RLC (1-20-2020) Part 2

Electricity and Magnetism Level pending

A DC voltage source excites an RLC network as shown. At time $t = 0$ , the inductors and capacitor are de-energized.

Let $I_{\infty}$ be the value of the current flowing out of the source as time $t$ approaches infinity. Let $I_M$ be the value of the first local maximum of the source current.

Enter your answer as the product of $I_{\infty}$ and $I_M$ .

Details and Assumptions:
1) $V_S = 10$
2) $R = 0.1$
3) $L_1 = 0.25$
4) $L_2 = C = 1$
5) Both current values are positive numbers

The answer is 1792.7.

1 solution

Steven Chase
Jan 21, 2020

The state variables are the inductor currents and the capacitor voltage $(I_{L1}, I_{L2}, V_{C})$ . Write the time-derivatives of the state variables in terms of the state variables and the forcing function. Let the voltage on the right side of $L_1$ be $V_P$ (across the two parallel branches). In the diagram, the plus and minus signs show the voltage polarity in relation to the current flow.

$V_{L1} = L_1 \dot{I}_{L1} \\ V_{L2} = L_2 \dot{I}_{L2} \\ I_C = C \dot{V}_C$

Expanding (with two intermediate quantities to begin with):

$I_C = I_{L1} - I_{L2} \\ V_P = V_C + R I_C \\ V_{S} - V_P = L_1 \dot{I}_{L1} \\ V_{P} - R I_{L2} = L_2 \dot{I}_{L2} \\ I_C = C \dot{V}_C$

All that remains is to isolate the derivatives and numerically integrate. The current plot is below. As expected, the source current begins at zero and asymptotically approaches $100$ . The first local current maximum has a value of about $18$ .

@Steven Chase sir please share your Numerical integraion code

A Former Brilliant Member - 11 months, 3 weeks ago

RLC (1-20-2020) Part 2

The answer is 1792.7.

1 solution

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