Transmission Line Matching - Generator and Load Mismatches
Self-guided learning
This example of transmission line matching contains a short description of the theory, detailed information on how to construct the model and the fully constructed model ready for you to download.
When working through the example, you may notice some small differences in your model compared to ours – this is usually simply due to the fact that you are using a different software version.
The Physics
A terminated transmission line is a good way to illustrate the concept of wave reflection which is a fundamental property of distributed systems.
As seen from the literature in [1] when a transmission line is terminated with an arbitrary load impedance Z_{L }it results in a reflection of the incident wave at the load that is characterized by the voltage reflection coefficient, Γ:
The superposition of the incident and reflected waves gives rise to standing waves, which can be prevented when Γ = 0 or when the load is matched to the line. When a line is mismatched not all of the available power from the generator is delivered to the load.
The above theory assumes that the generator was matched to the line resulting in zero reflection at the generator. However, both the generator and load may present mismatched impedances to the transmission line causing multiple reflections to occur. The best condition for maximum power transfer may involve standing waves on the line.
The literature in [1] provides a general model for a transmission line network with mismatched load and generator. The equations representing the input and load reflection coefficients, impedances and power delivered to the load are given in [1]. For this analysis, we assume the complex generator impedance Z_{g}, is fixed and consider three different cases of varying load impedance.
A basic schematic is shown in Figure 1.
The Model
Fig. 2 shows an ideal transmission line model in CST Studio Suite. The source and load impedances are modeled as external ports to an ideal electrical transmission line.
The ideal lossless transmission line (TL) block is designed for a characteristic impedance of 50 Ω at 3 GHz and an electrical length of 0.5λ. The following parameters are fixed values:
Parameter | Value | Description |
V_{g} | 8 volts | Source voltage |
Z_{g} | 20 + j5 Ω | Fixed generator impedance |
Z_{0} | 50 Ω | Line impedance of the ideal TL block |
Definition Frequency | 3 GHz | Frequency of operation |
Electrical Length | 0.5 | Length of transmission line |
Probes are placed at the input and output of the TL block to allow for determination of the input impedance of the line and power delivered to the load and are labeled as ‘In’ and ‘Out’ respectively. The following tasks are to be set up for analysis:
AC Analysis – to obtain the voltages and currents at the ports and probe points.
S-Parameter – to obtain the standing wave ratio and reflection coefficients on the line.
Template Based Post Processing – to calculate the input impedance to the line and obtain the power delivered to the load by specifying the formula:
Discussion of Results
Case 1: Load Matched to Line
In the case where we have Z_{l} = Z_{0} we expect the load reflection coefficient, Γ_{L} = 0 and the SWR = 1 from the relation:
which is seen in the S-parameters results.
The input impedance from the relation
reduces to Z_{in} = 50 Ω = Z_{0}. This is observed from the input probe voltage and current values from which a calculation of the complex impedance is done using a post-processing task.
Where is calculated for an electrical length of 0.5 λ at the operating frequency of 3 GHz.
The power delivered to the load given by the following relation and observed to be accurate in the CST Studio Suite simulation:
Case 2: Generator Matched to Loaded Line
In the case where Zin = Zg it is necessary is modify the line or load impedance to achieve this condition. The new load impedance that satisfies the condition of the generator matched to load is obtained from the relation,
which reduces to Z_{l} = 20 + j5 Ω. The load reflection coefficient, Γ_{L} may not be zero since this arrangement may lead to standing waves.
We confirm the complex input impedance satisfies the condition, Z_{in} = Z_{g} and obtain the power delivered to the load from the relation:
Case 3: Conjugate Matching
In the previous cases, the power delivered to the load was calculated from the absorbed power of the system which was less than the available power (P ≤ P_{avi}) which is the maximum deliverable power. In order to maximize P, we arrive at a condition known as conjugate matching which results in maximum power transfer to the load for a fixed generator impedance value.
Z_{in}=Z^{*}_{g}
Note that this value does not tell us that maximum power transfer that given input impedance Z_{in}, maximum power transfer occurs by setting the generator impedance as Z_{g} = Z^{*}_{in} but rather, when given a fixed source impedance Z_{g}, maximum power transfer occurs when Z_{in} = Z^{*}_{g}.
By setting the input impedance at Zin = 20 – j5 Ω, we obtain the required value for load impedance from the relation
Although this represents a case for a mismatched transmission line, the idea is that the reflections may still add in phase to deliver more power to the load. This is given by the relation:
Which indicates that the maximum available power in the system is extracted and transferred to the load.
Additional Tasks
- Does the conjugate matching condition provide the system with the best efficiency? Given the relation for P_{avl} it is seen that the efficiency of a system can be improved only by making R_{g} as small as possible and then applying conjugate matching for the given generator impedance Z_{g}. Try setting Z_{g} = 1 + j5 Ω and observe the power delivered.
Answer: The power efficiency is over 99% for this condition!
- Try adding an attenuation factor to the line resulting in a terminated lossy transmission line and observe its effect on the power transfer.
- Does the characteristic impedance of the line affect the power transfer?
References
[1] D.M. Pozar, Microwave Engineering, 4th Edition, John Wiley & Sons, pp. 77 - 79