Example: Formation of the coupling matrix (M matrix) for a simple filter with no cross-couplings
The goal of this example is to give you an introduction and understanding of how the M matrix is related to the circuit schematic and response of a filter by formulating in detail the M matrix for a simple filter with no cross-couplings. Many general features of the M matrix concept can be discussed within the framework of this simple example.
Consider the simple 2nd order low pass prototype filter shown in Figure 1. It consists of a pair of 1 henry (H) inductors connected in series through a K inverter (with a value K12 as labeled in the schematic). The filter response depends on the choice of K12 as well as the input and output impedance. If you dont know what a K inverter is or how it works, then please refer to Appendix A located at the end of this tutorial for a light introduction.
Figure 1 Simple 2nd order low pass prototype filter
We now connect a voltage source generating Vg volts and an internal impedance of Rin = 1 Ohm to the input side of filter and a load with impedance Rout = 1 Ohm to the output side of the filter as shown in Figure 2. The generator and load are each connected in series to the filter through K inverters with values Ks1 and K2L, respectively. The currents circulating through the series elements are labeled: is, i1, i2, and iL.
Figure 2 The 2nd order low pass prototype filter network including input/output couplings, generator, and load.
A convenient equation to find the transmission coefficient S21 for any two port network connected between a voltage source generating Vg volts with an internal impedance of Rin and a load impedance of Rout is [1]:
where it is assumed that the S-parameter reference impedances at port 1 and port 2 are equal to the generator and load impedances (Rin and Rout), respectively. In our filter shown in Figure 2 the impedance values are Rin = Rout = 1 Ohm. By Ohms law, the output voltage Vout across the load is equal to Vout = iL×(1 Ω), which is numerically equal to iL. With this, the transmission coefficient S21 for the filter is calculated to be:
For now we are mainly interested in the magnitude of the S-parameters, |S21| and |S11|. Because of this, all we need is |S21|, since we can find |S11| from |S21| by using the following convenient relation for a lossless network: |S21|2 + |S11|2 = 1.
To evaluate S21 using Eqn. (1), we must find the current across the load iL. To find iL we write down Kirchhoffs loop law for each of four closed circuit loops shown in Figure 2:
The system of four linear equations given by Eqns. (2a) (2d) can be rewritten in matrix form:
In principal, all four unknown current variables (is, i1, i2, and iL) can be obtained by inverting the matrix on the left hand side of Eqn. (3):
The current variable iL found using Eqn. (3b) can the be substituted into Eqn. (1) to find S21.
Let us now see how the M matrix is defined. Without much effort, Eqn. (3a) can be rearranged into the following matrix form (see Ref. [2] for a general treatment):
Where x denotes a column vector which contains the current variables:
b denotes a column vector which contains the source:
R denotes the termination impedance matrix:
Z denotes the impedance matrix taking into account the 1 H inductors:
and M denotes the M-matrix (also called the coupling matrix):
Note that the M matrix in Eqn. (4e) is symmetrical (M = MT). This is true in general since K inverters are reciprocal.
The M matrix is a very special matrix that has received a great deal of attention in the modern literature on microwave filters. This may seem strange at first since the M matrix is just one component of the full matrix equation given by Eqn. (4). So what makes the M matrix so special? There are actually many reasons.
For one, the M matrix in a sense characterizes the entire low pass prototype filter itself. If you are given only the M matrix specified in Eqn. (4e) (and nothing else) you have all the information necessary to reconstruct the filters circuit schematic shown in Figure 2 for this particular example. Of course, this requires you assume a priori that all inductors are 1 Henry and both source and generator impedances are 1 Ohm. Incidentally, this seems to be a common assumption in the literature. In other words, unless otherwise stated, it is safe to assume that the M matrix characterizes a lowpass prototype with a 1 Ohm source impedance, a 1 Ohm load impedance, and 1 H series inductors connected between K inverters with values of K corresponding to the elements of the M matrix. A notable caveat to this statement is that the circuit shown in Figure 2 has a dual circuit consisting of shunt 1 F capacitors connected together through corresponding admittance inverters, which overall yields the same S11 and S21 frequency response. Some people like working with the dual circuit. We wont do that here: Im a fan of inductors.
Another miraculous feature of the M matrix is that it can be scaled to account for different (i.e. non-unity) load/generator impedances and inductor values while preserving the same S11 and S21 frequency response. Let us suppose that instead of using 1 Ohm resistors and 1 H inductors, we want to use a generator with a source impedance of Rin, a load with an impedance of Rout, and inductor values on the source and load ends to be L1 and L2, respectively. It turns out that by appropriately scaling the values of the K inverters (i.e. the elements of the M matrix) we can account for these new values while preserving the same S11 and S21 frequency response as the original circuit shown in Figure 2. Figure 3 shows the corresponding filter schematic containing the new element values (Rin, Rout, L1, L2) with the updated (scaled) values of K.
Figure 3 2nd order low pass filter network with K inverter values scaled to yield the same S11 and S21 frequency response as the original circuit shown in Figure 2.
