Maximal effect in an AC circuit

David331 · Oct 11, 2021

Hello again, I have a solution for the question, but as it is my first time solving these types of exercises I am uncertain if I my solution is correct. The first picture represents the exercise with the circuit and the second is the just some rewrites of the circuit, in swedish. Picture 3 and 4 is my solution, in english. Any help is much appreciated.

Swedish AC circuit exercise: current source i0(t) = 10sin(1000t) mA feeding node A-B, with R1 = 1.8k in parallel with C1 = 1.75u and L1 = 2.5H; ideal transformer N1/N2 = 10 couples to R2 = 4.5 ohm in series with C3 = 0.25m, voltage u(t) across R2-C3.

Handwritten circuit derivation: an ideal transformer is replaced by an impedance transformation, with Z4 = (N1/N2)^2 (Z22 + Z23) and the equivalent circuit diagram labelled A, B, I0, Z1, Z21, Z22, Z23 and Z4.

Handwritten derivation of a Thevenin equivalent impedance for an AC circuit, with a small circuit schematic showing a source, two impedances and terminals A and B, followed by equations for maximum power transfer where Z2 equals the conjugate of Z1.

Handwritten working on lined paper: Req = (Re(1/(jωC1) - 1/(Z1+Z2) - 1/Z4))^-1 = (√2/200 - 0 - 450/(450²+400²))^-1 ≈ 171.5 Ω, then C2 = Im(...)/ω ≈ 6.486 mF.

alan123hk · Oct 11, 2021

This question requires long calculations to get the answer. My calculation result is R1=170 Ohm, C2=6.5uF.
I don’t know if I did something wrong that caused the calculation result to be inaccurate, but according to my calculation result is similar to yours, so I believe your calculation process is basically no problem.

David331 · Oct 12, 2021

Thanks for the answer! So, you got exact values of R1 and C2? Maybe there is just some rounding errors in prior computations? I will doublecheck my numbers once more.

alan123hk · Oct 12, 2021

This is probably due to rounding errors caused by my calculations being too sloppy, because in the calculation process, I used a very limited number of significant figures many times to write down the results displayed by the handheld calculator.

The Electrician · Oct 12, 2021

David331, here's what I got:

Mathematica input and output: parameter definitions for R1, R2, Zi, C1, C2, C3, L1 and N1N2, followed by a FindMinimum call minimising Abs[Conjugate[Zi] - 1/(...)] over C2 and R1, with the printed result.

Maximal effect in an AC circuit

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