Formula Sheet: AC Circuits

Table of Contents
1. Fundamental AC Concepts
2. Phasor Representation
3. AC Circuit Components
4. Impedance and Admittance
5. AC Circuit Analysis
6. Power in AC Circuits
7. Resonance in AC Circuits
8. Three-Phase Circuits
9. Coupled Circuits and Transformers
10. Frequency Response and Filters
11. Transient Response in AC Circuits
12. Balanced and Unbalanced Systems
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Fundamental AC Concepts

Sinusoidal Waveform Representation

• Instantaneous voltage: \[v(t) = V_m \sin(\omega t + \theta)\]
  • \(V_m\) = peak (maximum) voltage (V)
  • \(\omega\) = angular frequency (rad/s)
  • \(t\) = time (s)
  • \(\theta\) = phase angle (rad or degrees)
• Instantaneous current: \[i(t) = I_m \sin(\omega t + \phi)\]
  • \(I_m\) = peak (maximum) current (A)
  • \(\phi\) = phase angle (rad or degrees)
• Angular frequency: \[\omega = 2\pi f\]
  • \(f\) = frequency (Hz)
  • \(\omega\) = angular frequency (rad/s)
• Period: \[T = \frac{1}{f} = \frac{2\pi}{\omega}\]
  • \(T\) = period (s)

RMS (Root Mean Square) Values

• RMS voltage (sinusoidal): \[V_{rms} = \frac{V_m}{\sqrt{2}} = 0.707 V_m\]
  • \(V_{rms}\) = root mean square voltage (V)
  • \(V_m\) = peak voltage (V)
• RMS current (sinusoidal): \[I_{rms} = \frac{I_m}{\sqrt{2}} = 0.707 I_m\]
  • \(I_{rms}\) = root mean square current (A)
  • \(I_m\) = peak current (A)
• Peak voltage from RMS: \[V_m = \sqrt{2} \cdot V_{rms} = 1.414 V_{rms}\]
• Peak current from RMS: \[I_m = \sqrt{2} \cdot I_{rms} = 1.414 I_{rms}\]
• General RMS definition: \[V_{rms} = \sqrt{\frac{1}{T}\int_0^T v^2(t) \, dt}\]
  • Applies to any periodic waveform

Average and Peak-to-Peak Values

• Average value (full-wave rectified sine): \[V_{avg} = \frac{2V_m}{\pi} = 0.637 V_m\]
• Average value (general): \[V_{avg} = \frac{1}{T}\int_0^T v(t) \, dt\]
  • For pure sinusoid, average over full period = 0
• Peak-to-peak voltage: \[V_{pp} = 2V_m\]
  • \(V_{pp}\) = peak-to-peak voltage (V)
• Form factor: \[FF = \frac{V_{rms}}{V_{avg}}\]
  • For sinusoid: FF = 1.11
• Crest factor (peak factor): \[CF = \frac{V_m}{V_{rms}}\]
  • For sinusoid: CF = 1.414

Phasor Representation

Phasor Notation

• Rectangular form: \[\mathbf{V} = a + jb\]
  • \(a\) = real part (V)
  • \(b\) = imaginary part (V)
  • \(j = \sqrt{-1}\) (imaginary unit)
• Polar form: \[\mathbf{V} = V \angle \theta\]
  • \(V\) = magnitude (V)
  • \(\theta\) = phase angle (degrees or radians)
• Exponential form: \[\mathbf{V} = Ve^{j\theta}\]

Phasor Conversions

• Rectangular to polar (magnitude): \[V = \sqrt{a^2 + b^2}\]
• Rectangular to polar (angle): \[\theta = \tan^{-1}\left(\frac{b}{a}\right)\]
  • Must consider quadrant for correct angle
• Polar to rectangular (real part): \[a = V \cos(\theta)\]
• Polar to rectangular (imaginary part): \[b = V \sin(\theta)\]
• Euler's formula: \[e^{j\theta} = \cos(\theta) + j\sin(\theta)\]

Phasor Arithmetic

• Addition (rectangular form): \[(a_1 + jb_1) + (a_2 + jb_2) = (a_1 + a_2) + j(b_1 + b_2)\]
• Subtraction (rectangular form): \[(a_1 + jb_1) - (a_2 + jb_2) = (a_1 - a_2) + j(b_1 - b_2)\]
• Multiplication (polar form): \[(V_1 \angle \theta_1)(V_2 \angle \theta_2) = V_1V_2 \angle (\theta_1 + \theta_2)\]
• Division (polar form): \[\frac{V_1 \angle \theta_1}{V_2 \angle \theta_2} = \frac{V_1}{V_2} \angle (\theta_1 - \theta_2)\]
• Complex conjugate: \[\mathbf{V}^* = a - jb = V \angle (-\theta)\]

AC Circuit Components

Resistor in AC

• Resistance (frequency independent): \[R = \text{constant}\]
  • \(R\) = resistance (Ω)
• Voltage-current relationship: \[\mathbf{V}_R = R\mathbf{I}\]
  • Voltage and current are in phase (\(\theta = 0°\))
• Instantaneous power: \[p_R(t) = i^2(t)R = \frac{v^2(t)}{R}\]
• Average power: \[P_R = I_{rms}^2 R = \frac{V_{rms}^2}{R} = V_{rms}I_{rms}\]
  • \(P_R\) = average power dissipated (W)

