="/facts/Stanley_Farlow/AqRaEpSU">Farlow, S.J.</a> (1982). <i>Partial Differential Equations for Scientists and Engineers</i>. J. Wiley and Sons. p. 126. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-08639-8.</li>
<li>Kupershmidt, Boris A. (1998). "Remarks on random evolutions in Hamiltonian representation". <i>J. Nonlinear Math. Phys</i>. 5 (4): 383–395. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math-ph/9810020">math-ph/9810020</a>. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1998JNMP....5..483K">1998JNMP....5..483K</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2991%2Fjnmp.1998.5.4.10">10.2991/jnmp.1998.5.4.10</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:14771417">14771417</a>. Math-ph/9810020.</li>
<li><a href="https://web.archive.org/web/20110721090251/http://cktse.eie.polyu.edu.hk/eie403/">"Transmission line matching"</a>. Department of Electronic and Information Engineering. High Frequency Circuit Design. Hong Kong Polytechnic University. EIE403. Archived from <a href="http://cktse.eie.polyu.edu.hk/eie403/">the original</a> (PDF) on 2011-07-21. Retrieved 2005-09-24.</li>
<li>Wilson, B. (19 October 2005). <a href="https://web.archive.org/web/20060109065828/http://cnx.rice.edu/content/m1044/latest/">"Telegrapher's Equations"</a>. Connexions. Archived from <a href="http://cnx.rice.edu/content/m1044/latest/">the original</a> on 9 January 2006.</li>
<li>Wöhlbier, John Greaton (2000). <a href="https://web.archive.org/web/20060619072607/http://www.wildwestwohlbiers.org/john/files/ms_thesis.pdf"><i>Modeling and Analysis of a Traveling Wave under Multitone Excitation</i></a> (PDF). Electrical and Computer Engineering (M.S.). Madison, WI: University of Wisconsin. §"Fundamental Equation" and §"Transforming the Telegrapher's Equations". Archived from <a href="http://www.wildwestwohlbiers.org/john/files/ms_thesis.pdf">the original</a> (PDF) on 19 June 2006.</li>
<li><a href="http://www.keysight.com/find/nw_xline">"Wave Propagation along a Transmission Line"</a> (Educational Java Applet). Educational Resources. Keysight Technologies.[<i>permanent dead link</i>‍] (May need to add "<a href="http://www.keysight.com">http://www.keysight.com</a>" to your Java Exception Site list.)</li>
<li>Qian, Chunqi; Brey, William W. (2009). "Impedance matching with an adjustable, segmented transmission line". <i>Journal of Magnetic Resonance</i>. 199 (1): 104–110. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2009JMagR.199..104Q">2009JMagR.199..104Q</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.jmr.2009.04.005">10.1016/j.jmr.2009.04.005</a>. <a href="/facts/PMID_(identifier)/JlHAvMHt">PMID</a> <a href="https://pubmed.ncbi.nlm.nih.gov/19406676">19406676</a>.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li><a href="http://terahertz.tudelft.nl/Research/project.php?id=74&ti=27">"Transmission Line Calculator (Including radiation and surface-wave excitation losses)"</a>. <i>terahertz.tudelft.nl</i>. Delft, NL: <a href="/facts/Technical_University_of_Delft/wtPV7UmG">Technical University of Delft</a>.</li>
<li><a href="http://cecas.clemson.edu/cvel/emc/calculators/TL_Calculator/index.html">"Transmission Line Parameter Calculator"</a>. <i>cecas.clemson.edu/cvel</i>. Clemson, SC: <a href="/facts/Clemson_University/xPSgG0v1">Clemson University</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Jackman, Shawn M.; Matt Swartz; Marcus Burton; Thomas W. Head (2011). CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250. John Wiley & Sons. pp. Ch. 7. ISBN 978-1118041611. <a href="978-1118041611" target="_blank">978-1118041611</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Jackman, Shawn M.; Matt Swartz; Marcus Burton; Thomas W. Head (2011). CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250. John Wiley & Sons. pp. Ch. 7. ISBN 978-1118041611. <a href="978-1118041611" target="_blank">978-1118041611</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Oklobdzija, Vojin G.; Ram K. Krishnamurthy (2006). High-Performance Energy-Efficient Microprocessor Design. Springer