Transfer Function Solving Process w/ Example

Base KVL / KCL Equations:

KVL Equations:

1. Vin - Vc1 - Ir1*R1 - Ir2*R2 = 0
2. -Vc2 + Ir2*R2 = 0

KCL Equations:

1. -Ir1 + Ic1 = 0
2. -Ir2 + Ic1 - Ic2 = 0

Additional Equations:

1. Ic1 = s*C1*Vc1
2. Ic2 = s*C2*Vc2

MATLAB Code for Computing Reduced Equation Set:

syms Ir1 Ir2 Ic1 Ic2 Vc1 Vc2 Vin R1 R2 C1 C2 s

equs = [Vin - Vc1 - Ir1*R1 - Ir2*R2 == 0;
    -Vc2 + Ir2*R2 == 0;
    -Ir1 + Ic1 == 0;
    -Ir2 + Ic1 - Ic2 == 0];

vars = [Vin Vc1 Vc2 Ir1 Ir2 Ic1 Ic2];

[A, b] = equationsToMatrix(equs, vars);

equ1 = A(1,:)*vars.' == b(1,:);      %equ1 = Vin - Vc1 - Ir1*R1 - Ir2*R2 == 0;
equ2 = A(2,:)*vars.' == b(2,:);      %equ2 = - Vc2 + Ir2*R2 == 0;
equ3 = A(3,:)*vars.' == b(3,:);      %equ3 = - Ir1 + Ic1 == 0;
equ4 = A(4,:)*vars.' == b(4,:);      %equ4 = - Ir2 + Ic1 - Ic2 == 0;

equ5 = Ic1 == s*C1*Vc1;
equ6 = Ic2 == s*C2*Vc2;

MATLAB Code for Computing the Transfer Function:

sol = solve([equ1,equ2,equ3,equ4,equ5,equ6], [Ir1,Ir2,Ic1,Ic2,Vc1,Vc2])

Hout = simplify(sol.Vc2/Vin)

syms R C

Hout_Homo = subs(Hout, {R1,R2,C1,C2}, {R,R,C,C}) %Assuming R1=R2 and C1=C2.

Final Transfer function:

$H(s)=\frac{C_1{R}_{2}s}{C_1{C}_2{R}_1{R}_2{s}^2+C_1{R}_{1}s+C_1{R}_{2}s+C_2{R}_{2}s+1}$

Final Homogeneous Transfer Function:

$H_h{}_o{}_m{}_o(s)=\frac{CRs}{C^2{R}^2{s}^2+3CRs+1}$