Terminated Lossless line
Terminated Lossless Transmission Line: Reflection Coefficient, VSWR and Input Impedance
A terminated lossless transmission line is a transmission line with negligible series resistance and shunt conductance, terminated by a load impedance \(Z_L\). The analysis of a terminated line is important because the relationship between the load impedance and the characteristic impedance determines whether the incident wave is completely absorbed or partially reflected back toward the source. When the load is not matched to the characteristic impedance, a reflected wave is produced and combines with the incident wave to form a standing-wave pattern along the line. The resulting voltage and current distributions can be used to determine important transmission-line parameters such as the voltage reflection coefficient, VSWR, transmitted power, return loss, and input impedance.
Configuration of a Terminated Lossless Transmission Line
Consider a lossless transmission line having characteristic impedance \(Z_0\) and propagation constant \(\gamma=j\beta\). The transmission line is terminated by a load impedance \(Z_L\) at \(z=0\), while the source is located at \(z<0\). The coordinate \(z\) therefore increases from the source toward the load. Since the line is lossless, there is no attenuation of the traveling waves as they propagate along the transmission line.
Fig: Terminated Lossless Transmission Line
At the load, the incident wave arriving from the source interacts with the load impedance. If \(Z_L\) is different from \(Z_0\), the load cannot absorb all the incident power, and a portion of the wave is reflected toward the source. The total voltage and current on the line are therefore the combination of the forward-traveling incident wave and the backward-traveling reflected wave.
General Voltage and Current Equations
For a lossless transmission line, the propagation constant is purely imaginary and is given by:
\[ \gamma=j\beta \]
The general voltage distribution on the line is:
\[ V(z) = V_0^+e^{-j\beta z} + V_0^-e^{j\beta z} \]
Similarly, the current distribution is:
\[ I(z) = \frac{V_0^+}{Z_0}e^{-j\beta z} - \frac{V_0^-}{Z_0}e^{j\beta z} \]
Here, \(V_0^+\) is the complex amplitude of the forward-traveling voltage wave and \(V_0^-\) is the complex amplitude of the backward-traveling or reflected voltage wave. The first exponential term represents propagation toward the load, while the second represents propagation toward the source under the selected coordinate convention.
Voltage Reflection Coefficient
The voltage reflection coefficient describes the ratio of the reflected voltage wave to the incident voltage wave at the load. It is one of the most important quantities used to analyze a terminated transmission line. The reflection coefficient is determined by applying the load boundary condition at \(z=0\).
Boundary Condition at the Load
At the load position \(z=0\), the ratio of the total voltage to the total current must be equal to the load impedance \(Z_L\). Therefore:
\[ Z_L = \frac{V(0)}{I(0)} \]
At \(z=0\), the exponential terms become unity. Therefore, the total voltage at the load is:
\[ V(0) = V_0^+ + V_0^- \]
The total current at the load is:
\[ I(0) = \frac{V_0^+}{Z_0} - \frac{V_0^-}{Z_0} \]
or:
\[ I(0) = \frac{V_0^+-V_0^-}{Z_0} \]
Therefore, the load impedance becomes:
\[ Z_L = \frac{V_0^++V_0^-} {\dfrac{V_0^+-V_0^-}{Z_0}} \]
Multiplying both sides by the denominator gives:
\[ Z_L(V_0^+-V_0^-) = Z_0(V_0^++V_0^-) \]
Expanding both sides:
\[ Z_LV_0^+ - Z_LV_0^- = Z_0V_0^+ + Z_0V_0^- \]
Rearranging the terms containing \(V_0^+\) and \(V_0^-\):
\[ V_0^+(Z_L-Z_0) = V_0^-(Z_L+Z_0) \]
Therefore:
\[ \frac{V_0^-}{V_0^+} = \frac{Z_L-Z_0}{Z_L+Z_0} \]
Hence, the voltage reflection coefficient at the load is:
\[ \Gamma_L = \frac{V_0^-}{V_0^+} = \frac{Z_L-Z_0}{Z_L+Z_0} \]
The magnitude \(|\Gamma_L|\) indicates how much of the incident voltage wave is reflected, while the phase of \(\Gamma_L\) determines the phase relationship between the incident and reflected waves.
Standing Waves on a Lossless Transmission Line
When a reflected wave exists, it travels in the opposite direction to the incident wave. The superposition of these two waves produces a standing-wave pattern along the transmission line. Unlike a single traveling wave, the magnitude of the total voltage varies with position, producing alternating voltage maxima and minima.