Inductor in AC

• Inductive reactance: \[X_L = \omega L = 2\pi f L\]
  • \(X_L\) = inductive reactance (Ω)
  • \(L\) = inductance (H)
  • Reactance increases with frequency
• Impedance (phasor form): \[\mathbf{Z}_L = jX_L = j\omega L = X_L \angle 90°\]
• Voltage-current relationship (phasor): \[\mathbf{V}_L = jX_L\mathbf{I} = j\omega L\mathbf{I}\]
  • Voltage leads current by 90°
• Voltage-current relationship (time domain): \[v_L(t) = L\frac{di(t)}{dt}\]
• Current-voltage relationship (time domain): \[i_L(t) = \frac{1}{L}\int v(t) \, dt\]
• Reactive power: \[Q_L = I_{rms}^2 X_L = \frac{V_{rms}^2}{X_L} = V_{rms}I_{rms}\]
  • \(Q_L\) = reactive power (VAR, positive)
  • Average power dissipated = 0
• Energy stored: \[W_L = \frac{1}{2}Li^2\]
  • \(W_L\) = energy stored (J)

Capacitor in AC

• Capacitive reactance: \[X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}\]
  • \(X_C\) = capacitive reactance (Ω)
  • \(C\) = capacitance (F)
  • Reactance decreases with frequency
• Impedance (phasor form): \[\mathbf{Z}_C = -jX_C = \frac{1}{j\omega C} = X_C \angle -90°\]
• Voltage-current relationship (phasor): \[\mathbf{V}_C = -jX_C\mathbf{I} = \frac{\mathbf{I}}{j\omega C}\]
  • Voltage lags current by 90° (current leads voltage by 90°)
• Current-voltage relationship (time domain): \[i_C(t) = C\frac{dv(t)}{dt}\]
• Voltage-current relationship (time domain): \[v_C(t) = \frac{1}{C}\int i(t) \, dt\]
• Reactive power: \[Q_C = -I_{rms}^2 X_C = -\frac{V_{rms}^2}{X_C} = -V_{rms}I_{rms}\]
  • \(Q_C\) = reactive power (VAR, negative)
  • Average power dissipated = 0
• Energy stored: \[W_C = \frac{1}{2}Cv^2\]
  • \(W_C\) = energy stored (J)

Impedance and Admittance

Impedance

• General impedance definition: \[\mathbf{Z} = \frac{\mathbf{V}}{\mathbf{I}}\]
  • \(\mathbf{Z}\) = impedance (Ω)
  • Complex quantity with magnitude and phase
• Rectangular form: \[\mathbf{Z} = R + jX\]
  • \(R\) = resistance (real part, Ω)
  • \(X\) = reactance (imaginary part, Ω)
  • \(X > 0\) for inductive, \(X < 0\)="" for="">
• Polar form: \[\mathbf{Z} = Z \angle \theta = |Z| \angle \theta\]
  • \(Z\) or \(|Z|\) = impedance magnitude (Ω)
  • \(\theta\) = impedance angle (degrees or radians)
• Impedance magnitude: \[Z = |\mathbf{Z}| = \sqrt{R^2 + X^2}\]
• Impedance angle: \[\theta = \tan^{-1}\left(\frac{X}{R}\right)\]
  • Positive for inductive (lagging)
  • Negative for capacitive (leading)
• Resistance from impedance: \[R = Z\cos(\theta) = \text{Re}(\mathbf{Z})\]
• Reactance from impedance: \[X = Z\sin(\theta) = \text{Im}(\mathbf{Z})\]

Series Impedance Combinations

• Series R-L impedance: \[\mathbf{Z}_{RL} = R + jX_L = R + j\omega L\]
• Series R-C impedance: \[\mathbf{Z}_{RC} = R - jX_C = R - \frac{j}{\omega C}\]
• Series R-L-C impedance: \[\mathbf{Z}_{RLC} = R + j(X_L - X_C) = R + j\left(\omega L - \frac{1}{\omega C}\right)\]
• Total series impedance: \[\mathbf{Z}_{total} = \mathbf{Z}_1 + \mathbf{Z}_2 + \mathbf{Z}_3 + \cdots\]
  • Impedances add directly in series
• Total resistance (series): \[R_{total} = R_1 + R_2 + R_3 + \cdots\]
• Total reactance (series): \[X_{total} = X_1 + X_2 + X_3 + \cdots\]
  • Remember: inductive reactance positive, capacitive negative

Parallel Impedance Combinations

• Total parallel impedance (two elements): \[\mathbf{Z}_{total} = \frac{\mathbf{Z}_1 \mathbf{Z}_2}{\mathbf{Z}_1 + \mathbf{Z}_2}\]
• Total parallel impedance (general): \[\frac{1}{\mathbf{Z}_{total}} = \frac{1}{\mathbf{Z}_1} + \frac{1}{\mathbf{Z}_2} + \frac{1}{\mathbf{Z}_3} + \cdots\]
• Parallel R-L impedance: \[\mathbf{Z}_{RL} = \frac{R(jX_L)}{R + jX_L} = \frac{j\omega LR}{R + j\omega L}\]
• Parallel R-C impedance: \[\mathbf{Z}_{RC} = \frac{R(-jX_C)}{R - jX_C} = \frac{R}{1 + j\omega RC}\]

