Ch32: Alternating-Current Circuits

Alternating-Current (AC) Sources

• Voltage Source Definition: An AC circuit consists of circuit elements and a power source that provides an alternating voltage $\Delta v$.
• Time-Dependent Voltage: The output voltage of an AC source is harmonic and expressed as:   $\Delta v = \Delta V_{max} \sin(\omega t)$
• Angular Frequency $\omega$: Defined by the frequency $f$ or the period $T$:   $\omega = 2\pi f = \frac{2\pi}{T}$   - where $\Delta V_{max}$ is the maximum output voltage (voltage amplitude).

Resistors in an AC Circuit

• Circuit Equation: For a purely resistive circuit, the applied voltage is equal to the voltage drop across the resistor:   $\Delta v + \Delta v_R = 0 \implies \Delta v - i_R R = 0$$\Delta v = i_R R$
• Instantaneous Current:   $i_R = \frac{\Delta V_{max}}{R} \sin(\omega t) = I_{max} \sin(\omega t)$
• Phase Relationship: In a purely resistive circuit, the current and the voltage are in phase. They reach their maximum, minimum, and zero values simultaneously.
• Ohm's Law for Maximum Values:   $\Delta V_{max} = I_{max} R$
• Phasor Diagram for Resistors:   - A phasor is a vector whose length represents the maximum value ($I_{max}$ or $\Delta V_{max}$) and rotates at angular frequency $\omega$.   - The current and voltage phasors lie in the same direction because they are in phase.   - The projection of the phasor onto the vertical axis represents the instantaneous value: $\Delta v_R = \Delta V_{max} \sin(\omega t)$.

rms Current and Voltage

• Definition of rms: The root-mean-square (rms) value is used to describe the effective values of alternating current and voltage.
• rms Current ($I_{rms}$):   $I_{rms} = \sqrt{(i^2)_{avg}} = \frac{I_{max}}{\sqrt{2}} \approx 0.707 I_{max}$
• rms Voltage ($\Delta V_{rms}$):   $\Delta V_{rms} = \frac{\Delta V_{max}}{\sqrt{2}} \approx 0.707 \Delta V_{max}$
• Average Power: The average power dissipated in a resistor is given by:   $P_{avg} = I_{rms}^2 R$
• Example 32.1: Calculating rms Current:   - Given: $\Delta v = 200 \sin(\omega t)$ and $R = 47.0 \, \Omega$.   - Determine $\Delta V_{max} = 200 \, V$.   - Calculate $\Delta V_{rms} = \frac{200}{\sqrt{2}} \, V$.   - $I_{rms} = \frac{\Delta V_{rms}}{R} = \frac{200}{\sqrt{2} \times 47.0} = 3.01 \, A$.

Inductors in an AC Circuit

• Voltage across Inductor: $\Delta v_L = L \frac{di}{dt}$.
• Kirchhoff's Loop Rule:   $\Delta v + \Delta v_L = 0 \implies \Delta v - L \frac{di}{dt} = 0$$\Delta V_{max} \sin(\omega t) = L \frac{di}{dt}$
• Current Equation: By integrating the voltage:   $di = \frac{\Delta V_{max}}{L} \sin(\omega t) dt$$i_L = \frac{\Delta V_{max}}{\omega L} \sin(\omega t - \pi/2) = -I_{max} \cos(\omega t)$
• Phase Relationship: The current lags the voltage by $\pi/2$ radians ($90^\circ$), or one-fourth of a cycle.
• Inductive Reactance ($X_L$):   $X_L \equiv \omega L$   - Reactance is measured in Ohms ($\Omega$).   - $\Delta V_{max} = I_{max} X_L$ and $\Delta V_{rms} = I_{rms} X_L$.
• Quick Quiz 32.2: In an AC circuit with an inductor and a lightbulb, the lightbulb glows brightest at low frequencies. This occurs because $X_L = 2\pi f L$; as frequency decreases, reactance decreases, allowing more current to flow.
• Example 32.2: Purely Inductive Circuit:   - Given: $L = 25.0 \, mH$, $\Delta V_{rms} = 150 \, V$, $f = 60.0 \, Hz$.   - $X_L = 2\pi \times 60.0 \times 25.0 \times 10^{-3} = 9.42 \, \Omega$.   - $I_{rms} = \frac{150}{9.42} = 15.9 \, A$.   - If $f$ increases to $6.00 \, kHz$:     - $X_L = 2\pi \times 6.0 \times 10^{3} \times 25.0 \times 10^{-3} = 942 \, \Omega$.     - $I_{rms} = \frac{150}{942} = 0.159 \, A$.

Capacitors in an AC Circuit

• Charge and Voltage: $q = C \Delta V_{max} \sin(\omega t)$.
• Current Equation: Taking the derivative of charge:   $i_C = \frac{dq}{dt} = \omega C \Delta V_{max} \cos(\omega t)$$i_C = I_{max} \sin(\omega t + \pi/2)$
• Phase Relationship: The current leads the voltage by $\pi/2$ radians ($90^\circ$), or one-fourth of a cycle.
• Capacitive Reactance ($X_C$):   $X_C \equiv \frac{1}{\omega C}$   - Reactance is measured in Ohms ($\Omega$).   - $\Delta V_{max} = I_{max} X_C$ and $\Delta V_{rms} = I_{rms} X_C$.
• Quick Quiz 32.3 & 32.4: In an AC circuit with a capacitor and a lightbulb, the lightbulb glows brightest at high frequencies. This is because $X_C = \frac{1}{2\pi f C}$; as frequency increases, reactance decreases, leading to higher current.
• Example 32.3: Purely Capacitive Circuit:   - Given: $C = 8.00 \, \mu F$, $f = 60.0 \, Hz$, $\Delta V_{rms} = 150 \, V$.   - $X_C = \frac{1}{2\pi \times 60.0 \times 8.00 \times 10^{-6}} = 332 \, \Omega$.   - $I_{rms} = \frac{150}{332} = 0.452 \, A$.

