⚡️College Physics III – Thermodynamics, Electricity, and Magnetism Unit 15 – AC Circuits

Key Concepts and Definitions

• Alternating current (AC) involves the flow of electric charge that periodically reverses direction
• AC voltage sources produce sinusoidal waveforms with amplitude, frequency, and phase as key parameters
• Root mean square (RMS) values represent the equivalent DC value that would produce the same average power ($V_{RMS} = \frac{V_{peak}}{\sqrt{2}}$, $I_{RMS} = \frac{I_{peak}}{\sqrt{2}}$)
• Impedance ($Z$) represents the total opposition to current flow in an AC circuit, consisting of resistance, capacitive reactance, and inductive reactance
  • Measured in ohms ($\Omega$)
  • Calculated using complex numbers: $Z = R + jX$, where $R$ is resistance and $X$ is reactance
• Phasors are complex numbers that represent the amplitude and phase of sinusoidal waveforms
  • Used to simplify AC circuit analysis by converting time-domain sinusoids into frequency-domain vectors
• Power factor is the ratio of real power to apparent power in an AC circuit ($PF = \frac{P}{S}$)
  • Ranges from 0 to 1, with 1 indicating a purely resistive load and 0 indicating a purely reactive load
• Resonance occurs when the inductive and capacitive reactances are equal, resulting in a purely resistive circuit and maximum power transfer

Alternating Current Basics

• AC is generated by rotating machines called alternators or AC generators
  • Mechanical energy is converted into electrical energy through electromagnetic induction
• AC waveforms are characterized by their peak amplitude, frequency, and phase
  • Peak amplitude is the maximum voltage or current reached during each cycle
  • Frequency is the number of cycles per second, measured in hertz (Hz)
  • Phase refers to the relative timing between voltage and current waveforms
• The period ($T$) of an AC waveform is the time required for one complete cycle ($T = \frac{1}{f}$)
• AC allows for efficient long-distance power transmission due to the ability to step up and step down voltages using transformers
• Most household electrical systems use AC with a frequency of 50 Hz or 60 Hz
• AC circuits can include resistors, capacitors, and inductors, each affecting the current and voltage waveforms differently
• Kirchhoff's laws (KVL and KCL) still apply to AC circuits, but the calculations involve complex numbers due to the presence of reactance

Circuit Components in AC

• Resistors in AC circuits behave the same as in DC circuits, following Ohm's law ($V = IR$)
  • Voltage and current are in phase for purely resistive elements
• Capacitors in AC circuits introduce capacitive reactance ($X_C$), which opposes changes in voltage
  • $X_C = \frac{1}{2\pi fC}$, where $f$ is the frequency and $C$ is the capacitance
  • Voltage lags current by 90° in a purely capacitive circuit
• Inductors in AC circuits introduce inductive reactance ($X_L$), which opposes changes in current
  • $X_L = 2\pi fL$, where $f$ is the frequency and $L$ is the inductance
  • Current lags voltage by 90° in a purely inductive circuit
• The total impedance of a series RLC circuit is calculated using the formula: $Z = \sqrt{R^2 + (X_L - X_C)^2}$
• In parallel RLC circuits, the total impedance is found using the reciprocal formula: $\frac{1}{Z} = \frac{1}{Z_1} + \frac{1}{Z_2} + ... + \frac{1}{Z_n}$
• Transformers are used to step up or step down AC voltages and currents
  • Consist of two coupled coils (primary and secondary) wound around a common magnetic core
  • Voltage ratio depends on the turns ratio: $\frac{V_p}{V_s} = \frac{N_p}{N_s}$, where $N$ is the number of turns in each coil

AC Circuit Analysis

• Phasor diagrams represent the magnitude and phase relationships between voltage and current in AC circuits
  • Resistive elements have voltage and current in phase
  • Capacitive elements have voltage lagging current by 90°
  • Inductive elements have current lagging voltage by 90°
• Impedance triangle relates resistance, reactance, and impedance in a right-angled triangle ($Z^2 = R^2 + X^2$)
• Series RLC circuits can be analyzed using Kirchhoff's voltage law (KVL) and phasors
  • The sum of the voltage phasors equals the source voltage phasor
  • $\vec{V}_s = \vec{V}_R + \vec{V}_L + \vec{V}_C$
• Parallel RLC circuits can be analyzed using Kirchhoff's current law (KCL) and phasors
  • The sum of the current phasors equals the source current phasor
  • $\vec{I}_s = \vec{I}_R + \vec{I}_L + \vec{I}_C$
• Nodal analysis and mesh analysis techniques can be applied to AC circuits by using complex numbers to represent impedances and phasors
• Superposition theorem, Thevenin's theorem, and Norton's theorem can be used to simplify complex AC circuits