Admittance

• Admittance definition: \[\mathbf{Y} = \frac{1}{\mathbf{Z}} = \frac{\mathbf{I}}{\mathbf{V}}\]
  • \(\mathbf{Y}\) = admittance (S, siemens or mho)
• Rectangular form: \[\mathbf{Y} = G + jB\]
  • \(G\) = conductance (real part, S)
  • \(B\) = susceptance (imaginary part, S)
• Polar form: \[\mathbf{Y} = Y \angle (-\theta)\]
  • \(Y\) = admittance magnitude (S)
  • Angle is negative of impedance angle
• Conductance: \[G = \frac{R}{R^2 + X^2} = \frac{R}{Z^2}\]
• Susceptance: \[B = -\frac{X}{R^2 + X^2} = -\frac{X}{Z^2}\]
  • \(B > 0\) for capacitive
  • \(B < 0\)="" for="">
• Admittance magnitude: \[Y = \frac{1}{Z} = \sqrt{G^2 + B^2}\]
• Series admittances: \[\frac{1}{\mathbf{Y}_{total}} = \frac{1}{\mathbf{Y}_1} + \frac{1}{\mathbf{Y}_2} + \frac{1}{\mathbf{Y}_3} + \cdots\]
• Parallel admittances: \[\mathbf{Y}_{total} = \mathbf{Y}_1 + \mathbf{Y}_2 + \mathbf{Y}_3 + \cdots\]
  • Admittances add directly in parallel

AC Circuit Analysis

Ohm's Law for AC Circuits

• Phasor voltage-current relationship: \[\mathbf{V} = \mathbf{Z}\mathbf{I}\]
• Phasor current from voltage: \[\mathbf{I} = \frac{\mathbf{V}}{\mathbf{Z}} = \mathbf{Y}\mathbf{V}\]
• Magnitude relationship: \[V = IZ\]
• Phase relationship: \[\theta_V = \theta_I + \theta_Z\]
  • \(\theta_V\) = voltage phase angle
  • \(\theta_I\) = current phase angle
  • \(\theta_Z\) = impedance angle

Kirchhoff's Laws for AC Circuits

• Kirchhoff's Voltage Law (KVL): \[\sum \mathbf{V}_n = 0\]
  • Sum of phasor voltages around closed loop equals zero
  • Must use phasor addition
• Kirchhoff's Current Law (KCL): \[\sum \mathbf{I}_n = 0\]
  • Sum of phasor currents at node equals zero
  • Must use phasor addition

Voltage and Current Division

• Voltage divider (two impedances): \[\mathbf{V}_1 = \mathbf{V}_s \frac{\mathbf{Z}_1}{\mathbf{Z}_1 + \mathbf{Z}_2}\]
  • \(\mathbf{V}_s\) = source voltage
  • \(\mathbf{V}_1\) = voltage across \(\mathbf{Z}_1\)
• Voltage divider (general): \[\mathbf{V}_k = \mathbf{V}_s \frac{\mathbf{Z}_k}{\sum_{n=1}^N \mathbf{Z}_n}\]
• Current divider (two impedances): \[\mathbf{I}_1 = \mathbf{I}_s \frac{\mathbf{Z}_2}{\mathbf{Z}_1 + \mathbf{Z}_2}\]
  • \(\mathbf{I}_s\) = source current
  • \(\mathbf{I}_1\) = current through \(\mathbf{Z}_1\)
  • Note: opposite impedance in numerator
• Current divider (using admittance): \[\mathbf{I}_1 = \mathbf{I}_s \frac{\mathbf{Y}_1}{\mathbf{Y}_1 + \mathbf{Y}_2}\]
  • Often easier for parallel circuits

Mesh and Nodal Analysis

• Mesh analysis (general form):
  • Write KVL equations for each mesh
  • Express voltages in terms of mesh currents and impedances
  • Solve simultaneous equations for mesh currents
  • All quantities are phasors
• Nodal analysis (general form):
  • Write KCL equations at each node
  • Express currents in terms of node voltages and admittances
  • Solve simultaneous equations for node voltages
  • All quantities are phasors

Superposition in AC Circuits

• Superposition principle:
  • Response to multiple sources = sum of individual responses
  • Deactivate all sources except one at a time
  • Voltage sources → short circuit
  • Current sources → open circuit
  • Add phasor responses (all at same frequency)
  • Only valid for linear circuits
• Multiple frequencies:
  • Must analyze each frequency separately
  • Cannot use phasor addition across different frequencies
  • Final time-domain response = sum of time-domain responses

Thévenin and Norton Equivalent Circuits

• Thévenin impedance: \[\mathbf{Z}_{th} = \mathbf{Z}_{oc}\]
  • Impedance seen from terminals with all sources deactivated
  • Can also find as \(\mathbf{Z}_{th} = \frac{\mathbf{V}_{oc}}{\mathbf{I}_{sc}}\)
• Thévenin voltage: \[\mathbf{V}_{th} = \mathbf{V}_{oc}\]
  • Open-circuit voltage across terminals
• Norton current: \[\mathbf{I}_{N} = \mathbf{I}_{sc}\]
  • Short-circuit current through terminals
• Norton impedance: \[\mathbf{Z}_{N} = \mathbf{Z}_{th}\]
  • Same as Thévenin impedance
• Thévenin-Norton conversion: \[\mathbf{V}_{th} = \mathbf{I}_{N}\mathbf{Z}_{th}\] \[\mathbf{I}_{N} = \frac{\mathbf{V}_{th}}{\mathbf{Z}_{th}}\]

Power in AC Circuits

Instantaneous and Average Power

• Instantaneous power: \[p(t) = v(t) \cdot i(t)\]
  • \(p(t)\) = instantaneous power (W)
• Average (real) power: \[P = V_{rms}I_{rms}\cos(\theta)\]
  • \(P\) = average power (W)
  • \(\theta\) = phase angle between voltage and current (impedance angle)
  • \(\cos(\theta)\) = power factor
• Alternative real power formulas: \[P = I_{rms}^2 R = \frac{V_{rms}^2}{Z^2}R = I_{rms}^2 Z\cos(\theta)\]
• Real power (phasor form): \[P = \frac{1}{2}\text{Re}(\mathbf{V}\mathbf{I}^*)\]
  • \(\mathbf{I}^*\) = complex conjugate of current phasor
  • Phasors must be in peak values (not RMS)

