15.2 Simple AC Circuits

AC circuits are the backbone of our electrical grid, powering homes and businesses worldwide. They use alternating current, which changes direction periodically, unlike the steady flow of direct current. This constant change creates unique behaviors in circuit components.

Phasor diagrams and complex numbers help visualize and calculate these AC behaviors. Resistors, capacitors, and inductors each respond differently to AC, with varying phase relationships between voltage and current. Understanding these relationships is key to analyzing and designing efficient AC systems.

AC Circuit Components and Behavior

Phasor diagrams for AC circuits

• Represent sinusoidal AC quantities as vectors rotating counterclockwise at angular frequency $\omega$ (radians per second)
  • Voltage and current phasors represented by their peak values and phase angles (degrees or radians)
  • Phasor diagrams provide a visual representation of the phase relationships between voltage and current in AC circuits ($R$, $L$, $C$)
• Resistors in AC circuits exhibit voltage and current phasors in phase (phase angle difference is 0°)
  • Phasor representation: $V_R = I_R R$ where $V_R$ is the voltage across the resistor, $I_R$ is the current through the resistor, and $R$ is the resistance
  • Example: In a purely resistive circuit with a 100 Ω resistor and a 10 V peak voltage, the current phasor will be in phase with the voltage phasor and have a peak value of 0.1 A
• Capacitors in AC circuits have current leading voltage by 90° (voltage lags current by 90°)
  • Phasor representation: $I_C = j \omega C V_C$ or $V_C = -j \frac{1}{\omega C} I_C$ where $I_C$ is the current through the capacitor, $V_C$ is the voltage across the capacitor, $C$ is the capacitance, and $\omega$ is the angular frequency
  • Example: In a purely capacitive circuit with a 10 μF capacitor and a 5 V peak voltage at 1 kHz, the current phasor will lead the voltage phasor by 90° and have a peak value of approximately 0.31 mA
• Inductors in AC circuits have voltage leading current by 90° (current lags voltage by 90°)
  • Phasor representation: $V_L = j \omega L I_L$ or $I_L = -j \frac{1}{\omega L} V_L$ where $V_L$ is the voltage across the inductor, $I_L$ is the current through the inductor, $L$ is the inductance, and $\omega$ is the angular frequency
  • Example: In a purely inductive circuit with a 100 mH inductor and a 2 A peak current at 50 Hz, the voltage phasor will lead the current phasor by 90° and have a peak value of approximately 62.8 V
• Complex numbers are used to represent these phasor relationships mathematically

Reactance in simple AC circuits

• Reactance is the opposition to the flow of alternating current in a circuit caused by capacitors ($X_C$) and inductors ($X_L$)
  • Capacitive reactance: $X_C = \frac{1}{\omega C} = \frac{1}{2 \pi f C}$ where $X_C$ is the capacitive reactance, $\omega$ is the angular frequency, $f$ is the frequency in Hz, and $C$ is the capacitance
  • Inductive reactance: $X_L = \omega L = 2 \pi f L$ where $X_L$ is the inductive reactance, $\omega$ is the angular frequency, $f$ is the frequency in Hz, and $L$ is the inductance
• Impedance ($Z$) represents the total opposition to current flow in an AC circuit, consisting of resistance ($R$) and reactance ($X_L - X_C$)
  • $Z = \sqrt{R^2 + (X_L - X_C)^2}$ where $Z$ is the impedance, $R$ is the resistance, $X_L$ is the inductive reactance, and $X_C$ is the capacitive reactance
  • Phase angle between voltage and current: $\phi = \tan^{-1}(\frac{X_L - X_C}{R})$ where $\phi$ is the phase angle in radians
• Phase relationships between voltage and current depend on the circuit components
  • Purely resistive circuit: voltage and current are in phase (phase angle difference is 0°)
  • Purely capacitive circuit: current leads voltage by 90°
  • Purely inductive circuit: voltage leads current by 90°
  • Example: In an AC circuit with a 50 Ω resistor, a 100 μF capacitor, and a 10 mH inductor at 1 kHz, the impedance would be approximately 57.1 Ω, and the phase angle would be approximately -26.6°

Components in AC circuits

• Resistors resist the flow of current equally for both AC and DC and do not cause any phase shift between voltage and current
  • Dissipate energy as heat ($P = I^2R$)
  • Example: A 100 W light bulb acts as a resistor in an AC circuit, converting electrical energy into heat and light
• Capacitors store energy in an electric field and oppose changes in voltage by allowing current to lead voltage by 90°
  • Act as an open circuit at low frequencies and a short circuit at high frequencies
  • Example: A 10 μF capacitor in a 60 Hz AC circuit will have a reactance of approximately 265 kΩ, effectively acting as an open circuit
• Inductors store energy in a magnetic field and oppose changes in current by allowing voltage to lead current by 90°
  • Act as a short circuit at low frequencies and an open circuit at high frequencies
  • Example: A 100 mH inductor in a 10 kHz AC circuit will have a reactance of approximately 6.28 kΩ, effectively acting as an open circuit

AC Circuit Analysis and Characteristics

• Alternating current (AC) is characterized by its periodic change in direction and magnitude
• Power factor is the ratio of real power to apparent power in an AC circuit, indicating the efficiency of power transfer
• Resonance occurs in an AC circuit when inductive and capacitive reactances are equal, resulting in maximum power transfer
• RMS (root mean square) value represents the equivalent DC value that would produce the same heating effect as an AC signal
• Kirchhoff's laws are applied to analyze complex AC circuits, considering both magnitude and phase of voltages and currents