8.4: Arbitrary Tions

The Generalized Reflection Coefficient

A lossless transmission line excited at \(z = -l\) with a sinusoidal voltage source is now terminated at its other end at \(z =0\) with an arbitrary impedance \(Z_L\), which in general can be a complex number. Defining the load voltage and current at \(z =0\) as

\[ \begin{align} v\left ( z=0,t \right )&=v_{L}\left ( t \right )=\textrm{Re}\left ( V_{L}e^{j\omega t} \right ) \\i\left ( z=0,t \right )&=i_{L}\left ( t \right )=\textrm{Re}\left ( I_{L}e^{j\omega t} \right ),\quad I=V_{L}/Z_{L} \nonumber \end{align}  \]

where \(V_L\) and \(I_L\) are complex amplitudes, the boundary conditions at \(z =0\) are

\[ \textrm{V}_{+}+\textrm{V}_{-}=\textrm{V}_{L}\\
Y_{0}\left ( \textrm{V}_{+}-\textrm{V}_{-} \right )=I_{L}=\textrm{V}_{L}/Z_{L} \]

We define the reflection coefficient as the ratio

\[ \Gamma _{L}=\textrm{V}_{-}/\textrm{V}_{+} \]

and solve as

\[ \Gamma _{L}=\frac{Z_{L}-Z_{0}}{Z_{L}+Z_{0}} \]

Here in the sinusoidal steady state with reactive loads, \(\Gamma _{L}\) can be a complex number as \(Z_{L}\) may be complex. For transient pulse waveforms, \(\Gamma _{L}\) was only defined for resistive loads. For capacitative and inductive terminations, the reflections were given by solutions to differential equations in time. Now that we are only considering sinusoidal time variations so that time derivatives are replaced by \(j\omega \), we can generalize \(\Gamma _{L}\) for the sinusoidal steady state.

It is convenient to further define the generalized reflection coefficient as

\[ \Gamma\left ( z \right )=\frac{\textrm{V}_{-}e^{jkz}}{\textrm{V}_{+}e^{-jkz}}=\frac{\textrm{V}_{-}}{\textrm{V}_{+}}e^{2jkz}=\Gamma _{L}e^{2jkz} \]

where \(\Gamma _{L}\) is just \(\Gamma\left ( z = 0\right )\). Then the voltage and current on the line can be expressed as

\[ \hat{v}\left ( z \right )=\textrm{V}_{+}e^{-jkz}\left [ 1+\Gamma \left ( z \right ) \right ]\\
\hat{\imath }\left ( z \right )=Y_{0}\textrm{V}_{+}e^{-jkz}\left [ 1-\Gamma \left ( z \right ) \right ] \]

The advantage to this notation is that now the impedance along the line can be expressed as

\[ Z_{n}\left ( z \right )=\frac{Z\left ( z \right )}{Z_{0}}=\frac{\hat{v}\left ( z \right )}{\hat{\imath }\left ( z \right )Z_{0}}=\frac{1+\Gamma \left ( z \right )}{1-\Gamma \left ( z \right )} \]

where \(Z_{n}\) is defined as the normalized impedance. We can now solve (7) for \(\Gamma \left ( z \right )\) as

\[ \Gamma \left ( z \right )=\frac{Z_{n}\left ( z \right )-1}{Z_{n}\left ( z \right )+1} \]

Note the following properties of \(Z_{n}\left ( z \right )\) and \(\Gamma \left ( z \right )\) for passive loads:

1. \(Z_{n}\left ( z \right )\) is generally complex. For passive loads its real part is allowed over the range from zero to infinity while its imaginary part can extend from negative to positive infinity.
2. The magnitude of \(\Gamma \left ( z \right )\), \(\left | \Gamma_{L} \right |\) must be less than or equal to \(1\) for passive loads.
3. From (5), if \(z\) is increased or decreased by a half wavelength, \(\Gamma \left ( z \right )\) and hence \(Z_{n}\left ( z \right )\) remain unchanged. Thus, if the impedance is known at any position, the impedance of all-points integer multiples of a half wavelength away have the same impedance.
4. From (5), if \(z\) is increased or decreased by a quarter wavelength, \(\Gamma \left ( z \right )\) changes sign, while from (7) \(Z_{n}\left ( z \right )\) goes to its reciprocal \(\Rightarrow 1/Z_{n}\left ( z \right )=Y_{n}\left ( z \right )\).
5. If the line is matched, \(Z_L = Z_0\), then \(\Gamma _L= 0\) and \(Z_{n}\left ( z \right )= 1\). The impedance is the same everywhere along the line.

Simple Examples

(a) Load Impedance Reflected Back to the Source

Properties (iii)-(v) allow simple computations for transmission line systems that have lengths which are integer multiples of quarter or half wavelengths. Often it is desired to maximize the power delivered to a load at the end of a transmission line by adding a lumped admittance \(Y\) across the line. For the system shown in Figure 8-17a, the impedance of the load is reflected back to the generator and then added in parallel to the lumped reactive admittance \(Y\). The normalized load impedance of \(\left ( R_{L}+jX_{L} \right )/Z_{0}\) inverts when reflected back to the source by a quarter wavelength to \(Z_{0}/\left ( R_{L}+jX_{L} \right ) \). Since this is the normalized impedance the actual impedance is found by multiplying by \(Z_{0}\) to yield \(Z\left ( z=-\lambda /4 \right )=Z_{0}^{2}/\left ( R_{L}+jX_{L} \right )\). The admittance of this reflected load then adds in parallel to \(Y\) to yield a total admittance of \(Y+\left ( R_{L}+jX_{L} \right )/Z_{0}^{2}\). If \(Y\) is pure imaginary and of opposite sign to the reflected load susceptance with value \(-jX_{L}/Z_{0}^{2}\), maximum power is delivered to the line if the source resistance \(R_{S}\) also equals the resulting line input impedance, \(R_{S}=Z_{0}^{2}/R_{L}\). Since \(Y\) is purely 

Figure 8-17 The normalized impedance reflected back through a quarter-wave-long line inverts. (a) The time-average power delivered to a complex load can be maximized if \(Y\) is adjusted to just cancel the reactive admittance of the load reflected back to the source with \(R_{s}\) equaling the resulting input resistance. (b) If the length \(l_2\) of the second transmission line shown is a quarter wave long or an odd integer multiple of \(\lambda /4\) and its characteristic impedance is equal to the geometric average of \(Z_{1}\) and \(R_{L}\), the input impedance \(Z_{in}\) is matched to \(Z_{1}\).

reactive and the transmission line is lossless, half the time-average power delivered by the source is dissipated in the load:

\[ <P>=\frac{1}{8}\frac{V_{0}^{2}}{R_{S}}=\frac{1}{8}\frac{R_{L}V_{0}^{2}}{Z_{0}^{2}} \]

Such a reactive element \(Y\) is usually made from a variable length short circuited transmission line called a stub. As shown in Section 8-3-2a, a short circuited lossless line always has aipure reactive impedance.

To verify that the power in (9) is actually dissipated in the load, we write the spatial distribution of voltage and current along the line as

\[ \hat{v}\left ( z \right )=\textrm{V}_{+}e^{-jkz}\left [ 1+\Gamma \left ( z \right )e^{2jkz} \right ]\\
\hat{\imath }\left ( z \right )=Y_{0}\textrm{V}_{+}e^{-jkz}\left [ 1-\Gamma \left ( z \right )e^{2jkz} \right ] \]

where the reflection coefficient for this load is given by (4) as

\[ \Gamma _{L}=\frac{R_{L}+jX_{L}-Z_{0}}{R_{L}+jX_{L}+Z_{0}} \]