The total voltage is:
\[ V(z) = V_0^+e^{-j\beta z} + V_0^-e^{j\beta z} \]
Since:
\[ V_0^-=\Gamma_LV_0^+ \]
the voltage can also be written as:
\[ V(z) = V_0^+ \left( e^{-j\beta z} + \Gamma_Le^{j\beta z} \right) \]
The magnitude of the total voltage changes periodically along the line because the phase relationship between the incident and reflected waves changes with position.
Maximum and Minimum Voltage
The maximum voltage occurs when the incident and reflected voltage waves combine in phase. Therefore, the maximum voltage magnitude is:
\[ |V_{\max}| = |V_0^+| + |V_0^-| \]
Since:
\[ |V_0^-| = |\Gamma_L||V_0^+| \]
we obtain:
\[ |V_{\max}| = |V_0^+|(1+|\Gamma_L|) \]
The minimum voltage occurs when the incident and reflected waves combine out of phase. Thus:
\[ |V_{\min}| = |V_0^+| - |V_0^-| \]
Therefore:
\[ |V_{\min}| = |V_0^+|(1-|\Gamma_L|) \]
Voltage Standing Wave Ratio
The Voltage Standing Wave Ratio (VSWR) is defined as the ratio of the maximum voltage to the minimum voltage on the transmission line. It provides a convenient measure of the degree of impedance mismatch between the load and the transmission line.
\[ \mathrm{VSWR} = \frac{|V_{\max}|}{|V_{\min}|} \]
Substituting the expressions for maximum and minimum voltage:
\[ \mathrm{VSWR} = \frac{|V_0^+|(1+|\Gamma_L|)} {|V_0^+|(1-|\Gamma_L|)} \]
Canceling \(|V_0^+|\) gives:
\[ \boxed{ \mathrm{VSWR} = \frac{1+|\Gamma_L|} {1-|\Gamma_L|} } \]
The VSWR therefore depends only on the magnitude of the voltage reflection coefficient. A smaller reflection coefficient corresponds to a lower VSWR and a better impedance match.
Important VSWR Conditions
- Perfectly matched load: If \(Z_L=Z_0\), then \(\Gamma_L=0\) and \(\mathrm{VSWR}=1\). There is no reflected wave.
- Partial reflection: If \(0<|\Gamma_L|<1\), a standing-wave pattern is produced and the VSWR has a finite value greater than 1.
- Complete reflection: If \(|\Gamma_L|=1\), the minimum voltage becomes zero and the VSWR approaches infinity.
Power on a Lossless Transmission Line
For a lossless transmission line, there is no power dissipation along the line. The incident wave carries power toward the load, while the reflected wave carries power back toward the source. The net average power delivered to the load is therefore equal to the incident power minus the reflected power.
\[ P_{\mathrm{av}} = P_{\mathrm{incident}} - P_{\mathrm{reflected}} \]
For a real characteristic impedance \(Z_0\), the average power carried by the incident wave is:
\[ P_{\mathrm{incident}} = \frac{1}{2} \frac{|V_0^+|^2}{Z_0} \]
Similarly, the reflected-wave power is:
\[ P_{\mathrm{reflected}} = \frac{1}{2} \frac{|V_0^-|^2}{Z_0} \]
Therefore:
\[ P_{\mathrm{av}} = \frac{1}{2} \frac{|V_0^+|^2}{Z_0} - \frac{1}{2} \frac{|V_0^-|^2}{Z_0} \]
Using:
\[ |V_0^-| = |\Gamma_L||V_0^+| \]
gives:
\[ P_{\mathrm{av}} = \frac{1}{2} \frac{|V_0^+|^2}{Z_0} (1-|\Gamma_L|^2) \]
This expression shows that the fraction of incident power delivered to the load is related directly to the reflection coefficient. For a perfectly matched load, \(\Gamma_L=0\), so all incident power is delivered to the load. As the magnitude of the reflection coefficient increases, a larger fraction of the incident power is reflected back toward the source.
Return Loss
Return loss is another important parameter used to quantify the amount of reflected power resulting from an impedance mismatch. It is defined in terms of the ratio of reflected power to incident power:
\[ RL = -10\log_{10} \left( \frac{P_{\mathrm{reflected}}} {P_{\mathrm{incident}}} \right) \]
Since:
\[ \frac{P_{\mathrm{reflected}}} {P_{\mathrm{incident}}} = |\Gamma_L|^2 \]
the return loss becomes:
\[ RL = -10\log_{10}|\Gamma_L|^2 \]
Using the logarithm property:
\[ \boxed{ RL = -20\log_{10}|\Gamma_L| } \]
A perfectly matched load has \(|\Gamma_L|=0\), corresponding to theoretically infinite return loss. A larger return-loss value therefore indicates a smaller reflected wave and generally a better impedance match.