Science & Business Media. p. 297. ISBN 978-0387340470. <a href="978-0387340470" target="_blank">978-0387340470</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Oklobdzija, Vojin G.; Ram K. Krishnamurthy (2006). High-Performance Energy-Efficient Microprocessor Design. Springer Science & Business Media. p. 297. ISBN 978-0387340470. <a href="978-0387340470" target="_blank">978-0387340470</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Guru, Bhag Singh; Hüseyin R. Hızıroğlu (2004). Electromagnetic Field Theory Fundamentals, 2nd Ed. Cambridge Univ. Press. pp. 422–423. ISBN 978-1139451925. <a href="978-1139451925" target="_blank">978-1139451925</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Schmitt, Ron Schmitt (2002). Electromagnetics Explained: A Handbook for Wireless/ RF, EMC, and High-Speed Electronics. Newnes. pp. 153. ISBN 978-0080505237. <a href="978-0080505237" target="_blank">978-0080505237</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Carr, Joseph J. (1997). Microwave & Wireless Communications Technology. USA: Newnes. pp. 46–47. ISBN 978-0750697071. <a href="978-0750697071" target="_blank">978-0750697071</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Raisanen, Antti V.; Arto Lehto (2003). Radio Engineering for Wireless Communication and Sensor Applications. Artech House. pp. 35–37. ISBN 978-1580536691. <a href="978-1580536691" target="_blank">978-1580536691</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Jackman, Shawn M.; Matt Swartz; Marcus Burton; Thomas W. Head (2011). CWDP Certified Wireless Design Professional Official Study Guide: Exam PW0-250. John Wiley & Sons. pp. Ch. 7. ISBN 978-1118041611. <a href="978-1118041611" target="_blank">978-1118041611</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Raisanen, Antti V.; Arto Lehto (2003). Radio Engineering for Wireless Communication and Sensor Applications. Artech House. pp. 35–37. ISBN 978-1580536691. <a href="978-1580536691" target="_blank">978-1580536691</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Raisanen, Antti V.; Arto Lehto (2003). Radio Engineering for Wireless Communication and Sensor Applications. Artech House. pp. 35–37. ISBN 978-1580536691. <a href="978-1580536691" target="_blank">978-1580536691</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Weber, Ernst; Nebeker, Frederik (1994). The Evolution of Electrical Engineering. Piscataway, New Jersey: IEEE Press. ISBN 0-7803-1066-7. <a href="0-7803-1066-7" target="_blank">0-7803-1066-7</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Leon, Benjamin J.; Wintz, Paul A. (1970). Basic Linear Networks for Electrical and Electronics Engineer. US: Holt, Rinehart, and Winston. pp. 127 to 129. ISBN 0030783259. <a href="0030783259" target="_blank">0030783259</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Pozar, David M. (2013). Microwave Engineering (4th ed.). Hoboken, NJ, US: John Wiley & Sones, Inc. pp. 174, 175. ISBN 978-81-265-4190-4. <a href="978-81-265-4190-4" target="_blank">978-81-265-4190-4</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
<li id="fn:15"><p>Matthaei, George L.; Young, Leo; Jones, E. M. T. (1984). Microwave Filters, Impudence-Matching Networks, and Coupling Structures. 610 Washington Street, Dedham, Massachusetts, US: Artech House, Inc. (published 1985). p. 30. ISBN 0-89006-099-1.{{cite book}}:  CS1 maint: location (link) <a href="0-89006-099-1" target="_blank">0-89006-099-1</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li>
<li id="fn:16"><p>Drakos, Nikos; Hennecke, Marcus; Moore, Ross; Swan, Herb (November 22, 2013). "Transmission Line". Quite Universal Circuit Simulator (Qucs). <a href="https://qucs.sourceforge.net/tech/node61.html" target="_blank">https://qucs.sourceforge.net/tech/node61.html</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li>
<li id="fn:17"><p>Leon, Benjamin J.; Wintz, Paul A. (1970). Basic Linear Networks for Electrical and Electronics Engineer. US: Holt, Rinehart, and Winston. pp. 127 to 129. ISBN 0030783259. <a href="0030783259" target="_blank">0030783259</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li>
