Chapter 31 — Alternating Current

13 points · 3 extra credit

Chapter 31 — Alternating Current

Core: 13 points · Extra credit: 3 points. Show your reasoning; partial credit is given.

1. 1. (1 pt •) A household outlet supplies a sinusoidal voltage with amplitude $V_0=170\,\text{V}$ at $f=60\,\text{Hz}$.

   (a) Find the rms voltage.

   (b) A resistive heater draws a current of amplitude $I_0=2.5\,\text{A}$. Find the rms current.

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2. 2. (2 pts ••) A $0.25\,\text{H}$ inductor and, separately, an $8.0\,\mu\text{F}$ capacitor are each driven at $f=60\,\text{Hz}$.

   (a) Find the inductive reactance $X_L$ and the capacitive reactance $X_C$.

   (b) If each element alone is connected across a $120\,\text{V}$ (rms) source at this frequency, find the rms current through each.

   (c) Without recomputing, state how each reactance changes if the frequency is doubled.

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3. 3. (2 pts ••) In a series circuit driven at a single frequency, the voltage amplitudes measured across the resistor, inductor, and capacitor are $V_R=45\,\text{V}$, $V_L=80\,\text{V}$, and $V_C=40\,\text{V}$.

   (a) Using a phasor diagram, find the amplitude of the source voltage.

   (b) Find the phase angle of the source voltage relative to the current, and state whether the current leads or lags the source.

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4. 4. (3 pts •••) A series R–L–C circuit has $R=200\,\Omega$, $L=0.40\,\text{H}$, and $C=5.0\,\mu\text{F}$, driven by a source of rms voltage $120\,\text{V}$ at $f=60\,\text{Hz}$.

   (a) Find $X_L$, $X_C$, and the impedance $Z$.

   (b) Find the rms current.

   (c) Find the phase angle $\phi$, and state whether the circuit is inductive or capacitive.

   (d) Find the average power delivered by the source, and check it against $I_\text{rms}^2 R$.

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5. 5. (2 pts ••) For the same components as Problem 4 ($L=0.40\,\text{H}$, $C=5.0\,\mu\text{F}$, $R=200\,\Omega$, source rms voltage $120\,\text{V}$):

   (a) Find the resonance angular frequency $\omega_0$ and the corresponding frequency $f_0$.

   (b) Find the rms current and the average power delivered at resonance, and explain why the power is larger than in Problem 4.

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6. 6. (2 pts ••) An AC motor operating from a $240\,\text{V}$ (rms) line draws an rms current of $5.0\,\text{A}$ with a power factor of $0.80$ (current lagging).

   (a) Find the average power consumed by the motor.

   (b) Find the phase angle between the line voltage and the current.

   (c) The apparent power is $V_\text{rms}I_\text{rms}$. Find it, and explain in one sentence why it exceeds the average power.

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7. 7. (1 pt •) An ideal transformer has a primary of $800$ turns and a secondary of $100$ turns, with $120\,\text{V}$ (rms) applied to the primary.

   (a) Find the secondary voltage.

   (b) The secondary drives a $3.0\,\Omega$ resistive load. Find the secondary current and the primary current.

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8. Extra Credit (3 pts •••)

   A series R–L–C circuit is driven by a source $v(t)=V_0\cos\omega t$, and carries current $i(t)=I_0\cos(\omega t-\phi)$, where $\phi$ is the phase by which the current lags the voltage.

   (a) Starting from the instantaneous power $p(t)=v(t)\,i(t)$, integrate over one full period $T=2\pi/\omega$ to derive the average power

   $$P_\text{av}=\tfrac{1}{2}V_0 I_0\cos\phi = V_\text{rms}I_\text{rms}\cos\phi.$$

   (b) Write $Z(\omega)=\sqrt{R^2+\left(\omega L-\dfrac{1}{\omega C}\right)^2}$. By setting $dZ/d\omega=0$, prove that the impedance is a minimum (and therefore the current amplitude a maximum) exactly at $\omega_0=1/\sqrt{LC}$.

   (c) Using part (a), show that the average power dissipated is maximized at $\omega_0$, and give its value there in terms of $V_0$ and $R$.

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