Transfer function in a 2L2C circuit

circuit analysisimpedancekirchhoffs-laws

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.

enter image description here

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

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

schematic

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.

Best Answer

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.

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