Transmission Lines and Impedance Matching

Distributed Line Model

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Transmission line and distributed model.

A transmission line guides electromagnetic energy between locations and is modeled by resistance $R$, inductance $L$, conductance $G$ and capacitance $C$ distributed continuously per unit length.

The primary line parameters are

Symbol	Meaning	SI unit
$R$	series resistance per unit length	$\Omega/\mathrm{m}$
$L$	series inductance per unit length	$\mathrm{H/m}$
$G$	shunt conductance per unit length	$\mathrm{S/m}$
$C$	shunt capacitance per unit length	$\mathrm{F/m}$

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Telegrapher equations.

The telegrapher equations are the two coupled differential equations that govern voltage and current along a uniform transmission line. They may be written in the time or frequency domain.

Time-Domain Form

For instantaneous line voltage $v(z,t)$ and current $i(z,t)$,

$$\begin{aligned} \frac{\partial v(z,t)}{\partial z} & =-R\,i(z,t)-L\frac{\partial i(z,t)}{\partial t}, \\ \frac{\partial i(z,t)}{\partial z} & =-G\,v(z,t)-C\frac{\partial v(z,t)}{\partial t}. \end{aligned}$$

Phasor Form

For sinusoidal steady state, $\partial/\partial t\rightarrow j\omega$, where $j^2=-1$ and $\omega=2\pi f$. The voltage and current phasors $V(z)$ and $I(z)$ therefore satisfy

$$\frac{dV}{dz}=-(R+j\omega L)I, \qquad \frac{dI}{dz}=-(G+j\omega C)V.$$

Distributed model of a uniform transmission line.

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Derivation: Telegrapher equations from the distributed model

Consider the differential line section of length $\Delta z$ shown above.

Series-path KVL. The input voltage equals the output voltage plus the resistive and inductive drops:

$$v(z,t)-v(z+\Delta z,t) =R\Delta z\,i(z,t) +L\Delta z\frac{\partial i(z,t)}{\partial t}.$$

Divide by $\Delta z$ and take the limit $\Delta z\to0$:

$$\frac{\partial v(z,t)}{\partial z} =-R\,i(z,t)-L\frac{\partial i(z,t)}{\partial t}.$$

Node KCL. The input current divides between the output path and the shunt conductance and capacitance:

$$\begin{aligned} i(z,t)={} & i(z+\Delta z,t) +G\Delta z\,v(z+\Delta z,t) \\ & +C\Delta z\frac{\partial v(z+\Delta z,t)}{\partial t}. \end{aligned}$$

Rearrange, divide by $\Delta z$, and take $\Delta z\to0$:

$$\frac{\partial i(z,t)}{\partial z} =-G\,v(z,t)-C\frac{\partial v(z,t)}{\partial t}.$$

Finally, replacing $\partial/\partial t$ by $j\omega$ gives

$$\frac{dV}{dz}=-(R+j\omega L)I, \qquad \frac{dI}{dz}=-(G+j\omega C)V.$$

Propagation Constant and Characteristic Impedance

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Transmission-line propagation constant.

The complex propagation constant $\gamma=\alpha+j\beta$ describes attenuation through $\alpha$ and phase change through $\beta$ per unit distance along a line.

It is

$$\gamma=\alpha+j\beta= \sqrt{(R+j\omega L)(G+j\omega C)}.$$

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Characteristic impedance.

Characteristic impedance $Z_0$ is the voltage-to-current ratio of a single travelling wave on the line; it depends on the distributed line parameters rather than on line length.

It is

$$Z_0=\sqrt{\frac{R+j\omega L}{G+j\omega C}}.$$

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Lossless transmission line.

A lossless line has zero series resistance and shunt conductance, $R=G=0$, so waves propagate without attenuation.

For a lossless line,

$$Z_0=\sqrt{\frac{L}{C}}, \qquad \beta=\omega\sqrt{LC}, \qquad v=\frac{1}{\sqrt{LC}}.$$

Travelling Waves

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Note

In the context of a transmission line, it refers to an electrical voltage or current signal that propagates continuously along the physical length of the line.

[!NOTE]

Incident and reflected travelling waves.

Line voltage and current are superpositions of a forward incident wave and a backward reflected wave; the current sign reverses for the backward wave because its direction of power travel is reversed.

Thus,

$$V(z)=V_0^+e^{-\gamma z}+V_0^-e^{\gamma z},$$ $$I(z)=\frac{V_0^+}{Z_0}e^{-\gamma z}-\frac{V_0^-}{Z_0}e^{\gamma z}.$$

Load Reflection

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Load reflection coefficient.

