Welcome back. In this web lecture, we're going to talk about microwave networks. In applications, you often find that you have many more ports and only one or two ports that we considered up to now. For example, in phased arrays, which are used in 5G wireless communication in base stations, we often have systems which can even have 100s or 1000s of ports, input-output ports. We need to come up with a concept to describe such more complex microwave systems. In this lecture, the objective is to introduce such a K-port microwave network. We're going to talk about impedance matrix and scattering matrix. We're going to introduce the input and output reflection coefficient in particular for a 2-port network but it can be generalized to any size of the network and we're going to discuss an example. Let us first look at a general K-port microwave network. We have K-ports and each port is characterized by a port voltage V_k and a port of current I_k and all these ports, they interact with each other. A way to describe that is well-known from circuit theory, is to use the impedance matrix. Let us take a 2-port as an example. But we can generalize that to the K-port case of course. Now the relation between the port voltages and port currents is given by this matrix relation where the first element in the impedance matrix, set 11 can be found by determining the voltage and the current in port 1 when port 2 is open circuited, so there's no current flowing there. The other elements of the impedance matrix can be found in the similar way and in that way we can describe the properties of this network. Similarly, it's also possible to make a relation between the port currents and port voltages, and that's called the admittance matrix, also well-known from circuit theory. In this case, we do not leave the ports open to determine the elements of the matrix, but we short-circuit them and we get similar relations as we have seen for the impedance matrix as shown here. Now, the issue is at microwave frequencies is that it's not so easy or almost not possible even to measure the voltage and the currents directly at port, so that's not possible, what is possible is to measure the complex amplitude. The amplitude and phase of the incident and reflected waves going into ports using a vector network analyser, also called VNA. Now an example of a relative low cost vector network analyser is shown here, this is the FieldFox which we use also in our classes, which can measure a 2-port network and can measure the incident and reflected waves fairly accurately. Therefore, we need to describe a 2-port network in terms of the incident and reflected waves. For historical reasons, it's very common to use new symbols, in this case, a and b. A and b are described, let's say the complex amplitude of the incident and the reflected waves at each of the ports. They are related to the complex voltage amplitudes that we have introduced in the previous lectures in this way. A is the normalized voltage amplitude of the incident wave. Normalized by square root of Z_0 and Z_0 is the reference for frequency. It can also be expressed in the current amplitudes. For B similarly, it's the voltage amplitude of the reflected wave normalized to the square root of Z_0. Now for port 2, we can do the same. Z_0 was the reference impedance often equal to 50-ohm. For port 2, we can describe it in a similar way. We have a_2 and b_2 in this case and note that a_2 is again directed towards the input of port 2 and b_2 is directed outward the port. Now the scattering matrix describes the relation between these quantities, between a_1, b_1 and a_2, b_2 and it is given in this form. The reflected quantities b_1 and b_2 are related to a_1 and a_2 by the scattering matrix. For example, the first element of this scattering matrix S_11 can be determined by this equation. It's b_1 reflected wave divided by the incident wave in case that we terminate port 2 with a matched load. That means that a_2, there's no reflection coming back into the two-port network. Often we use a 50-ohm system, so then we terminate with 50-ohm. The other elements of the scattering matrix can be found in a similar way. Now let's take an example. Let's look at an example and try to apply the scattering matrix. A very simple example is, of course, we just have a two-port network, which is a lumped element 50-ohm resistor. Whereat the input, we have a_1 reflected wave b_1 and at port 2 we have a_2 going into the network and b_2. Now let's try to determine the scattering matrix for this 50-ohm lumped element. We will assume that the reference impedance is again 50-ohm. For this particular case, we will first start to look into the S_11. We attach a 50-ohm termination to port 2 and the circuit that we get then is just simply a series connection of two 50-ohm loads. Which means that the input impedance looking into this circuit is just simply 100-ohm, which means that S_11 is b_1 the reflected wave divided by the incident wave can be simply calculated by using the reflection coefficient equation that we derived in one of the previous web lectures. It is an equal to the input impedance, 100-ohm minus the reference impedance is 50-ohm. We find that S_11 is equal to one-third. S_21 can be determined by again terminating port 2 with a 50-ohm load and by adding a source impedance 50-ohm to port 1. The source voltage is V_s and we can now determine the S_21 by first determining b_2. What is going into the 50-ohm termination divided by a_1 and a_1 is, let's say the amplitude going into the two-port network. Now b_2, since we have a match termination, it can be found from V_2 plus, because V_2 minus is zero. We assume that normalized and a_1 can be found in a way from V_1 and the current going into the network by using the equations that we have derived for the voltage and the current, and we can rewrite it in this form. We can rewrite a_1 in terms of the total voltage and the total current. This is a nice exercise to do yourself. It's quite straightforward. Now, this equation then simplifies into the equation we see here, where we can also note that VS, the source voltage by using Kirchhoff's voltage law, can be simply found from V_1 plus Z_0, which is 50 ohm, times the current, which gives us this equation. It's 2 times V_2 plus, where V_2 plus is equal to V_2, as we've already seen. V_2 can simply be found by looking at the circuit and observing that it is a voltage divider. At the end, we see that S_21 is equal to two-thirds. Now, because of the symmetry of the network, the other elements can be found in a similar way. That gives us the total scattering matrix or this simple example as shown here, which might not be what you would have expected for such a simple 50 ohm resistor network. Now, this concept of the scattering matrix to describe the two port microwave network can be generalized to this configuration, where often we have a source at the input, could be a frequency oscillator or something like that, a source with the source impedance, and at the output we have a load which could be an antenna, for example. Now, let's look at the input reflection coefficient of this particular system. The input reflection coefficient can be found by first looking into the relation between b_1 and b_2 and the a_1 and a_2, which is the scattering matrix. Next to that, at the termination, we have a relation between a_2 and b_2 because that termination defines this reflection coefficient at the load. That's simply equal to Z_L minus Z_0 divided by Z_L plus Z_0, as we've seen in the previous web lectures. We have three equations and because of that we can define a relation between b_1 and a_1, which is in fact the input reflection coefficient. We work these three equations, we find that Gamma in, the input reflection coefficient, is then equal to S_11 plus a term which is determined by the other parameters of the scattering matrix and Gamma L. Only in the case that Gamma L is zero, in that case, the input reflection coefficient looking into this network is equal to S_11. Well, in a similar way, we can define the output reflection coefficient when we look from port two into this network, and then we find the similar equation. Also in this case, only when Gamma S is zero, in only that case, we see that Gamma out is equal to S_22. Now, next to the reflection coefficients think of a power amplifier. We also can define the forward gain, which is the absolute value of S_21 squared. Similarly, we can also define the reverse gain in the other direction. Now, to summarize this web lecture, we've introduced microwave networks, the general case, we've looked in particular into the example of two-port networks, we defined impedance admittance matrices, then, defined the scattering matrix, which is more common to use at microwave frequencies, we've shown you a very simple example of a 50 ohm resistor, we defined the input and output reflection coefficient and we looked into the forward and reverse gain. Thank you for watching.