Reactive Power

• Reactive power: \[Q = V_{rms}I_{rms}\sin(\theta)\]
  • \(Q\) = reactive power (VAR, volt-ampere reactive)
  • \(\theta\) = impedance angle
  • Positive for inductive (lagging)
  • Negative for capacitive (leading)
• Alternative reactive power formulas: \[Q = I_{rms}^2 X = \frac{V_{rms}^2}{Z^2}X = I_{rms}^2 Z\sin(\theta)\]
  • \(X\) = net reactance (Ω)
• Reactive power (phasor form): \[Q = \frac{1}{2}\text{Im}(\mathbf{V}\mathbf{I}^*)\]
• Inductive reactive power: \[Q_L = I_{rms}^2 X_L = \frac{V_{rms}^2}{X_L}\]
  • Positive value
• Capacitive reactive power: \[Q_C = -I_{rms}^2 X_C = -\frac{V_{rms}^2}{X_C}\]
  • Negative value

Apparent Power and Complex Power

• Apparent power: \[S = V_{rms}I_{rms}\]
  • \(S\) = apparent power (VA, volt-ampere)
  • Magnitude of complex power
• Apparent power from P and Q: \[S = \sqrt{P^2 + Q^2}\]
• Complex power: \[\mathbf{S} = P + jQ\]
  • \(\mathbf{S}\) = complex power (VA)
  • Complex quantity
• Complex power (phasor form): \[\mathbf{S} = \mathbf{V}_{rms}\mathbf{I}_{rms}^* = \frac{1}{2}\mathbf{V}\mathbf{I}^*\]
  • First form uses RMS phasors
  • Second form uses peak phasors
• Complex power from impedance: \[\mathbf{S} = I_{rms}^2 \mathbf{Z} = \frac{V_{rms}^2}{\mathbf{Z}^*}\]
• Magnitude of complex power: \[|\mathbf{S}| = S = V_{rms}I_{rms}\]
• Angle of complex power: \[\angle \mathbf{S} = \theta = \angle \mathbf{Z}\]

Power Factor

• Power factor definition: \[pf = \cos(\theta) = \frac{P}{S}\]
  • \(pf\) = power factor (dimensionless, 0 to 1)
  • \(\theta\) = angle between voltage and current
• Power factor from P and Q: \[pf = \frac{P}{\sqrt{P^2 + Q^2}}\]
• Power factor from R and Z: \[pf = \frac{R}{Z} = \frac{R}{\sqrt{R^2 + X^2}}\]
• Leading vs. lagging:
  • Lagging power factor: current lags voltage (inductive, \(\theta > 0\), \(Q > 0\))
  • Leading power factor: current leads voltage (capacitive, \(\theta < 0\),="" \(q=""><>
  • Unity power factor: current in phase with voltage (resistive, \(\theta = 0\), \(Q = 0\))
• Real power from power factor: \[P = S \cdot pf = V_{rms}I_{rms} \cdot pf\]
• Phase angle from power factor: \[\theta = \cos^{-1}(pf)\]
  • Sign determined by leading/lagging specification

Power Factor Correction

• Capacitance for power factor correction: \[C = \frac{Q_C}{\omega V_{rms}^2} = \frac{P(\tan\theta_1 - \tan\theta_2)}{\omega V_{rms}^2}\]
  • \(C\) = capacitance needed (F)
  • \(Q_C\) = reactive power to be compensated (VAR)
  • \(\theta_1\) = original power factor angle
  • \(\theta_2\) = desired power factor angle
  • Applied in parallel with load
• Reactive power correction: \[Q_C = Q_1 - Q_2 = P(\tan\theta_1 - \tan\theta_2)\]
  • \(Q_1\) = original reactive power
  • \(Q_2\) = desired reactive power
• Alternative form (using power factors): \[Q_C = P\left(\sqrt{\frac{1}{pf_1^2} - 1} - \sqrt{\frac{1}{pf_2^2} - 1}\right)\]
  • \(pf_1\) = original power factor
  • \(pf_2\) = desired power factor

Maximum Power Transfer

• Maximum power transfer condition: \[\mathbf{Z}_L = \mathbf{Z}_{th}^*\]
  • \(\mathbf{Z}_L\) = load impedance
  • \(\mathbf{Z}_{th}^*\) = complex conjugate of Thévenin impedance
  • \(R_L = R_{th}\) and \(X_L = -X_{th}\)
• Maximum power delivered: \[P_{max} = \frac{V_{th(rms)}^2}{4R_{th}} = \frac{|V_{th}|^2}{8R_{th}}\]
  • First form uses RMS voltage
  • Second form uses peak voltage
• Efficiency at maximum power transfer: \[\eta = 50\%\]
  • Half the power dissipated in source, half in load