The load reflection coefficient $\Gamma_L$ is the complex ratio of reflected to incident voltage at the load and measures the magnitude and phase of mismatch between $Z_L$ and $Z_0$.

At the load,

$$\Gamma_L=\frac{Z_L-Z_0}{Z_L+Z_0}.$$

Reference directions for a line terminated in a load.

Special cases:

Load	Reflection coefficient
Matched load, $Z_L=Z_0$	$\Gamma_L=0$
Open circuit	$\Gamma_L=+1$
Short circuit	$\Gamma_L=-1$

Standing Waves, VSWR and Return Loss

Standing Waves

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Standing wave.

A standing wave is the stationary amplitude pattern produced when incident and reflected waves of the same frequency travel in opposite directions along a transmission line.

A load mismatch produces the reflected wave. Its interference with the incident wave creates fixed voltage maxima and minima along the line. Maxima are called antinodes; a minimum that reaches zero is called a node.

VSWR

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Voltage standing-wave ratio.

VSWR measures mismatch by comparing the maximum and minimum voltage magnitudes in the standing-wave envelope.

For load reflection coefficient $\Gamma_L$,

$$\mathrm{VSWR} =\frac{V_{\max}}{V_{\min}} =\frac{1+|\Gamma_L|}{1-|\Gamma_L|}.$$

A matched load has $|\Gamma_L|=0$ and $\mathrm{VSWR}=1$. As the mismatch increases, VSWR increases; complete reflection gives $|\Gamma_L|=1$ and $\mathrm{VSWR}\to\infty$.

Return Loss

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Return loss.

Return loss is a positive logarithmic measure of the ratio of incident power $P_i$ to reflected power $P_r$. A larger return loss means a smaller reflected fraction and therefore a better impedance match.

$$\mathrm{RL} =10\log_{10}\left(\frac{P_i}{P_r}\right) =-20\log_{10}|\Gamma_L|\ \mathrm{dB}.$$

An ideal match has $|\Gamma_L|=0$ and $\mathrm{RL}\to\infty$. Complete reflection has $|\Gamma_L|=1$ and $\mathrm{RL}=0\ \mathrm{dB}$.

Voltage standing-wave envelope on a mismatched line.

Comparison

Quantity	Meaning	Ideal match	Complete reflection
Standing-wave envelope	Voltage variation	Flat	Maximum
$\lvert\Gamma_L\rvert$	Voltage reflection magnitude	$0$	$1$
VSWR	$V_{\max}/V_{\min}$	$1$	$\infty$
Return loss (dB)	Logarithmic power ratio	$\infty$	$0$

Input Impedance

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Input impedance.

Input impedance $Z_{\mathrm{in}}$ is the voltage-to-current ratio seen looking into a finite transmission-line section; it depends on $Z_0$, load impedance, propagation constant and distance from the load.

For a lossless line of length $l$, the terminal voltage and current are related to the load quantities by

$$V_{\text{in}}=V_L\cos\beta l+jZ_0I_L\sin\beta l,$$ $$I_{\text{in}}=j\frac{V_L}{Z_0}\sin\beta l+I_L\cos\beta l.$$

Substituting $V_L=Z_LI_L$, taking $Z_{\text{in}}=V_{\text{in}}/I_{\text{in}}$, and dividing numerator and denominator by $\cos\beta l$ gives the input-impedance transformation below.

For a lossless line of length $l$ terminated by $Z_L$,

$$Z_{\text{in}}=Z_0 \frac{Z_L+jZ_0\tan\beta l}{Z_0+jZ_L\tan\beta l}.$$

Important special lengths:

Length	Input impedance
$l=\lambda/2$	$Z_{\text{in}}=Z_L$
$l=\lambda/4$	$Z_{\text{in}}=\dfrac{Z_0^2}{Z_L}$

Impedance Matching

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Impedance matching.

Impedance matching transforms the load seen by a transmission line so that $Z_L=Z_0$ at the reference plane. An exact match gives $\Gamma_L=0$, $\mathrm{VSWR}=1$ and no reflected power.

Quarter-Wave Transformer

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A quarter-wave transformer is a transmission-line section whose characteristic impedance transforms one real resistance into another.

• Insert a section of characteristic impedance $Z_t$ between the main line and a real load $R_L$.

• At the design frequency $f_0$, make its electrical length $90^\circ$, or $l=\lambda_t/4$, where $\lambda_t$ is the wavelength in the transformer.

• The match is exact at $f_0$ but is generally narrowband. A complex load must first be transformed to a real resistance for this simple formula to apply.

