Transfer function in a 2L2C circuit

Question

I have to find the transfer function of the following circuit (AC, sinusoidal), using complex numbers, Kirchoff point rule with potentials or Millman's theorem.

The transfer function is \$\underline{H}=\frac{\underline{i_1}}{\underline{u}}\$ .

Here is what I did for now. Schematic using complex impedances:

^{simulate this circuit – Schematic created using CircuitLab}

Then we have : \$-(V_B-V_A)\frac{1}{Z_L+Z_C}-(V_B-V_A)\frac{1}{Z_C}+(V_C-V_A)\frac{1}{Z_L}+u(t)\times i_1=0\$ ^-- False, check in comments for the right one

But what to do now, how to find i1 and u? Thank you in advance.

Answer 1

One way of seeing the question is that it demands for net conductance of the circuit. Which is inverse of net impedance offered by the circuit u/i1. As i1 signifies net current drawn by ckt. The following explanation is based on that methodology. I assume you have to find transfer function of the=is ckt. One simple method might be.

• Replace Z by Laplace equivalents in circuit.
• Find impedence(net) of the circuit.
• Znet = (sL)+inv(sC + inv(sL + 1/sC)) which is u/i1
• Take inverse and you have your answer.

Just to warn nodal might go slightly long. Second is just adaptation of a visible fact(not too much of ideal nodal analysis) it is use (u-Va)/zl = i1 . Then you find Va interms of Z and u replace it in this second eq. And then u should have ur answer. Hop this ans satisfies your query.