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I've been given the following circuit and been asked to give the differential equation showing the relationship between \$u_e(t)\$ and \$u_a(t)\$

circuit diagram

The differential equation should be in the form $$\scriptstyle u_{a}^{\prime}(t) + k_1\cdot u_{a}(t)= k_2\cdot u_{e}(t) + k_3\cdot u_{e}^{\prime}(t) + k_4\cdot u_{e}^{\prime\prime}(t)$$

Some of the coefficients can be zero.

I can see that I'm going to have to integrate the inductor term in order to get an expression for the current, but I'm getting really confused about where I need to start and how to proceed.

I tried applying Kirschoff's laws as far as I could but I don't see how to fit them into each other to get a meaningful result.

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2 Answers 2

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Hints:

Voltage at left voltage divider (between R1 and R2) is just input voltage scaled down by a constant factor \$\frac{R_2}{R_1+R_2}\$.

Voltage at right terminal (between R3 and R4) is R4 times current through inductor \$I_L\$. Get current through inductor by setting up an appropriate diffenrential equation using \$U_L=-L \frac{dI_L}{dt}\$ and KVL and Ohm's law.

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Hints:

  • It's better to solve the circuit by transforming the circuit to s domain
  • Apply Kirchoff's Voltage Law to three loops and you will get 3 equation for 3 loop current.
  • get expression for current in loop containing inductor.
  • Apply inverse Laplace transform to get needed differential equation
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