For a quarter-wave transformer matching a real load $R_L$ to a line $Z_0$,

$$Z_t=\sqrt{Z_0R_L}, \qquad l=\frac{\lambda_t}{4}.$$

This condition follows directly from the input impedance of the transformer section:

$$Z_{\text{in}}=Z_t \frac{R_L+jZ_t\tan\beta_tl} {Z_t+jR_L\tan\beta_tl}.$$

At $l=\lambda_t/4$, $\beta_tl=\pi/2$, so the tangent terms dominate and

$$Z_{\text{in}}=\frac{Z_t^2}{R_L}.$$

Requiring the main line to see $Z_{\text{in}}=Z_0$ gives $Z_t=\sqrt{Z_0R_L}$. The quarter-wave section is therefore an impedance inverter at its design frequency.

Quarter-wave transformer matching section.

L-Network Matching

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An L-network uses one series reactance and one shunt reactance to transform between unequal impedances at a chosen frequency.

• Ideal inductors and capacitors store energy without dissipating average power, so the network can provide a lossless conjugate match.

• The series and shunt arrangement depends on which side has the higher resistance; the shunt element is placed on the high-resistance side in the topology shown below.

• A series inductor with a shunt capacitor gives a low-pass match. Interchanging them gives the dual high-pass form.

• L-networks are simple and useful when lumped components remain practical, but their match is frequency-sensitive.

Low-pass L-match topology.

Single-Stub Matching

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A single-stub match uses a short- or open-circuited line section to cancel the reactive part of the transformed load admittance or impedance.

• For the shunt-stub arrangement shown, choose a distance $d$ from the load at which the normalized admittance is $y(d)=1+jb$.

• Choose the stub length $l_s$ so that it supplies susceptance $-jb$; the total normalized admittance then becomes $1$, producing a match.

• Open and shorted stubs are both possible. Stub matching is convenient at radio and microwave frequencies, where distributed lines replace lumped reactances.

• The possible solutions repeat every $\lambda/2$ and are generally narrowband.

Single shunt-stub matching schematic.

Double-Stub Matching

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A double-stub match uses two adjustable stubs separated by a fixed line length.

• Only the two stub lengths need adjustment, so the method is useful when a single stub cannot be placed at its calculated distance from the load.

• The first stub moves the admittance to a value that the second stub can transform to the chart centre.

• Certain loads cannot be matched for some fixed stub spacings; these form a forbidden region. Double-stub matches are also frequency-sensitive.

Smith-Chart Design

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The Smith chart is a graphical calculation tool rather than a separate matching network. It represents normalized impedance, admittance and reflection coefficient on one diagram.

• Normalize the load as $z_L=Z_L/Z_0$; for a shunt-stub design, convert it to admittance $y_L=1/z_L$.

• Move clockwise toward the generator on the constant-$|\Gamma|$ circle until the $g=1$ conductance circle is reached. The outer scale gives the distance $d$ from the load.

• Move along the $g=1$ circle by adding the stub susceptance until the chart centre, $y=1+j0$, is reached. The corresponding scale reading gives $l_s$.

Smith chart path for shunt-stub matching.

Matching Method Comparison

Feature	Quarter-wave transformer	Single stub	Double stub
Elements	one $\lambda/4$ line section	one open or short stub	two fixed-position stubs
Direct load	real in its basic form	real or complex	real or complex, with forbidden regions
Adjustment	section impedance and length	stub length and position	two stub lengths
Installation	needs a different line impedance	needs access at a calculated point	useful when connection points are fixed
Bandwidth	narrowband	narrowband	narrowband

All three methods make the impedance seen by the main line equal to $Z_0$ at the design frequency. The choice is governed mainly by load type, available line impedances, accessible connection points and tuning needs.

Worked Example

A $50\ \Omega$ line is terminated by a real $100\ \Omega$ load. The load reflection coefficient is

$$\Gamma_L=\frac{100-50}{100+50}=\frac{1}{3}=0.333.$$

Therefore

$$\mathrm{VSWR}=\frac{1+1/3}{1-1/3}=2, \qquad \mathrm{RL}=-20\log_{10}(1/3)=9.54\ \mathrm{dB}.$$

The reflected power fraction is $\lvert\Gamma_L\rvert^2=1/9$, or about $11.1\%$. To match the same load with a quarter-wave section,

$$Z_t=\sqrt{(50)(100)}=70.7\ \Omega.$$

At the design frequency, this section presents $Z_{\text{in}}=Z_t^2/R_L=50\ \Omega$ to the main line, eliminating the reflection there.