# How to find load impedance when input impedance and reflection coefficient over range of frequencies is known

I know that there's a relationship between input and load impedances and reflection coefficient and it's given by following expression:

However, the problem is that this cannot be used when we only know that Zsource = 50ohms and when it comes to reflection coefficient, we know it's graphic representation over a range of frequencies. Let's say it looks something like this:

Now, it is possible to determine load impedance when we choose value of reflection coefficient for one particular point on this graph (meaning one particular frequency), however, considering the fact that load impedance can be complex value, i cannot find a way to determine unique value for load impedance this way. I need to create impedance matching circuit for load that produces reflection coefficient shown in picture for source with 50ohms resistance. Any help is appreciated!

• Would you mind putting that into a log-frequency chart? Since reflection (regular and diffuse) is related to transmission (regular and diffuse) and absorption, such that $1=a+r+t$, if may help to trigger imaginations to also see the $1-\Gamma$ version, as well.
– jonk
Jan 30, 2021 at 22:11
• @jonk Thanks for your comment, the thing is, this diagram is obtained by measuring reflection coefficient of an antenna and then it was recreated in matlab simply by choosing particular points on diagram and inserting them into corresponding vector. My point is that i don't have an mathematical expression that gives me reflection coefficient as a function of frequency. Therefore, im not quite sure how should i go about creating this diagram in log scale. Technically, i could use semilogx instead of plot in matlab, but that just makes no sense to me. Jan 31, 2021 at 17:19

No, there is no way to determine two numbers, ( magnitude, phase) , from one data point.

However, perhaps you know more : if you know the physical nature of the load, you can model it as a circuit. You can find values of the circuit that give you the same reflection coefficient magnitude as your data. Then you can calculate the phase of the reflection coefficient ( or load impedance) of your circuit model.

Of course, there is not only one circuit that leads to a particular magnitude response. However, lots of times you can see that, for example, this trace is acting as an inductor, this structure is creating some capacitive coupling, and so on. You likely should start with a ladder type structure, as this topology is familiar to most engineers. I would start with a lossless ladder terminated in a single resistor. Then you find the values of the inductors, capacitors, and resistor that match your magnitude data, and from that you can find the phase.

Off the top of my head, you have some kind of band pass network at 68Mhz terminated in either 17 ohms or 150 ohms. Start from there and go to it with some nice computer program.

• Thanks for your answer, i see what you're trying to say, and i like the idea. However, i am not quite sure how can i tell whether some trace, or some part of trace is acting like an inductor or a capacitor. Jan 31, 2021 at 17:22
• To some extent, this comes from experience. Wires with current flowing in them are like inductors. Big conductors that have no current in them, but have voltage on them, are like capacitors. Wires close to ground are like capacitors or transmission lines. Currents flowing through things that create voltages are inductive. Voltages across things that make currents flow are capacitive. Jan 31, 2021 at 17:57