The RLC Series Circuit

• Configuration: A resistor ($R$), inductor ($L$), and capacitor ($C$) connected in series with an AC source $\Delta v = \Delta V_{max} \sin(\omega t)$.
• Current: The instantaneous current is the same for all elements:   $i = I_{max} \sin(\omega t - \phi)$   - where $\phi$ is the phase angle.
• Instantaneous Voltages:   - $\Delta v_R = I_{max} R \sin(\omega t)$   - $\Delta v_L = I_{max} X_L \sin(\omega t + \pi/2)$   - $\Delta v_C = I_{max} X_C \sin(\omega t - \pi/2)$
• Impedance ($Z$): The total resistance to current in an AC circuit:   $Z \equiv \sqrt{R^2 + (X_L - X_C)^2}$   - General Ohm's Law relationship: $\Delta V_{max} = I_{max} Z$.
• Phase Angle ($\phi$):   $\tan(\phi) = \frac{X_L - X_C}{R}$   - If $X_L > X_C$: $\phi$ is positive (circuit is more inductive, current lags voltage).   - If $X_L < X_C$: $\phi$ is negative (circuit is more capacitive, current leads voltage).   - If $X_L = X_C$: $\phi = 0$ (circuit is in resonance).

Analysis of a Series RLC Circuit (Example 32.4)

• Parameters: $R = 425 \, \Omega$, $L = 1.25 \, H$, $C = 3.50 \, \mu F$, $f = 60.0 \, Hz$, $\Delta V_{max} = 150 \, V$.
• Reactances and Impedance:   - $\omega = 2\pi f = 377 \, s^{-1}$   - $X_L = \omega L = 377 \times 1.25 = 471 \, \Omega$   - $X_C = \frac{1}{\omega C} = \frac{1}{377 \times 3.50 \times 10^{-6}} = 758 \, \Omega$   - $Z = \sqrt{425^2 + (471 - 758)^2} = 513 \, \Omega$
• Maximum Current:   - $I_{max} = \frac{\Delta V_{max}}{Z} = \frac{150}{513} = 0.293 \, A$
• Phase Angle:   - $\phi = \arctan\left(\frac{471 - 758}{425}\right) = -34.0^\circ$ (Current leads voltage).
• Maximum Voltages:   - $\Delta V_R = I_{max} R = 0.293 \times 425 = 124 \, V$   - $\Delta V_L = I_{max} X_L = 0.293 \times 471 = 138 \, V$   - $\Delta V_C = I_{max} X_C = 0.293 \times 758 = 222 \, V$
• Note on Voltage Addition: The sum of maximum voltages ($124 + 138 + 222 = 484 \, V$) is greater than the source maximum ($150 \, V$). This sum is not physically meaningful because the individual voltages are not in phase.
• Engineering Modification: To change the phase angle to $\phi = -30.0^\circ$, a new inductance $L$ is calculated using:   $L = \frac{1}{\omega} [R \tan(\phi) + X_C] = 1.36 \, H$.

Power in an AC Circuit

• Instantaneous Power: $p = i \Delta v = I_{max} \Delta V_{max} \sin(\omega t) \sin(\omega t - \phi)$.
• Average Power ($P_{avg}$):   $P_{avg} = \frac{1}{2} I_{max} \Delta V_{max} \cos(\phi) = I_{rms} \Delta V_{rms} \cos(\phi)$
• Power Factor: The term $\cos(\phi)$ is called the power factor.   - For a purely resistive load, $\phi = 0$ and $\cos(\phi) = 1$.   - In terms of resistance and impedance: $\cos(\phi) = \frac{R}{Z}$.
• Alternate Power Formula:   $P_{avg} = I_{rms} \Delta V_{rms} \left(\frac{R}{Z}\right) = I_{rms} \left(\frac{\Delta V_{rms}}{Z}\right) R = I_{rms}^2 R$
• Example 32.5: For the RLC circuit in Example 32.4:   - $\Delta V_{rms} = \frac{150}{\sqrt{2}} = 106 \, V$   - $I_{rms} = \frac{0.293}{\sqrt{2}} = 0.207 \, A$   - $P_{avg} = I_{rms} \Delta V_{rms} \cos(\phi) = (0.207)(106) \cos(-34.0^\circ) = 18.2 \, W$.

Resonance in a Series RLC Circuit

• Resonance Condition: Resonance occurs when the current is at its maximum value, which happens when the impedance is minimized ($Z = R$).   - This requires $X_L = X_C$.
• Resonant Frequency (\omega_0):   $\omega_0 L = \frac{1}{\omega_0 C} \implies \omega_0 = \frac{1}{\sqrt{LC}}$
• Power at Resonance: Average power is maximized at resonance because the impedance is at its minimum value $R$:   $P_{avg} = \frac{(\Delta V_{rms})^2 R}{R^2 + (X_L - X_C)^2}$
• Example 32.6: Resonating Series RLC Circuit:   - Given: $R = 150 \, \Omega$, $L = 20.0 \, mH$, $\Delta V_{rms} = 20.0 \, V$, $\omega = 5000 \, s^{-1}$.   - Find $C$ for maximum current (resonance):   - $C = \frac{1}{\omega^2 L} = \frac{1}{(5000)^2 \times 20.0 \times 10^{-3}} = 2.00 \times 10^{-6} \, F = 2.00 \, \mu F$.