Premed · Premed · Physics 2
Lecture 13: AC Circuits
Physics II — Electromagnetism, Optics & Modern Physics
Learning Objectives
By the end of this lecture, students will be able to:
- Describe alternating current and voltage and define RMS values
- Analyze the behavior of resistors, capacitors, and inductors in AC circuits
- Define impedance, reactance, and phase angle for RLC series circuits
- Explain resonance in RLC circuits and calculate the resonant frequency
- Calculate average power dissipation and the power factor in AC circuits
Lecture Content
I. Alternating Current Fundamentals
Alternating current (AC) is current that varies sinusoidally with time. The voltage and current can be written as v(t) = V_max sin(omega t) and i(t) = I_max sin(omega t + phi), where phi is the phase difference between current and voltage. The key parameters are the amplitude (peak value) V_max or I_max, the angular frequency omega = 2 pi f in rad/s, the frequency f in Hz (60 Hz in North America, 50 Hz in most of the world), and the period T = 1/f.
RMS (root-mean-square) values represent the effective DC equivalent of an AC quantity. They are V_rms = V_max / sqrt(2), which is approximately 0.707 V_max, and I_rms = I_max / sqrt(2). The standard household outlet provides V_rms = 120 V in North America, corresponding to a peak voltage of approximately 170 V. Voltmeters and ammeters display RMS values when measuring AC.
II. AC Circuit Elements
A resistor in AC has voltage and current that are in phase (phi = 0): V_R = I_max R sin(omega t). The resistance R is independent of frequency.
A capacitor in AC has current that leads voltage by 90 degrees (pi/2 radians). The mnemonic "ICE" helps remember this: I leads C (capacitor) which has E (voltage) lagging. The opposition to current flow is characterized by the capacitive reactance X_C = 1/(omega C) = 1/(2 pi f C), so that V_C,max = I_max X_C. Capacitive reactance decreases with increasing frequency, meaning a capacitor passes high-frequency signals easily. Reactance has units of ohms.
An inductor in AC has voltage that leads current by 90 degrees (pi/2 radians). The mnemonic "ELI" captures this: E (voltage) leads I in an L (inductor). The inductive reactance is X_L = omega L = 2 pi f L, so that V_L,max = I_max X_L. Inductive reactance increases with increasing frequency, meaning an inductor blocks high-frequency signals. Reactance again has units of ohms.
<image>Three panels showing voltage and current waveforms for each AC circuit element. Panel A (Resistor): Voltage (solid line) and current (dashed line) oscillate perfectly in phase, with v_R and i_R reaching their peaks simultaneously. Panel B (Capacitor): Current (dashed) leads voltage (solid) by 90 degrees — current peaks one quarter cycle before voltage. The capacitive reactance X_C = 1/(omega C) is noted. Panel C (Inductor): Voltage (solid) leads current (dashed) by 90 degrees — voltage peaks one quarter cycle before current. The inductive reactance X_L = omega L is noted. Each panel labels the phase relationship clearly.</image>
III. Series RLC Circuit
A series RLC circuit contains a resistor R, inductor L, and capacitor C driven by an AC source. The current is the same through all elements (since they are in series), but the voltages differ in phase: V_R is in phase with I, V_L leads I by 90 degrees, and V_C lags I by 90 degrees.
Because the voltage phasors point in different directions, the total voltage is the phasor sum (not the algebraic sum) of the individual voltages: V_max = I_max sqrt(R^2 + (X_L - X_C)^2). The impedance Z is the AC equivalent of resistance, defined as Z = sqrt(R^2 + (X_L - X_C)^2), so that Ohm's law for AC becomes V_max = I_max Z. Impedance has units of ohms.
The phase angle phi between the total voltage and the current is given by tan(phi) = (X_L - X_C) / R. When phi > 0, the voltage leads the current and the circuit is inductive. When phi < 0, the current leads the voltage and the circuit is capacitive. When phi = 0, the circuit is at resonance.
IV. Phasor Diagrams
A phasor is a rotating vector that represents an AC quantity, with its length equal to the amplitude and its angle representing the phase. The projection of the phasor onto the vertical (or horizontal) axis gives the instantaneous value.
For a series RLC circuit, the phasor diagram places the current phasor I_max along the reference (horizontal) direction. The resistor voltage V_R lies along I (in phase). The inductor voltage V_L points 90 degrees ahead (upward), and the capacitor voltage V_C points 90 degrees behind (downward). The total voltage phasor is the vector sum, with a horizontal component equal to V_R and a vertical component equal to V_L - V_C. The angle between the total voltage phasor and the current phasor is the phase angle phi. The corresponding impedance triangle has hypotenuse Z, horizontal side R, and vertical side (X_L - X_C).
<image>A phasor diagram for a series RLC circuit. The horizontal axis represents the reference direction (current phasor I). V_R is drawn horizontally along I. V_L is drawn vertically upward (90 degrees ahead of I). V_C is drawn vertically downward (90 degrees behind I). The net reactive voltage (V_L - V_C) is shown as a vertical component. The total voltage phasor V_max is drawn as the diagonal of the rectangle formed by V_R and (V_L - V_C), making angle phi with the horizontal. The impedance triangle is shown alongside: hypotenuse Z, horizontal side R, vertical side (X_L - X_C), with angle phi.</image>
V. Resonance in RLC Circuits
Resonance occurs when the inductive and capacitive reactances are equal (X_L = X_C), so that omega_0 L = 1/(omega_0 C). Solving for the resonant frequency gives omega_0 = 1/sqrt(LC), or equivalently f_0 = 1/(2 pi sqrt(LC)).
At resonance, the impedance reaches its minimum value of Z = R, and the current reaches its maximum value of I = V_max / R. The phase angle is zero, meaning voltage and current are in phase. Remarkably, the individual voltages across the inductor and capacitor can each be much larger than the source voltage, a phenomenon known as voltage amplification.
The quality factor Q measures the sharpness of the resonance peak: Q = omega_0 L / R = 1/(omega_0 CR) = (1/R) sqrt(L/C). A high Q corresponds to a narrow, sharp resonance peak, while a low Q gives a broad peak. Resonant RLC circuits are used in radio tuning circuits to select a specific broadcast frequency from among many, in bandpass filters, and in MRI systems where the resonant frequency of hydrogen nuclei determines the operating frequency.
VI. Power in AC Circuits
The instantaneous power in an AC circuit is p(t) = v(t) i(t). The average power dissipated is P_avg = (1/2) V_max I_max cos(phi) = V_rms I_rms cos(phi), where cos(phi) is the power factor.
Only the resistor dissipates power on average. Capacitors and inductors alternately store and release energy, so their average power consumption is zero. The average power can therefore also be written as P_avg = I_rms^2 R. The apparent power is S = V_rms I_rms, measured in volt-amperes (VA), and the reactive power is Q_reactive = V_rms I_rms sin(phi), measured in VAR.
Power factor correction is important in industrial settings. Many industrial loads, such as motors, are inductive, giving a power factor less than 1 (phi > 0). Adding capacitors in parallel brings phi closer to zero, improving the power factor. This reduces the current needed for the same power delivery, thereby lowering I^2R losses in the transmission lines. At resonance, the power factor equals 1, and maximum power transfer occurs.