Power in AC Circuits

• Real power ($P$) is the average power consumed by resistive elements, measured in watts (W)
  • $P = V_{RMS}I_{RMS}\cos{\phi}$, where $\phi$ is the phase angle between voltage and current
• Reactive power ($Q$) is the power absorbed and released by reactive elements (capacitors and inductors), measured in volt-ampere reactive (VAR)
  • $Q = V_{RMS}I_{RMS}\sin{\phi}$
• Apparent power ($S$) is the total power in an AC circuit, measured in volt-amperes (VA)
  • $S = V_{RMS}I_{RMS} = \sqrt{P^2 + Q^2}$
• Power factor ($PF$) is the ratio of real power to apparent power ($PF = \frac{P}{S} = \cos{\phi}$)
  • Unity power factor (1) indicates a purely resistive load, while a power factor of 0 indicates a purely reactive load
• Power triangle relates real power, reactive power, and apparent power in a right-angled triangle ($S^2 = P^2 + Q^2$)
• Power factor correction involves adding capacitors or inductors to improve the power factor of a load
  • Reduces the amount of reactive power drawn from the source and minimizes power losses in transmission lines

Resonance and Filters

• Resonance occurs when the inductive and capacitive reactances are equal ($X_L = X_C$)
  • At resonance, the impedance is purely resistive, and the current reaches its maximum value
  • Resonant frequency: $f_r = \frac{1}{2\pi\sqrt{LC}}$
• Series resonance results in a minimum impedance and maximum current at the resonant frequency
  • Commonly used in voltage amplification and selectivity applications
• Parallel resonance results in a maximum impedance and minimum current at the resonant frequency
  • Commonly used in current suppression and filtering applications
• Filters are circuits designed to pass or block specific frequency ranges
  • Low-pass filters allow low frequencies to pass while attenuating high frequencies
  • High-pass filters allow high frequencies to pass while attenuating low frequencies
  • Band-pass filters allow a specific range of frequencies to pass while attenuating frequencies outside that range
  • Band-stop (notch) filters attenuate a specific range of frequencies while allowing frequencies outside that range to pass
• Quality factor ($Q$) is a measure of the sharpness of the resonance peak
  • Higher $Q$ values indicate a narrower bandwidth and more selective filtering
  • $Q = \frac{f_r}{\Delta f}$, where $\Delta f$ is the bandwidth (the frequency range where the power is at least half of its peak value)

Applications and Real-World Examples

• Power generation and distribution systems rely on AC for efficient long-distance transmission
  • Transformers are used to step up voltages for transmission and step down voltages for distribution and end-user consumption
• Household electrical systems use AC power (120V/60Hz in North America, 220-240V/50Hz in many other countries)
  • Outlets, appliances, and lighting systems are designed to work with AC power
• Electric motors and generators often use AC due to its ability to create rotating magnetic fields
  • Induction motors, synchronous motors, and alternators are common examples
• Electronic devices, such as power supplies and battery chargers, often include AC-to-DC conversion stages
  • Rectifiers, filters, and voltage regulators are used to convert AC to the required DC voltages
• Audio and radio frequency (RF) circuits extensively use AC principles
  • Filters, amplifiers, and oscillators are designed using RLC components and resonance principles
• Wireless power transfer systems, such as those used in smartphones and electric vehicle charging, rely on resonant coupling between coils
  • The transmitter and receiver coils are tuned to the same resonant frequency to maximize power transfer efficiency

Problem-Solving Strategies

• Identify the type of AC circuit (series, parallel, or combination) and the components involved (resistors, capacitors, inductors, and sources)
• Convert time-domain sinusoidal quantities to phasors using the amplitude and phase information
• Calculate the impedance of each component using the appropriate formulas ($R$, $X_L = 2\pi fL$, $X_C = \frac{1}{2\pi fC}$)
• Apply Kirchhoff's laws (KVL and KCL) using phasors to set up circuit equations
  • For series circuits, use KVL: the sum of the voltage phasors equals the source voltage phasor
  • For parallel circuits, use KCL: the sum of the current phasors equals the source current phasor
• Solve the circuit equations using complex arithmetic to find the unknown voltages, currents, or impedances
• If required, use circuit theorems (superposition, Thevenin's, or Norton's) to simplify the circuit and solve for the desired quantities
• Calculate power (real, reactive, and apparent) using the appropriate formulas ($P = V_{RMS}I_{RMS}\cos{\phi}$, $Q = V_{RMS}I_{RMS}\sin{\phi}$, $S = V_{RMS}I_{RMS}$)
• For resonance problems, determine the resonant frequency using $f_r = \frac{1}{2\pi\sqrt{LC}}$ and analyze the circuit behavior at resonance
• Verify the results by checking the consistency of the solutions and ensuring that the power calculations balance (power generated = power consumed)