Resonance in AC Circuits

Series Resonance

• Resonant frequency: \[f_0 = \frac{1}{2\pi\sqrt{LC}}\]
  • \(f_0\) = resonant frequency (Hz)
  • \(L\) = inductance (H)
  • \(C\) = capacitance (F)
• Resonant angular frequency: \[\omega_0 = \frac{1}{\sqrt{LC}} = 2\pi f_0\]
  • \(\omega_0\) = resonant angular frequency (rad/s)
• Condition for resonance: \[X_L = X_C\] \[\omega_0 L = \frac{1}{\omega_0 C}\]
  • Inductive and capacitive reactances are equal
• Impedance at resonance: \[\mathbf{Z}_0 = R\]
  • Impedance is purely resistive (minimum)
  • Reactances cancel
• Current at resonance: \[I_0 = \frac{V}{R}\]
  • Maximum current in series RLC
• Quality factor (series): \[Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 RC} = \frac{1}{R}\sqrt{\frac{L}{C}}\]
  • \(Q\) = quality factor (dimensionless)
  • Measure of selectivity
• Bandwidth (series): \[BW = \frac{f_0}{Q} = \frac{R}{2\pi L}\]
  • \(BW\) = bandwidth (Hz)
  • Range of frequencies where power ≥ 50% of maximum
• Half-power frequencies: \[f_1 = f_0 - \frac{BW}{2}\] \[f_2 = f_0 + \frac{BW}{2}\]
  • \(f_1\) = lower half-power frequency (Hz)
  • \(f_2\) = upper half-power frequency (Hz)
• Voltage magnification: \[V_L = V_C = QV\]
  • Voltage across L or C at resonance
  • \(V\) = source voltage

Parallel Resonance

• Resonant frequency (ideal parallel LC): \[f_0 = \frac{1}{2\pi\sqrt{LC}}\]
  • Same as series resonance for ideal components
• Resonant frequency (practical, with resistance): \[f_0 = \frac{1}{2\pi}\sqrt{\frac{1}{LC} - \frac{R^2}{L^2}}\]
  • \(R\) = series resistance in inductor branch
  • Approximation: \(f_0 \approx \frac{1}{2\pi\sqrt{LC}}\) when \(Q >> 1\)
• Condition for resonance: \[B_L = B_C\]
  • Inductive and capacitive susceptances are equal
• Impedance at resonance (ideal): \[\mathbf{Z}_0 = \infty\]
  • Maximum impedance (purely resistive)
  • For practical circuits: \(Z_0 = \frac{L}{RC}\) (very large)
• Current at resonance: \[I_0 = \frac{V}{Z_0}\]
  • Minimum current from source in parallel RLC
• Quality factor (parallel): \[Q = \omega_0 RC = \frac{R}{\omega_0 L} = R\sqrt{\frac{C}{L}}\]
  • \(R\) = parallel resistance
  • For tank circuit with series R in L branch: \(Q = \frac{\omega_0 L}{R}\)
• Bandwidth (parallel): \[BW = \frac{f_0}{Q} = \frac{1}{2\pi RC}\]
  • For parallel resistance R
• Current magnification: \[I_L = I_C = QI\]
  • Current through L or C at resonance
  • \(I\) = source current

General Resonance Properties

• Phase angle at resonance: \[\theta = 0°\]
  • Voltage and current in phase
  • Power factor = 1 (unity)
• Selectivity:
  • Higher Q → narrower bandwidth → more selective
  • Lower Q → wider bandwidth → less selective
• Energy considerations:
  • At resonance, energy oscillates between L and C
  • No reactive power from source
  • All source power is real power

Three-Phase Circuits

Three-Phase Voltage Relationships

• Phase sequence notation:
  • Positive (abc) sequence: Phase a leads b by 120°, b leads c by 120°
  • Negative (acb) sequence: Phase a leads c by 120°, c leads b by 120°
• Balanced three-phase voltages (abc sequence): \[\mathbf{V}_a = V_p \angle 0°\] \[\mathbf{V}_b = V_p \angle -120°\] \[\mathbf{V}_c = V_p \angle -240° = V_p \angle 120°\]
  • \(V_p\) = phase voltage magnitude (V)
• Sum of balanced voltages: \[\mathbf{V}_a + \mathbf{V}_b + \mathbf{V}_c = 0\]
• Sum of balanced currents: \[\mathbf{I}_a + \mathbf{I}_b + \mathbf{I}_c = 0\]

Wye (Y) Connection

• Line-to-line voltage (Wye): \[V_L = \sqrt{3} V_p\]
  • \(V_L\) = line-to-line voltage (line voltage, V)
  • \(V_p\) = line-to-neutral voltage (phase voltage, V)
• Phase voltage from line voltage (Wye): \[V_p = \frac{V_L}{\sqrt{3}}\]
• Line current equals phase current (Wye): \[I_L = I_p\]
  • \(I_L\) = line current (A)
  • \(I_p\) = phase current (A)
• Line voltage leads phase voltage by 30°: \[\mathbf{V}_{ab} = \sqrt{3} V_p \angle 30°\]
  • If \(\mathbf{V}_a = V_p \angle 0°\)
• Line-to-line voltage relationships (Wye): \[\mathbf{V}_{ab} = \mathbf{V}_a - \mathbf{V}_b\] \[\mathbf{V}_{bc} = \mathbf{V}_b - \mathbf{V}_c\] \[\mathbf{V}_{ca} = \mathbf{V}_c - \mathbf{V}_a\]

Delta (Δ) Connection

• Line voltage equals phase voltage (Delta): \[V_L = V_p\]
• Line current (Delta): \[I_L = \sqrt{3} I_p\]
  • \(I_p\) = current through each phase impedance (A)
• Phase current from line current (Delta): \[I_p = \frac{I_L}{\sqrt{3}}\]
• Line current lags phase current by 30°: \[\mathbf{I}_a = \sqrt{3} I_p \angle -30°\]
  • If \(\mathbf{I}_{ab} = I_p \angle 0°\)
• Line current relationships (Delta): \[\mathbf{I}_a = \mathbf{I}_{ab} - \mathbf{I}_{ca}\] \[\mathbf{I}_b = \mathbf{I}_{bc} - \mathbf{I}_{ab}\] \[\mathbf{I}_c = \mathbf{I}_{ca} - \mathbf{I}_{bc}\]