Input Impedance of a Terminated Lossless Line
The input impedance is the impedance seen by the source when looking into the transmission line toward the load. Even when the load impedance is fixed, the input impedance changes with the electrical length of the transmission line because the incident and reflected waves undergo a phase change as they propagate.
Let the input point be a distance \(l\) from the load toward the source. Since the load is at \(z=0\), the input point is located at:
\[ z=-l \]
The input impedance is therefore:
\[ Z_{\mathrm{in}} = \frac{V(-l)}{I(-l)} \]
Voltage at the Input Point
Starting with the general voltage equation:
\[ V(z) = V_0^+e^{-j\beta z} + V_0^-e^{j\beta z} \]
At \(z=-l\):
\[ V(-l) = V_0^+e^{j\beta l} + V_0^-e^{-j\beta l} \]
Using \(V_0^-=\Gamma_LV_0^+\):
\[ V(-l) = V_0^+ \left( e^{j\beta l} + \Gamma_Le^{-j\beta l} \right) \]
Current at the Input Point
The current equation is:
\[ I(z) = \frac{V_0^+}{Z_0}e^{-j\beta z} - \frac{V_0^-}{Z_0}e^{j\beta z} \]
At \(z=-l\):
\[ I(-l) = \frac{V_0^+}{Z_0}e^{j\beta l} - \frac{V_0^-}{Z_0}e^{-j\beta l} \]
Substituting \(V_0^-=\Gamma_LV_0^+\):
\[ I(-l) = \frac{V_0^+}{Z_0} \left( e^{j\beta l} - \Gamma_Le^{-j\beta l} \right) \]
Derivation of the Input Impedance
By definition:
\[ Z_{\mathrm{in}} = \frac{V(-l)}{I(-l)} \]
Substituting the voltage and current expressions:
\[ Z_{\mathrm{in}} = Z_0 \frac{ e^{j\beta l} + \Gamma_Le^{-j\beta l} }{ e^{j\beta l} - \Gamma_Le^{-j\beta l} } \]
Dividing the numerator and denominator by \(e^{j\beta l}\) gives:
\[ Z_{\mathrm{in}} = Z_0 \frac{ 1+\Gamma_Le^{-j2\beta l} }{ 1-\Gamma_Le^{-j2\beta l} } \]
This form clearly shows that the input impedance depends on both the load reflection coefficient and the electrical length of the line.
Substituting the load reflection coefficient:
\[ \Gamma_L = \frac{Z_L-Z_0}{Z_L+Z_0} \]
and simplifying the exponential terms using trigonometric identities gives the standard input-impedance expression:
\[ \boxed{ Z_{\mathrm{in}} = Z_0 \frac{ Z_L+jZ_0\tan(\beta l) }{ Z_0+jZ_L\tan(\beta l) } } \]
This is the fundamental input impedance equation for a terminated lossless transmission line. It shows that the impedance observed at the input is not necessarily equal to the physical load impedance. Instead, the transmission line transforms the load impedance according to its characteristic impedance and electrical length.
Special Cases of Input Impedance
The input-impedance equation becomes particularly useful for special electrical lengths. These cases explain why transmission lines can be used for impedance transformation and matching.
Matched Load
When the load is matched to the characteristic impedance:
\[ Z_L=Z_0 \]
the reflection coefficient becomes zero:
\[ \Gamma_L=0 \]
Therefore:
\[ Z_{\mathrm{in}}=Z_0 \]
Thus, a matched lossless transmission line appears as its characteristic impedance regardless of its physical length.
Quarter-Wavelength Line
For a quarter-wavelength line:
\[ l=\frac{\lambda}{4} \]
Since:
\[ \beta l = \frac{2\pi}{\lambda} \frac{\lambda}{4} = \frac{\pi}{2} \]
the tangent term becomes very large. The input impedance reduces to the well-known impedance-transforming relationship:
\[ \boxed{ Z_{\mathrm{in}} = \frac{Z_0^2}{Z_L} } \]
This property makes a quarter-wavelength transmission line particularly useful for impedance transformation and matching applications.
Half-Wavelength Line
For a half-wavelength line:
\[ l=\frac{\lambda}{2} \]
then:
\[ \beta l=\pi \]
and:
\[ \tan(\beta l)=0 \]
Therefore:
\[ \boxed{ Z_{\mathrm{in}}=Z_L } \]
Thus, a lossless transmission line having an electrical length of one-half wavelength repeats the load impedance at its input.