Phase Coordinates Method: An Example of Calculating a Transfer Matrix

Calculation of transfer matrices of multipoles of various shapes is quite simple. Transfer matrices are a mathematical description of the circuit under consideration, which defines the relationship between the input quantities and the output values of the multipole.

Consider an example of calculating the transfer matrix for the circuit in Fig. 1.

Fig. 1. The considered scheme

To determine the transfer matrix for the scheme under consideration, equations are compiled according to the laws of Kirchhoff. Preliminary designate the currents and set arbitrarily the direction of currents and voltages (Fig. 2).

Fig. 2. Setting the directions of currents and voltages

For the circuit, equations are compiled according to the first and second laws of Kirchhoff:

These equations are easily solved using the Matlab software package. Below is a script for calculating a system of equations and extracting values at the input U₁ and I₁ depending on the output values for the given example.

>> syms I1 I2 U1 U2 Z1 Z2 Z3
>> I3 = I2-I1;
>> res = solve(U1==I1*Z1+I3*Z3,-I3*Z3+I2*Z2+U2==0,U1,I1);
>> collect(res.U1,U2)
 
ans =
 
(-(Z1 - Z3)/Z3)*U2 + (I2*Z1*Z3 - I2*Z1*Z2 + I2*Z2*Z3)/Z3
 
>> collect(res.I1,U2)
 
ans =
 (-1/Z3)*U2 - (I2*Z2 - I2*Z3)/Z3

We present the obtained equations in the form of a matrix equation connecting the input current I₁ and input voltage U₁ and output parameters of the multipole:

.

The resulting coupling matrix of the input and output quantities is called the direct transfer matrix of the multipole A. In the general case, the equation of a multipole form A is written as

.

Form A equations are applied if multipoles are connected in cascade (Fig. 3).

Fig. 3. Cascade connection of multipoles

In this case, the equivalent direct transfer matrix of the entire circuit will be determined by the expression

A = A₁ ∙ A₂ ∙ … ∙ A_N.

In addition to form A equations, form Y equations are widely used. These equations are applied in the case of parallel connection of multipoles (Fig. 4).

Fig. 4. Parallel connection of multipoles

For such a scheme, the equivalent form Y matrix will be determined by the formula

Y = Y₁ + Y₂ + … + Y_N.

The form matrix A and the form matrix Y are interconnected. The attached files recount these matrices from one form to another in the Matlab language:

• Form A → form Y: A_to_Y.m
• Form Y → form A: Y_to_A.m