At \(z = -l = -\lambda /4\) we have the boundary condition

\[ \begin{align} \hat{v}\left ( z=-l \right )=V_{0}/2&=\textrm{V}_{+}e^{jkl}\left ( 1+\Gamma _{L}e^{-2jkl} \right ) \\ & =j\textrm{V}_{+}\left ( 1-\Gamma _{L} \right ) \nonumber \end{align} \]

which allows us to solve for \(\textrm{V}_{+}\) as

\[ \textrm{V}_{+}=\frac{-jV_{0}}{2\left ( 1-\Gamma _{L} \right )}=\frac{-jV_{0}}{4Z_{0}}\left ( R_{L}+jX_{L}+Z_{0} \right ) \]

The time-average power dissipated in the load is then

\[ \begin{align}<P_{L}>&=\frac{1}{2}\textrm{Re}\left [ \hat{v}\left ( z=0 \right )\hat{\imath ^{\ast }}\left ( z=0 \right ) \right ] \\ &
=\frac{1}{2}\left | \hat{\imath}\left ( z=0 \right )  \right |^{2}R_{L} \nonumber \\ &
=\frac{1}{2}\left | \textrm{V}_{+} \right |^{2}\left | 1-_{L} \right |^{2}Y_{0}^{2}R_{L} \nonumber \\ &
=\frac{1}{8}V_{0}^{2}Y_{0}^{2}R_{L} \nonumber \end{align} \]

which agrees with (9).

(b) Quarter Wavelength Matching

It is desired to match the load resistor \(R_{L}\) to the transmission line with characteristic impedance \(Z_{1}\) for any value of its length \(l_{1}\). As shown in Figure 8-17b, we connect the load to \(Z_{1}\) via another transmission line with characteristic impedance \(Z_{2}\). We wish to find the values of \(Z_{2}\) and \(l_{2}\) necessary to match \(R_{L}\) to \(Z_{1}\).

This problem is analogous to the dielectric coating problem of Example 7-1, where it was found that reflections could be eliminated if the coating thickness between two different dielectric media was an odd integer multiple of a quarter wavelength and whose wave impedance was equal to the geometric average of the impedance in each adjacent region. The normalized load on \(Z_{2}\) is then \(Z_{n2}=R_{L}/Z_{2}\). If \(l_{2}\) is an odd integer multiple of a quarter wavelength long, the normalized impedance \(Z_{n2}\) reflected back to the first line inverts to \(Z_{2}/R_{L}\). The actual impedance is obtained by multiplying this normalized impedance by \(Z_{2}\) to give \(Z_{2}^{2}/R_{L}\). For \(Z_{in}\) to be matched to \(Z_{1}\) for any value of \(l_{1}\), this impedance must be matched to \(Z_{1}\):

\[ Z_{1}=Z_{2}^{2}/R_{L}\Rightarrow Z_{2}=\sqrt{Z_{1}R_{L}} \]

The Smith Chart

Because the range of allowed values of \(\Gamma _{L}\) must be contained within a unit circle in the complex plane, all values of \(Z_{n}\left ( z \right )\) can be mapped by a transformation within this unit circle using (8). This transformation is what makes the substitutions of (3)-(8) so valuable. A graphical aid of this mathematical transformation was developed by P. H. Smith in 1939 and is known as the Smith chart. Using the Smith chart avoids the tedium in problem solving with complex numbers.

Let us define the real and imaginary parts of the normalized impedance at some value of \(z\) as

\[ Z_{n}\left ( z \right )=r+jx \]

The reflection coefficient similarly has real and imaginary parts given as

\[\Gamma \left ( z \right )=\Gamma _{r}+j\Gamma _{i} \]

Using (7) we have

\[r+jx=\frac{1+\Gamma _{r}+j\Gamma _{i}}{1-\Gamma _{r}-j\Gamma _{i}} \]

Multiplying numerator and denominator by the complex conjugate of the denominator \(\left (1-\Gamma _{r}+j\Gamma _{i}  \right )\) and separating real and imaginary parts yields

\[ r=\frac{1-\Gamma _{r}^{2}-\Gamma _{i}^{2}}{\left (1-\Gamma _{r}  \right )^{2}+\Gamma _{i}^{2}}\\
r=\frac{2\Gamma _{i}}{\left (1-\Gamma _{r}  \right )^{2}+\Gamma _{i}^{2}} \]