Three-Phase Power

• Total real power (balanced three-phase): \[P_{3\phi} = 3V_p I_p \cos(\theta) = \sqrt{3} V_L I_L \cos(\theta)\]
  • \(P_{3\phi}\) = total three-phase real power (W)
  • \(\theta\) = angle of load impedance (angle between phase voltage and phase current)
• Total reactive power (balanced three-phase): \[Q_{3\phi} = 3V_p I_p \sin(\theta) = \sqrt{3} V_L I_L \sin(\theta)\]
  • \(Q_{3\phi}\) = total three-phase reactive power (VAR)
• Total apparent power (balanced three-phase): \[S_{3\phi} = 3V_p I_p = \sqrt{3} V_L I_L\]
  • \(S_{3\phi}\) = total three-phase apparent power (VA)
• Complex power (three-phase): \[\mathbf{S}_{3\phi} = P_{3\phi} + jQ_{3\phi} = \sqrt{3} V_L I_L \angle \theta\]
• Power per phase: \[P_{phase} = V_p I_p \cos(\theta) = \frac{P_{3\phi}}{3}\]
• Power factor (three-phase): \[pf = \cos(\theta) = \frac{P_{3\phi}}{S_{3\phi}}\]
• Alternative power formulas (Wye): \[P_{3\phi} = 3I_L^2 R_{phase} = \frac{3V_p^2}{Z_{phase}}\cos(\theta)\]
  • \(R_{phase}\) = resistance per phase (Ω)
  • \(Z_{phase}\) = impedance per phase (Ω)
• Alternative power formulas (Delta): \[P_{3\phi} = 3I_p^2 R_{phase} = \frac{3V_L^2}{Z_{phase}}\cos(\theta)\]

Wye-Delta Transformations

• Wye to Delta impedance conversion: \[\mathbf{Z}_{\Delta} = 3\mathbf{Z}_Y\]
  • \(\mathbf{Z}_{\Delta}\) = impedance per phase in Delta (Ω)
  • \(\mathbf{Z}_Y\) = impedance per phase in Wye (Ω)
  • For balanced loads only
• Delta to Wye impedance conversion: \[\mathbf{Z}_Y = \frac{\mathbf{Z}_{\Delta}}{3}\]
• General Wye to Delta (unbalanced): \[\mathbf{Z}_{ab} = \frac{\mathbf{Z}_a\mathbf{Z}_b + \mathbf{Z}_b\mathbf{Z}_c + \mathbf{Z}_c\mathbf{Z}_a}{\mathbf{Z}_c}\] \[\mathbf{Z}_{bc} = \frac{\mathbf{Z}_a\mathbf{Z}_b + \mathbf{Z}_b\mathbf{Z}_c + \mathbf{Z}_c\mathbf{Z}_a}{\mathbf{Z}_a}\] \[\mathbf{Z}_{ca} = \frac{\mathbf{Z}_a\mathbf{Z}_b + \mathbf{Z}_b\mathbf{Z}_c + \mathbf{Z}_c\mathbf{Z}_a}{\mathbf{Z}_b}\]
• General Delta to Wye (unbalanced): \[\mathbf{Z}_a = \frac{\mathbf{Z}_{ab}\mathbf{Z}_{ca}}{\mathbf{Z}_{ab} + \mathbf{Z}_{bc} + \mathbf{Z}_{ca}}\] \[\mathbf{Z}_b = \frac{\mathbf{Z}_{ab}\mathbf{Z}_{bc}}{\mathbf{Z}_{ab} + \mathbf{Z}_{bc} + \mathbf{Z}_{ca}}\] \[\mathbf{Z}_c = \frac{\mathbf{Z}_{bc}\mathbf{Z}_{ca}}{\mathbf{Z}_{ab} + \mathbf{Z}_{bc} + \mathbf{Z}_{ca}}\]

Balanced Three-Phase Analysis

• Per-phase analysis method:
  • Convert Delta loads to equivalent Wye if necessary
  • Analyze one phase (usually phase a) with neutral reference
  • Use single-phase circuit analysis techniques
  • Results for other phases differ by ±120° phase shift
  • Multiply single-phase power by 3 for total power
• Neutral current (balanced system): \[I_N = 0\]
  • No current flows in neutral wire for balanced loads
• Phase voltage in terms of line voltage: \[\mathbf{V}_{an} = \frac{\mathbf{V}_{ab}}{\sqrt{3}} \angle -30°\]
  • For Wye connection, positive sequence

Coupled Circuits and Transformers

Mutual Inductance

• Mutual inductance definition: \[M = k\sqrt{L_1 L_2}\]
  • \(M\) = mutual inductance (H)
  • \(k\) = coupling coefficient (0 ≤ k ≤ 1)
  • \(L_1, L_2\) = self-inductances (H)
• Coupling coefficient: \[k = \frac{M}{\sqrt{L_1 L_2}}\]
  • \(k = 1\) for perfect coupling
  • \(k = 0\) for no coupling
• Voltage induced by mutual inductance: \[v_2(t) = M\frac{di_1(t)}{dt}\]
  • \(v_2\) = voltage induced in coil 2 (V)
  • \(i_1\) = current in coil 1 (A)
• Phasor voltage (mutual coupling): \[\mathbf{V}_2 = j\omega M\mathbf{I}_1\]

Dot Convention for Coupled Coils

• Series aiding (dots at same end): \[L_{total} = L_1 + L_2 + 2M\]
• Series opposing (dots at opposite ends): \[L_{total} = L_1 + L_2 - 2M\]
• Voltage equations with dot convention:
  • Current entering dot on one coil induces voltage positive at dot on other coil
  • Current leaving dot on one coil induces voltage negative at dot on other coil