<li id="fn:18"><p>Pozar, David M. (2013). Microwave Engineering (4th ed.). Hoboken, NJ, US: John Wiley & Sones, Inc. pp. 174, 175. ISBN 978-81-265-4190-4. <a href="978-81-265-4190-4" target="_blank">978-81-265-4190-4</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li>
<li id="fn:19"><p>Pozar, David M. (1998). Microwave Engineering (2nd ed.). Canada: John Wiley & Sons, Inc. p. 192. ISBN 0-471-17096-8. <a href="0-471-17096-8" target="_blank">0-471-17096-8</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li>
<li id="fn:20"><p>Matthaei, George L.; Young, Leo; Jones, E. M. T. (1984). Microwave Filters, Impudence-Matching Networks, and Coupling Structures. 610 Washington Street, Dedham, Massachusetts, US: Artech House, Inc. (published 1985). p. 30. ISBN 0-89006-099-1.{{cite book}}:  CS1 maint: location (link) <a href="0-89006-099-1" target="_blank">0-89006-099-1</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li>
<li id="fn:21"><p>Drakos, Nikos; Hennecke, Marcus; Moore, Ross; Swan, Herb (November 22, 2013). "Transmission Line". Quite Universal Circuit Simulator (Qucs). <a href="https://qucs.sourceforge.net/tech/node61.html" target="_blank">https://qucs.sourceforge.net/tech/node61.html</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li>
<li id="fn:22"><p>Matthaei, George L.; Young, Leo; Jones, E. M. T. (1984). Microwave Filters, Impudence-Matching Networks, and Coupling Structures. 610 Washington Street, Dedham, Massachusetts, US: Artech House, Inc. (published 1985). p. 30. ISBN 0-89006-099-1.{{cite book}}:  CS1 maint: location (link) <a href="0-89006-099-1" target="_blank">0-89006-099-1</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li>
<li id="fn:23"><p>University of Texas at Austin (December 14, 2015). "Microsoft Word - dissertation_def_rev.doc - ch_2.pdf" (PDF). <a href="http://weewave.mer.utexas.edu/MED_files/Former_Students/thesis_dssrtns/Friar_R_diss/ch_2.pdf" target="_blank">http://weewave.mer.utexas.edu/MED_files/Former_Students/thesis_dssrtns/Friar_R_diss/ch_2.pdf</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li>
<li id="fn:24"><p>"2.3: Scattering Parameters - Engineering LibreTexts". Engineering LibreTexts. October 21, 2020. <a href="https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_III_-_Networks_(Steer)/02%3A_Chapter_2/2.3%3A_Scattering_Parameters" target="_blank">https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_III_-_Networks_(Steer)/02%3A_Chapter_2/2.3%3A_Scattering_Parameters</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li>
<li id="fn:25"><p>Pozar, David M. (1998). Microwave Engineering (2nd ed.). John Wiley and Sons, Inc. pp. 383–388. ISBN 0-471-17096-8. <a href="0-471-17096-8" target="_blank">0-471-17096-8</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li>
<li id="fn:26"><p>Maththaei, George L.; Young, Leo; E. M. T., Jones (1964). Microwave filters, impedance-matching networks, and coupling structures. Dedham, MA, US: Artech House Books. pp. 174–196. ISBN 0-89006-099-1. <a href="0-89006-099-1" target="_blank">0-89006-099-1</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li>
<li id="fn:27"><p>Rhea, Randall W. (1995). HF Filter Design and Computer Simulation. New York, NY, US: McGraw-Hill. p. 85. ISBN 0-07-052055-0. <a href="0-07-052055-0" target="_blank">0-07-052055-0</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li>
<li id="fn:28"><p>"5.6: Formulas for Impedance of Coupled Microstrip Lines". Engineering LibreTexts. October 21, 2022. <a href="https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.06%3A_Formulas_for_Impedance_of_Coupled_Microstrip_Lines" target="_blank">https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.06%3A_Formulas_for_Impedance_of_Coupled_Microstrip_Lines</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li>
<li id="fn:29"><p>Drakos, Nikos; Hennecke, Marcus; Moore, Ross; Herb, Swan (November 22, 2013). "Parallel coupled microstrip lines". Quite universal circuit simulator. <a href="https://qucs.sourceforge.net/tech/node77.html" target="_blank">https://qucs.sourceforge.net/tech/node77.html</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li>
