⚡Electrical Circuits and Systems I Unit 10 – AC Power Analysis

Key Concepts and Definitions

• AC (Alternating Current) involves the flow of electric charge that periodically reverses direction
• Voltage and current in AC circuits vary sinusoidally with time
• Frequency ($f$) represents the number of cycles per second and is measured in Hertz (Hz)
• Period ($T$) is the time required for one complete cycle and is the reciprocal of frequency ($T = 1/f$)
  • For example, if the frequency is 60 Hz, the period is $1/60$ seconds or approximately 16.67 milliseconds
• Angular frequency ($\omega$) is the rate of change of the phase angle and is related to frequency by $\omega = 2\pi f$
• Phasors are complex numbers that represent the amplitude and phase of sinusoidal waveforms
• Impedance ($Z$) is the total opposition to current flow in an AC circuit and consists of resistance, inductance, and capacitance

AC Waveforms and Characteristics

• AC waveforms are typically sinusoidal and can be described by their amplitude, frequency, and phase
• Peak amplitude ($V_p$ or $I_p$) is the maximum value of the waveform
• Peak-to-peak amplitude ($V_{pp}$ or $I_{pp}$) is the difference between the maximum and minimum values of the waveform
• RMS (Root Mean Square) value is the equivalent DC value that would produce the same heating effect and is equal to the peak value divided by $\sqrt{2}$ for sinusoidal waveforms
  • $V_{rms} = V_p / \sqrt{2}$ and $I_{rms} = I_p / \sqrt{2}$
• Phase difference ($\phi$) is the angular difference between two waveforms and is measured in degrees or radians
• Leading and lagging waveforms occur when one waveform reaches its peak before or after the other
  • For example, in an inductive circuit, the current lags the voltage by 90°

Phasors and Complex Numbers

• Phasors simplify AC circuit analysis by representing sinusoidal waveforms as complex numbers
• The magnitude of a phasor represents the RMS value of the waveform
• The angle of a phasor represents the phase shift relative to a reference
• Complex numbers consist of a real part and an imaginary part ($a + jb$)
  • $j$ is the imaginary unit, defined as $j^2 = -1$
• Polar form of a complex number is $A\angle\theta$, where $A$ is the magnitude and $\theta$ is the angle
• Rectangular form of a complex number is $a + jb$, where $a$ is the real part and $b$ is the imaginary part
• Conversion between polar and rectangular forms:
  • $a = A\cos\theta$ and $b = A\sin\theta$
  • $A = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}(b/a)$

RLC Circuit Analysis

• RLC circuits contain resistors (R), inductors (L), and capacitors (C)
• Impedance of resistors ($Z_R$) is equal to the resistance ($R$)
• Impedance of inductors ($Z_L$) is $j\omega L$, where $L$ is the inductance
• Impedance of capacitors ($Z_C$) is $1/(j\omega C)$, where $C$ is the capacitance
• Total impedance in series RLC circuits is the sum of individual impedances: $Z_{total} = Z_R + Z_L + Z_C$
• Total impedance in parallel RLC circuits is the reciprocal of the sum of reciprocals: $1/Z_{total} = 1/Z_R + 1/Z_L + 1/Z_C$
• Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL) apply to AC circuits using phasors

Power in AC Circuits

• Instantaneous power ($p(t)$) is the product of instantaneous voltage and current: $p(t) = v(t) \times i(t)$
• Active power ($P$) is the average power consumed by the circuit and is measured in watts (W)
  • $P = V_{rms} I_{rms} \cos\phi$, where $\phi$ is the phase difference between voltage and current
• Reactive power ($Q$) is the power exchanged between the source and reactive components (inductors and capacitors) and is measured in volt-ampere reactive (VAR)
  • $Q = V_{rms} I_{rms} \sin\phi$
• Apparent power ($S$) is the total power supplied to the circuit and is measured in volt-ampere (VA)
  • $S = V_{rms} I_{rms}$ and $S^2 = P^2 + Q^2$
• Power factor ($PF$) is the ratio of active power to apparent power and represents the efficiency of the circuit
  • $PF = P / S = \cos\phi$
• In purely resistive circuits, the power factor is 1, and all the power is active power
• In purely reactive circuits (inductors or capacitors), the power factor is 0, and all the power is reactive power

Resonance and Filters

• Resonance occurs when the inductive and capacitive reactances are equal in magnitude, causing the impedance to be purely resistive
• Series resonance occurs when the total impedance is at a minimum and the current is at a maximum
  • At series resonance, $X_L = X_C$ and $f_r = 1 / (2\pi\sqrt{LC})$
• Parallel resonance occurs when the total impedance is at a maximum and the current is at a minimum
  • At parallel resonance, $X_L = X_C$ and $f_r = 1 / (2\pi\sqrt{LC})$
• Quality factor ($Q$) represents the sharpness of the resonance and is the ratio of the resonant frequency to the bandwidth
  • $Q = f_r / BW$, where $BW$ is the bandwidth (the frequency range where the power is at least half the maximum value)
• Filters are circuits that allow certain frequencies to pass while attenuating others
• Low-pass filters allow low frequencies to pass and attenuate high frequencies
• High-pass filters allow high frequencies to pass and attenuate low frequencies
• Band-pass filters allow a specific range of frequencies to pass and attenuate frequencies outside that range
• Band-stop filters (notch filters) attenuate a specific range of frequencies and allow frequencies outside that range to pass

Applications and Real-World Examples

• Power systems use AC because it is easier to transform voltages using transformers and transmit power over long distances
  • Most household appliances operate on 120V or 240V AC at 50Hz or 60Hz
• Audio systems use AC signals to represent sound waves
  • Microphones convert sound waves into AC signals, which are then processed, amplified, and converted back into sound waves by speakers
• Communication systems use AC signals to transmit information
  • Radio and television signals are AC signals that are modulated to carry information and transmitted through the air
• Induction motors use the principles of AC and magnetism to convert electrical energy into mechanical energy
  • The rotating magnetic field created by the AC current in the stator windings interacts with the rotor, causing it to rotate
• Transformers use AC to step up or step down voltages for power transmission and distribution
  • The primary and secondary windings are coupled through a magnetic core, allowing energy to be transferred between the windings

Common Mistakes and Tips

• Remember that phasors represent RMS values, not peak values
• Pay attention to the units of frequency (Hz) and angular frequency (rad/s)
• When using complex numbers, be consistent with the use of $j$ or $i$ as the imaginary unit
• In series RLC circuits, the voltages across the components are not in phase with each other, but the current is the same through all components
• In parallel RLC circuits, the currents through the components are not in phase with each other, but the voltage is the same across all components
• When calculating power, use RMS values of voltage and current, not peak values
• Remember that the power factor is the cosine of the phase difference between voltage and current, not the phase difference itself
• When analyzing resonant circuits, identify whether it is series or parallel resonance and use the appropriate formulas
• When designing filters, consider the desired cutoff frequencies, attenuation, and component values
• Always double-check your calculations and units to avoid errors in your analysis