Coupled Circuit Equations

• Primary voltage equation: \[\mathbf{V}_1 = (R_1 + j\omega L_1)\mathbf{I}_1 \pm j\omega M\mathbf{I}_2\]
  • \(+\) for aiding, \(-\) for opposing
  • \(R_1\) = primary resistance (Ω)
• Secondary voltage equation: \[\mathbf{V}_2 = \pm j\omega M\mathbf{I}_1 + (R_2 + j\omega L_2)\mathbf{I}_2\]
  • \(R_2\) = secondary resistance (Ω)
• Impedance matrix form: \[\begin{bmatrix} \mathbf{V}_1 \\ \mathbf{V}_2 \end{bmatrix} = \begin{bmatrix} \mathbf{Z}_{11} & \mathbf{Z}_{12} \\ \mathbf{Z}_{21} & \mathbf{Z}_{22} \end{bmatrix} \begin{bmatrix} \mathbf{I}_1 \\ \mathbf{I}_2 \end{bmatrix}\]
  • \(\mathbf{Z}_{11} = R_1 + j\omega L_1\)
  • \(\mathbf{Z}_{22} = R_2 + j\omega L_2\)
  • \(\mathbf{Z}_{12} = \mathbf{Z}_{21} = \pm j\omega M\)

Ideal Transformer Relationships

• Turns ratio: \[n = \frac{N_1}{N_2}\]
  • \(n\) = turns ratio
  • \(N_1\) = number of primary turns
  • \(N_2\) = number of secondary turns
• Voltage ratio (ideal transformer): \[\frac{V_1}{V_2} = \frac{N_1}{N_2} = n\]
  • \(V_1\) = primary voltage (V)
  • \(V_2\) = secondary voltage (V)
• Current ratio (ideal transformer): \[\frac{I_1}{I_2} = \frac{N_2}{N_1} = \frac{1}{n}\]
  • \(I_1\) = primary current (A)
  • \(I_2\) = secondary current (A)
• Power conservation (ideal transformer): \[S_1 = S_2\] \[V_1 I_1 = V_2 I_2\]
  • No power loss in ideal transformer
• Impedance transformation (ideal): \[\mathbf{Z}_1 = n^2 \mathbf{Z}_2\]
  • \(\mathbf{Z}_1\) = impedance referred to primary (Ω)
  • \(\mathbf{Z}_2\) = load impedance on secondary (Ω)
• Load impedance referred to primary: \[\mathbf{Z}_{in} = \left(\frac{N_1}{N_2}\right)^2 \mathbf{Z}_L = n^2 \mathbf{Z}_L\]
• Secondary voltage (step-down, n > 1): \[V_2 = \frac{V_1}{n}\]
  • Voltage decreases
  • Current increases
• Secondary voltage (step-up, n <> \[V_2 = \frac{V_1}{n}\]
  • Voltage increases
  • Current decreases

Non-Ideal Transformer Model

• Efficiency: \[\eta = \frac{P_{out}}{P_{in}} = \frac{P_{out}}{P_{out} + P_{losses}}\]
  • \(\eta\) = efficiency (0 to 1 or percentage)
  • \(P_{losses}\) = copper losses + core losses (W)
• Copper losses (winding resistance): \[P_{Cu} = I_1^2 R_1 + I_2^2 R_2\]
  • \(R_1, R_2\) = primary and secondary winding resistances (Ω)
• Core losses: \[P_{core} = P_{hysteresis} + P_{eddy}\]
  • Hysteresis losses ∝ frequency
  • Eddy current losses ∝ frequency²
• Voltage regulation: \[VR = \frac{V_{2,NL} - V_{2,FL}}{V_{2,FL}} \times 100\%\]
  • \(VR\) = voltage regulation (%)
  • \(V_{2,NL}\) = secondary voltage at no load (V)
  • \(V_{2,FL}\) = secondary voltage at full load (V)

Frequency Response and Filters

Transfer Function

• Transfer function definition: \[\mathbf{H}(\omega) = \frac{\mathbf{V}_{out}}{\mathbf{V}_{in}}\] or \[\mathbf{H}(j\omega) = \frac{\mathbf{V}_{out}}{\mathbf{V}_{in}}\]
  • \(\mathbf{H}\) = transfer function (complex, dimensionless for voltage ratio)
  • Can also be defined for current or other quantities
• Magnitude (gain): \[|\mathbf{H}(\omega)| = \frac{V_{out}}{V_{in}}\]
• Phase: \[\phi(\omega) = \angle \mathbf{H}(\omega)\]
  • \(\phi\) = phase shift (rad or degrees)
• Gain in decibels: \[G_{dB} = 20\log_{10}|\mathbf{H}(\omega)|\]
  • \(G_{dB}\) = gain in decibels (dB)

First-Order RC Low-Pass Filter

• Transfer function: \[\mathbf{H}(\omega) = \frac{1}{1 + j\omega RC}\]
  • Output across capacitor
• Cutoff frequency: \[f_c = \frac{1}{2\pi RC}\] or \[\omega_c = \frac{1}{RC}\]
  • \(f_c\) = cutoff (corner, -3dB) frequency (Hz)
  • Frequency where gain = 0.707 (-3 dB)
• Magnitude response: \[|\mathbf{H}(\omega)| = \frac{1}{\sqrt{1 + (\omega RC)^2}}\]
• Phase response: \[\phi(\omega) = -\tan^{-1}(\omega RC)\]
• At cutoff frequency: \[|\mathbf{H}(\omega_c)| = \frac{1}{\sqrt{2}} = 0.707\] \[\phi(\omega_c) = -45°\]