<li id="fn:30"><p>Garg, Ramesh; Bahl, Inder; Bozzi, Maurizio (2013). Microstrip Lines and Slotlines (3rd ed.). Boston, London: Artech House. pp. 462–473. ISBN 978-1-60807-535-5. <a href="978-1-60807-535-5" target="_blank">978-1-60807-535-5</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li>
<li id="fn:31"><p>"5.9: Models of Parallel Coupled Lines - Engineering LibreTexts". Libre Texts. October 21, 2020. <a href="https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.09%3A_Models_of_Parallel_Coupled_Lines" target="_blank">https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_II_-_Transmission_Lines_(Steer)/05%3A_Coupled_Lines_and_Applications/5.09%3A_Models_of_Parallel_Coupled_Lines</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li>
<li id="fn:32"><p>Syed V. Ahamed, Victor B. Lawrence, Design and engineering of intelligent communication systems, pp.130–131, Springer, 1997 ISBN 0-7923-9870-X. <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li>
<li id="fn:33"><p>Evaluating Microphone Cable Performance & Specifications Archived 2016-05-09 at the Wayback Machine <a href="http://www.canare.com/UploadedDocuments/A%20Technical%20Paper%20-%20Evaluating%20Microphone%20Cable%20Perfrmance%20and%20Specifications.pdf" target="_blank">http://www.canare.com/UploadedDocuments/A%20Technical%20Paper%20-%20Evaluating%20Microphone%20Cable%20Perfrmance%20and%20Specifications.pdf</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li>
<li id="fn:34"><p>How Starquad Works Archived 2016-11-12 at the Wayback Machine <a href="http://www.belden.com/blog/broadcastav/How-Starquad-Works.cfm" target="_blank">http://www.belden.com/blog/broadcastav/How-Starquad-Works.cfm</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li>
<li id="fn:35"><p>Lampen, Stephen H. (2002). Audio/Video Cable Installer's Pocket Guide. McGraw-Hill. pp. 32, 110, 112. ISBN 978-0071386210. <a href="978-0071386210" target="_blank">978-0071386210</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li>
<li id="fn:36"><p>Rayburn, Ray (2011). Eargle's The Microphone Book: From Mono to Stereo to Surround – A Guide to Microphone Design and Application (3 ed.). Focal Press. pp. 164–166. ISBN 978-0240820750. <a href="978-0240820750" target="_blank">978-0240820750</a> <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li>
<li id="fn:37"><p>Qian, Chunqi; Brey, William W. (2009). "Impedance matching with an adjustable segmented transmission line". Journal of Magnetic Resonance. 199 (1): 104–110. Bibcode:2009JMagR.199..104Q. doi:10.1016/j.jmr.2009.04.005. PMID 19406676. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li>
<li id="fn:38"><p>"Microwaves101 | Parasitics". Microwave Encyclopedia. Retrieved April 2, 2024. <a href="https://www.microwaves101.com/encyclopedias/parasitics" target="_blank">https://www.microwaves101.com/encyclopedias/parasitics</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li>
<li id="fn:39"><p>"2.12: Richards's Transformation - Engineering LibreTexts". Engineering LibreTexts. February 1, 2021. <a href="https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_IV%3A_Modules_(Steer)/02%3A_Filters/2.12%3A_Richardss_Transformation" target="_blank">https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/Microwave_and_RF_Design_IV%3A_Modules_(Steer)/02%3A_Filters/2.12%3A_Richardss_Transformation</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li>
<li id="fn:40"><p>Rhea, Randall W. (1995). HF Filter Design and Computer Simulation. McGraw-Hill, Inc. pp. 86–89. ISBN 0-07-052055-0. <a href="0-07-052055-0" target="_blank">0-07-052055-0</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li>
<li id="fn:41"><p>Rhea, Randall W. "A Multimode High-Frequency Inductor Model", Applied Microwaves & Wireless, November/December 1997, pp. 70-72, 74, 76-78, 80, Noble Publishing, Atlanta, Georgia, <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li>
</ol>

Transmission line open-in-new

Transmission line