## Related questions with answers

Question

Suppose the circuit described by the equation below

$\frac{dI}{dt}+\frac{R}{L}I= \frac{1}{L}V(t)$

is driven by a sinusoidal voltage source $V(t)=V \sin \omega t$ (where $V$ and $\omega$ are constant).

(a) Show that

$I(t)=\frac{V}{R^2+L^2 \omega^2}(R \sin \omega t-L \omega \cos \omega t)+C e^{-(R / L) t}$

(b) Let $Z=\sqrt{R^2+L^2 \omega^2}$. Choose $\theta$ so that $Z \cos \theta=R$ and $Z \sin \theta=$ $L \omega$. Use the addition formula for the sine function to show that

$I(t)=\frac{V}{Z} \sin (\omega t-\theta)+C e^{-(R / L) t}$

This shows that the current in the circuit varies sinusoidally apart from a DC term (called the transient current in electronics) that decreases exponentially.

Solution

VerifiedStep 1

1 of 8We are given:

$V(t)=V\sin \omega t$

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