The elements of the M matrix for the circuit shown in Figure 3 can be read directly off the schematic:
Note how the value of each K inverter is scaled by the square root of its adjoining element values. For example, take the K inverter at the source end of Figure 3 which couples the generator with impedance Rin to the inductor with inductance L1. The corresponding value of this scaled K inverter is:
where Ks1 is the old value. The next K inverter couples the two inductors together. The corresponding value of this scaled K inverter is:
where K12 is the old value. The same principal holds for the K inverter at the generator end. The termination impedance matrix R and impedance matrix Z of the circuit shown in Figure 3 can also be read directly off the schematic:
We will now utilize a linear transformation to prove that the circuit in Figure 3 yields the same S11 and S21 frequency response as the original circuit shown in Figure 2. However, be prepared: the proof is a bit long and mathematical. First, for convenience, let us rewrite again Eqn. (4), which governs the response of the original circuit:
We now introduce (by definition) a new column vector x that is related to the original column vector x via the following linear transform:
where x contains a set of different unknowns, which we shall denote by priming the original unknowns (the prime does not mean derivative here it is just a notation for a set of different variables):
and P is the transformation matrix with nonzero entries solely along the diagonal:
where P1, P2, P3, and P4 are all assumed to be real numbers. One has the freedom to choose a different transformation matrix P, but the particular form given in Eqn. (8b) is especially suited for our purposes here. Notably, by choosing such a simple form of P, we set up the problem so that the elements of the new variable x are linearly proportional to the corresponding elements of the original variable x:
Substituting Eqn. (8) for x into Eqn. (4) yields:
In principal, one can solve Eq. (10) for the new vector x and then reconstruct the original (i.e. physical) current elements of the original vector x by utilizing Eq. (9). However, this doesnt really buy us anything. Let us go one step further into the mathematical void and multiply both sides of Eqn. (10) by P (trust me, this will all be worth it in the end):
We can expand the left hand side of Eqn. (11):
The matrix-vector product Pb on the right hand side of Eq. (12) can be simplified nicely:
Substituting Eq. (13) into Eq. (12) yields:
Dividing both sides of Eqn. (14) by P1 yields:
For simplification, let us absorb P1 into the column vector of unknowns:
where x contains yet another set of unknowns defined by the relation:
Substituting Eqn. (17) into Eqn. (8) yields the following relationship between the elements of the new variable x and the elements of the original variable x:
Now let us group together the three sets of matrix products that appear on the left hand side of Eqn. (16) as new matrices each denoted with a double prime:
where R denotes an equivalent termination impedance matrix given by:
Z denotes an equivalent impedance matrix given by:
and M denotes an equivalent impedance matrix given by:
Eqn. (19) is exactly the current loop matrix equation describing the response of the particular circuit shown in Figure 4 below (compare Eqn. (19) with Eqn. (4)). If you dont believe me, write down Kirchhoffs loop law for each of four closed circuit loops shown in Figure 4 and recast these equations in matrix form (like what we did going from Eqn. (2) to Eqn. (3)). You will recover Eqn. (19).
Figure 4 2nd order low pass filter network with the same S11 and S21 frequency response as the original circuit shown in Figure 2.
The transmission coefficient S21 of the circuit shown in Figure 4 is:
where we used the relation
from Eq. (18). Note that S21 given by Eqn. (20) for the circuit shown in Figure 4 is equal to S21 for the original circuit shown in Figure 2 given by Eqn. (1). Hence both circuits have the same S11 and S21 frequency response. Furthermore, we see that the filter shown in Figure 3 is equivalent to the filter shown in Figure 4 if we simply rename the variables P1, P2, P3, and P4 as follows:
Substituting Eqns. (21a) (21d) into the schematic shown in Figure 4 leads to the schematic shown in Figure 3.
Appendix A: An Introduction to the K inverter
The ideal impedance (K) inverter is a linear reciprocal two port circuit element completely characterized by its impedance K. Figure A.1 shows the conventional circuit symbol of an impedance inverter. This circuit symbol is basically a two-terminal box labeled with the value K.
Figure A.1 The conventional circuit diagram of an impedance (K) inverter
By definition, the ideal impedance inverter links or couples the voltage across one set of terminals to the current flowing through the other set by the following very simple linear relations:
where V1 and I1 represent the voltage and current at port 1, and V2 and I2 represent the voltage and current at port 2. Here we use a sign convention specifying that a positive I1 flows into port 1, while a positive I2 flows out of port 2 as illustrated in Figure 1. Note that the unit of K is in ohms [Ω]. Ideally, the impedance K does not change with frequency, temperature, or anything else, i.e., an ideal impedance inverter would work perfectly from DC to gamma ray frequencies through a nuclear winter. Many simple circuits like mutually coupled inductors behave as impedance inverters over wide bandwidths. For more details see the chapters on filters in Refs [3] and [4].
The impedance inverter gets its name by the fact it provides an input impedance that is the inverse of a load impedance. Figure A.2 shows a complex load with impedance ZL connected in tandem to port 2 of the impedance inverter. In this situation, the input impedance Zin looking into port 1 is given by:
where we utilized the relations given in Eqs. (A.1) and (A.2) and the fact that V2/I2=ZL.
Figure A.2 An impedance inverter doing what it does best inverting an impedance.
And thats about all you need to know about K inverters for this tutorial!
References
[1] G. Gonzalez, Microwave
Transistor Amplifiers: Analysis and Design 2nd Ed.,
[2] R. J. Cameron, C. M. Kudsia, and R. R. Mansour, Microwave
Filters for Communication Systems: Fundamentals, Design, and Applications,
[3] D.M. Pozar,
Microwave Engineering 3rd Ed. ,
[4] R.E. Collin, Foundations
for Microwave Engineering 2nd Ed. ,