Since we wish to plot (19) in the \(\Gamma _{r}-\Gamma _{i}\) plane we rewrite these equations as

\[ \left ( \Gamma _{r}-\frac{r}{1+r} \right )^{2}+\Gamma _{i}^{2}=\frac{1}{\left ( 1+r \right )^{2}}\\
\left (\Gamma _{r}-1\right )^{2}+\left ( \Gamma _{i}-\frac{1}{x} \right )^{2}=\frac{1}{x^{2}} \]

Both equations in (20) describe a family of orthogonal circles.The upper equation is that of a circle of radius \(1/(1 +r)\) whose center is at the position \(\Gamma _{i}=0,\,\Gamma _{r}=r/\left ( 1+r \right )\). The lower equation is a circle of radius \(\left | 1/x \right |\) centered at the position \(\Gamma _{r}=1,\,\Gamma _{i}=1/x\). Figure 8-18a illustrates these circles for a particular value of \(r\) and \(x\), while Figure 8-1 8b shows a few representative values of \(r\) and \(x\). In Figure 8-19, we have a complete Smith chart. Only those parts of the circles that lie within the unit circle in the \(\Gamma\) plane are considered for passive

Figure 8-18 For passive loads the Smith chart is constructed within the unit circle in the complex \(\Gamma\) plane. (a) Circles of constant normalized resistance \(r\) and reactance \(x\) are constructed with the centers and radii shown. (b) Smith chart construction for various values of \(r\) and \(x\).

resistive-reactive loads. The values of \(\Gamma\left ( z \right )\) themselves are usually not important and so are not listed, though they can be easily found from (8). Note that all circles pass through the point \(\Gamma _{r}=1,\,\Gamma _{i}=0\).

The outside of the circle is calibrated in wavelengths toward the generator, so if the impedance is known at any point on the transmission line (usually at the load end), the impedance at any other point on the line can be found using just a compass and a ruler. From the definition of \(\Gamma\left ( z \right )\) in (5) with \(z\) negative, we move clockwise around the Smith chart when heading towards the source and counterclockwise when moving towards the load.

Figure 8-18

In particular, consider the transmission line system in Figure 8-20a. The normalized load impedance is \(Z_{n}=1+j\). Using the Smith chart in Figure 8-20b, we find the load impedance at position \(A\). The effective impedance reflected back to \( z = -l\) must lie on the circle of constant radius returning to \( A\) whenever \(l\) is an integer multiple of a half wavelength. The table in Figure 8-20 lists the impedance at \(z = -l\) for various line lengths. Note that at point \( C\), where \(l=\lambda /4\), that the normalized impedance is the reciprocal of

Figure 8-19 A complete Smith chart.

that at \(A\). Similarly the normalized impedance at \(B\) is the reciprocal of that at \(D\).

The current from the voltage source is found using the equivalent circuit shown in Figure 8-20c as

\[ i=\left | \hat{I} \right |\sin \left ( wt-\phi  \right ) \]

where the current magnitude and phase angle are

\[ \left | \hat{I} \right |=\frac{\textrm{V}_{0}}{\left | 50+Z\left ( z=-l \right ) \right |},\quad \phi =\tan ^{-1}\frac{\textrm{Im}\left [ Z\left ( z=-l \right ) \right ]}{50+\textrm{Re}\left [ Z\left ( z=-l \right ) \right ]} \]

Representative numerical values are listed in Figure 8-20.

Figure 8-20 (a) The load impedance at \(z = 0\) reflected back to the source is found using the (b) Smith chart for various line lengths. Once this impedance is known the source current is found by solving the simple series circuit in (c).

Standing Wave Parameters

The impedance and reflection coefficient are not easily directly measured at microwave frequencies. In practice, one slides an ac voltmeter across a slotted transmission line and measures the magnitude of the peak or rms voltage and not its phase angle.