First-Order RC High-Pass Filter

• Transfer function: \[\mathbf{H}(\omega) = \frac{j\omega RC}{1 + j\omega RC}\]
  • Output across resistor
• Cutoff frequency: \[f_c = \frac{1}{2\pi RC}\] \[\omega_c = \frac{1}{RC}\]
• Magnitude response: \[|\mathbf{H}(\omega)| = \frac{\omega RC}{\sqrt{1 + (\omega RC)^2}}\]
• Phase response: \[\phi(\omega) = 90° - \tan^{-1}(\omega RC)\]
• At cutoff frequency: \[|\mathbf{H}(\omega_c)| = \frac{1}{\sqrt{2}} = 0.707\] \[\phi(\omega_c) = 45°\]

First-Order RL Filters

• RL low-pass (output across R): \[\mathbf{H}(\omega) = \frac{1}{1 + j\omega L/R}\] \[\omega_c = \frac{R}{L}\]
• RL high-pass (output across L): \[\mathbf{H}(\omega) = \frac{j\omega L/R}{1 + j\omega L/R}\] \[\omega_c = \frac{R}{L}\]

Second-Order RLC Filters

• Series RLC bandpass (output across R): \[\mathbf{H}(\omega) = \frac{j\omega RC}{1 + j\omega RC + (j\omega)^2 LC}\]
• Center frequency (bandpass): \[f_0 = \frac{1}{2\pi\sqrt{LC}}\]
  • Same as resonant frequency
• Bandwidth: \[BW = \frac{R}{2\pi L}\]
• Quality factor: \[Q = \frac{f_0}{BW} = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 RC}\]
• Lower cutoff frequency: \[f_1 = f_0\left(\sqrt{1 + \frac{1}{4Q^2}} - \frac{1}{2Q}\right)\]
  • Approximation for high Q: \(f_1 \approx f_0 - \frac{BW}{2}\)
• Upper cutoff frequency: \[f_2 = f_0\left(\sqrt{1 + \frac{1}{4Q^2}} + \frac{1}{2Q}\right)\]
  • Approximation for high Q: \(f_2 \approx f_0 + \frac{BW}{2}\)

Bode Plot Characteristics

• First-order low-pass roll-off:
  • -20 dB/decade above \(f_c\)
  • -6 dB/octave above \(f_c\)
• First-order high-pass roll-off:
  • +20 dB/decade below \(f_c\)
  • +6 dB/octave below \(f_c\)
• Decade:
  • Frequency ratio of 10:1
• Octave:
  • Frequency ratio of 2:1
• Second-order roll-off:
  • -40 dB/decade (low-pass)
  • +40 dB/decade (high-pass)

Transient Response in AC Circuits

RL Circuit Transient Response

• Time constant (RL): \[\tau = \frac{L}{R}\]
  • \(\tau\) = time constant (s)
• Current growth (RL): \[i(t) = \frac{V}{R}(1 - e^{-t/\tau})\]
  • For DC voltage source V applied at t = 0
• Current decay (RL): \[i(t) = I_0 e^{-t/\tau}\]
  • \(I_0\) = initial current (A)
  • After removing source
• Voltage across inductor (growth): \[v_L(t) = Ve^{-t/\tau}\]
• Voltage across resistor (growth): \[v_R(t) = V(1 - e^{-t/\tau})\]

RC Circuit Transient Response

• Time constant (RC): \[\tau = RC\]
• Voltage growth (RC): \[v_C(t) = V(1 - e^{-t/\tau})\]
  • For DC voltage source V applied at t = 0
• Voltage decay (RC): \[v_C(t) = V_0 e^{-t/\tau}\]
  • \(V_0\) = initial voltage (V)
  • Capacitor discharging through resistor
• Charging current: \[i(t) = \frac{V}{R}e^{-t/\tau}\]
• Voltage across resistor (charging): \[v_R(t) = Ve^{-t/\tau}\]

General Transient Properties

• Time to reach steady state:
  • Approximately 5τ (99.3% of final value)
  • After 1τ: 63.2% of final value
  • After 2τ: 86.5% of final value
  • After 3τ: 95.0% of final value
  • After 4τ: 98.2% of final value
• Initial conditions:
  • Inductor current cannot change instantaneously
  • Capacitor voltage cannot change instantaneously

Balanced and Unbalanced Systems

Symmetrical Components

• Positive sequence component: \[\mathbf{V}_1 = \frac{1}{3}(\mathbf{V}_a + a\mathbf{V}_b + a^2\mathbf{V}_c)\]
  • \(a = 1\angle 120° = e^{j2\pi/3}\) (operator)
  • \(a^2 = 1\angle 240° = 1\angle -120°\)
• Negative sequence component: \[\mathbf{V}_2 = \frac{1}{3}(\mathbf{V}_a + a^2\mathbf{V}_b + a\mathbf{V}_c)\]
• Zero sequence component: \[\mathbf{V}_0 = \frac{1}{3}(\mathbf{V}_a + \mathbf{V}_b + \mathbf{V}_c)\]
• Operator properties: \[a^3 = 1\] \[1 + a + a^2 = 0\]
• Reconstruction of phase voltages: \[\mathbf{V}_a = \mathbf{V}_0 + \mathbf{V}_1 + \mathbf{V}_2\] \[\mathbf{V}_b = \mathbf{V}_0 + a^2\mathbf{V}_1 + a\mathbf{V}_2\] \[\mathbf{V}_c = \mathbf{V}_0 + a\mathbf{V}_1 + a^2\mathbf{V}_2\]