From (6) the magnitude of the voltage and current at any position \(z\) is

\[ \left | \hat{v}\left ( z \right ) \right |=\left | \textrm{V}_{+} \right |\left | 1+\Gamma \left ( z \right ) \right |\\
\left | \hat{\imath }\left ( z \right ) \right |=Y_{0}\left | \textrm{V}_{+} \right |\left | 1-\Gamma \left ( z \right ) \right | \]

From (23), the variations of the voltage and current magnitudes can be drawn by a simple construction in the F plane, as shown in Figure 8-21. Note again that \(\left | \textrm{V}_{+} \right |\) is just a real number independent of \(z\) and that \(\left |\Gamma \left ( z \right )  \right |\leq 1\) for a passive termination. We plot \(\left |1+\Gamma \left ( z \right )  \right |\) and \(\left |1-\Gamma \left ( z \right )  \right |\) since these terms are proportional to the voltage and current magnitudes, respectively. The following properties from this con-

Figure 8-21 The voltage and current magnitudes along a transmission line are respectively proportional to the lengths of the vectors \(\left |1+\Gamma \left ( z \right )  \right |\) and \(\left |1-\Gamma \left ( z \right )  \right |\) in the complex \(\Gamma \) plane.

struction are apparent:

1. The magnitude of the current is smallest and the voltage magnitude largest when \(\Gamma \left ( z \right )=1\) at point \(A\) and vice versa when \(\Gamma \left ( z \right )=-1\) at point \(B\).
2. The voltage and current are in phase at the points of maximum or minimum magnitude of either at points \(A\) or \(B\).
3. A rotation of \(\Gamma \left ( z \right )\) by an angle \(\pi \) corresponds to a change of \(\lambda /4\) in \(z\), thus any voltage (or current) maximum is separated by \(\lambda /4\) from its nearest minima on either side.

By plotting the lengths of the phasors \(\left |1\pm \Gamma \left ( z \right ) \right |\), as in Figure 8-22, we obtain a plot of what is called the standing wave pattern on the line. Observe that the curves are not sinusoidal. The minima are sharper than the maxima so the minima are usually located in position more precisely by measurement than the maxima.

From Figures 8-21 and 8-22, the ratio of the maximum voltage magnitude to the minimum voltage magnitude is defined as the voltage standing wave ratio, or \(\textrm{VSWR}\) for short:

\[ \frac{\left | \hat{v}\left ( z \right ) \right |_{max}}{\left | \hat{v}\left ( z \right ) \right |_{min}}=\frac{1+\left | \Gamma _{L} \right |}{1-\left | \Gamma _{L} \right |}=\textrm{VSWR} \]

The \(\textrm{VSWR}\) is measured by simply recording the largest and smallest readings of a sliding voltmeter. Once the \(\textrm{VSWR}\) is measured, the reflection coefficient magnitude can be calculated from (24) as

\[ \left | \Gamma _{L} \right |=\frac{\textrm{VSWR}-1}{\textrm{VSWR}+1} \]

The angle \(\phi \) of the reflection coefficient

\[  \Gamma _{L} =\left | \Gamma _{L} \right |e^{j\phi } \]

can also be determined from these standing wave measurements. According to Figure 8-21, \(\Gamma \left ( z \right )\) must swing clockwise through an angle \(\phi +\pi \) as we move from the load at \(z =0\) toward the generator to the first voltage minimum at \(B\). The shortest distance \(d_{min}\) that we must move to reach the first voltage minimum is given by

\[ 2kd_{min}=\phi +\pi  \]

or

\[ \frac{\phi }{\pi }=4\frac{d_{min}}{\lambda }-1 \]

Figure 8-22 Voltage and current standing wave patterns plotted for various values of the VSWR.

A measurement of \(d_{min}\), as well as a determination of the wavelength (the distance between successive minima or maxima is \(\lambda /2\)) yields the complex reflection coefficient of the load using (25) and (28). Once we know the complex reflection coefficient we can calculate the load impedance from (7). These standing wave measurements are sufficient to determine the terminating load impedance \(Z_L\). These measurement properties of the load reflection coefficient and its relation to the load impedance are of great importance at high frequencies where the absolute measurement of voltage or current may be difficult. Some special cases of interest are:

1. Matched line --- If \(\Gamma _{L}=0\), then \(\textrm{VSWR} =1\). The voltage magnitude is constant everywhere on the line.
2. Short or open circuited line --- If \(\Gamma _{L}=1\), then \(\textrm{VSWR} =\infty \). The minimum voltage on the line is zero.
3. The peak normalized voltage \(\left | \hat{v}\left ( z \right )/V_{+} \right |\) is \(1+\left | \Gamma _{L} \right |\) while the minimum normalized voltage is \(1-\left | \Gamma _{L} \right |\).
4. The normalized voltage at \(z =0\) is \(\) while the normalized current \(\left | \hat{i}\left ( z \right )/Y_{0}V_{+} \right |\) at \(z = 0\) is \(1-\left | \Gamma _{L} \right |\).
5. If the load impedance is real \(\left ( Z_L+R_L \right )\), then (4) shows us that rL is real. Then evaluating (7) at \(z =0\), where \(\Gamma \left ( z=0 \right )=\Gamma _{L}\), we see that when \(Z_{L}> Z_{0}\) that \(\textrm{VSWR} =Z_{L}/ Z_{0} \) while if \(Z_{L}<Z_{0}\), \(\textrm{VSWR}=Z_{0}/Z_{L}\).

For a general termination, if we know the \(\textrm{VSWR}\) and \(d_{min}\), we can calculate the load impedance from (7) as

\[ \begin{align}Z_{L}&=Z_{0}\frac{1+\left | \Gamma _{L} \right |e^{j\phi}}{1-\left | \Gamma _{L} \right |e^{j\phi}} \\ &
=Z_{0}\frac{\left [ \textrm{VSWR}+1+\left ( \textrm{VSWR}-1 \right )e^{j\phi} \right ]}{\left [ \textrm{VSWR}+1-\left ( \textrm{VSWR}-1 \right )e^{j\phi} \right ]} \nonumber \end{align} \]

Multiplying through by \(e^{j\phi /2}\) and then simplifying yields

\[ \begin{align}Z_{L}&=\frac{Z_{0}\left [ \textrm{VSWR}-j\tan \left ( \phi /2 \right ) \right ]}{\left [ 1-j\textrm{VSWR}\tan \left ( \phi /2 \right ) \right ]} \\ &
=\frac{Z_{0}\left [ \textrm{VSWR}-j\tan \left ( \phi /2 \right ) \right ]}{\left [ 1-j\textrm{VSWR}\tan \left ( \phi /2 \right ) \right ]} \nonumber \end{align} \]

Example \(\PageIndex{1}\) EXAMPLE 8-2 VOLTAGE STANDING WAVE RATIO

The \(\textrm{VSWR}\) on a \(50-\textrm{Ohm}\) (characteristic impedance) transmission line is \(2\). The distance between successive voltage minima is \(40\,\textrm{cm}\) while the distance from the load to the first minima is \(10\,\textrm{cm}\). What is the reflection coefficient and load impedance?

SOLUTION

We are given

\[\textrm{VSWR}=2\\
kd_{min}=\frac{2\pi \left ( 10 \right )}{2\left ( 40 \right )}=\frac{\pi }{4} \nonumber \]

The reflection coefficient is given from (25)-(28) as

\[ \Gamma _{L}=\frac{1}{3}e^{-j\pi /2}=\frac{-j}{3}  \nonumber \]

while the load impedance is found from (30) as

\[ Z_{L}=\frac{50\left ( 1-2j \right )}{2-j} \nonumber \]
\[=40-30j\,\textrm{ohm} \nonumber  \]