AC Circuits

Analysis of circuits with time-varying sinusoidal voltages and currents.

Definition

Alternating Current (AC) circuits involve voltages and currents that vary sinusoidally with time. AC is the standard for power distribution due to efficient voltage transformation using transformers.

AC vs DC

Property Direct Current (DC) Alternating Current (AC)
Direction Flows in one direction only Reverses direction periodically
Magnitude Constant Varies with time
Current flow +ve to −ve terminal Alternates between +ve→−ve and −ve→+ve
Electron flow −ve to +ve terminal Alternates direction
Lamp brightness Constant Flickering (at line frequency)
Transmission High losses over long distances Efficient for long-distance power transmission
Voltage conversion Requires complex electronics Easy with transformers

[!note] Why AC won the "War of Currents" In the late 1800s, Thomas Edison advocated for DC while Nikola Tesla pioneered AC. AC became the global standard because transformers enable efficient voltage step-up for transmission and step-down for consumption, dramatically reducing power losses ($P_{\text{loss}} = I^2R$) over long distances.

graph TB
    subgraph dc["Direct Current (DC)"]
        D1["Direction: Flows in one direction only"]
        D2["Magnitude: Constant"]
        D3["Transmission: High losses over long distances"]
        D4["Voltage conversion: Requires complex electronics"]
    end

    subgraph ac["Alternating Current (AC)"]
        A1["Direction: Reverses direction periodically"]
        A2["Magnitude: Varies with time"]
        A3["Transmission: Efficient for long-distance power transmission"]
        A4["Voltage conversion: Easy with transformers"]
    end

    dc ~~~ ac

    classDef dcStyle fill:#ffe3e3,stroke:#c92a2a,stroke-width:2px
    classDef acStyle fill:#d3f9d8,stroke:#2f9e44,stroke-width:2px
    class D1,D2,D3,D4 dcStyle
    class A1,A2,A3,A4 acStyle

Sinusoidal AC Signals

AC voltage and current are described by sinusoidal functions. The general forms are:

$$I(t) = I_0 \sin(\omega t)$$

$$V(t) = V_0 \sin(\omega t)$$

Where:

  • $I(t)$, $V(t)$ : instantaneous current and voltage
  • $I_0$, $V_0$ : peak (maximum) current and voltage
  • $\omega$ : angular frequency (rad/s)
  • $t$ : time (s)

[!note] Sinusoidal AC can also be written using a cosine function, which is simply a phase-shifted sine wave.

Frequency relations:

$$\omega = \frac{2\pi}{T} = 2\pi f$$

  • $T$ : period — time for one complete cycle (s)
  • $f$ : frequency — number of complete cycles per second (Hz)

Writing Equations from Graphs

Step 1: Identify the peak value ($I_0$ or $V_0$) and period ($T$) from the graph.
Step 2: Calculate angular frequency: $\omega = \frac{2\pi}{T}$.
Step 3: Substitute into the general equation.

Average & RMS Values

Average Value

The mathematical average of a sinusoidal AC signal over a complete cycle is zero, because the positive and negative half-cycles cancel exactly. While the average is zero, power is still delivered — this is why average value is not useful for power analysis in AC circuits.

[!example] Analogy: If a basketball bounces up and down, its average height might be zero, but it is still moving and doing work.

Root Mean Square (RMS)

RMS provides the effective value of an AC signal — the equivalent DC value that would deliver the same power to a resistive load.

The RMS process:

  1. Square all instantaneous values (eliminates negative signs)
  2. Take the Mean (average) of those squared values over one cycle
  3. Take the square Root of that mean

For a pure sinusoid:

$$I_{\text{rms}} = \frac{I_{\max}}{\sqrt{2}} \approx 0.707, I_{\max}$$

$$V_{\text{rms}} = \frac{V_{\max}}{\sqrt{2}} \approx 0.707, V_{\max}$$

[!tip] Real-world context

  • Your home power supply is rated at 230V RMS; the actual peak voltage is about 325V.
  • Electric bills are calculated using RMS power consumption (kWh).
  • All AC meters and appliance labels display RMS values.

Power in AC Circuits

For resistive loads, average power is calculated using RMS values:

$$P = V_{\text{rms}} I_{\text{rms}}$$

Given peak values:

$$P = \frac{V_0 I_0}{2}$$

Key Concepts

  • Sinusoidal Waveforms — $v(t) = V_m \sin(\omega t + \phi)$
  • RMS Values — $V_{rms} = \frac{V_m}{\sqrt{2}}$, effective values for power
  • Phasors — rotating vectors representing AC quantities
  • Impedance — $Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0} = \sqrt{R^2 + X^2}$, opposition to AC flow; scalar quantity in ohms ($\Omega$); in DC circuits it behaves like resistance
  • Reactance — opposition to AC flow
    • Capacitive: $X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$
    • Inductive: $X_L = \omega L = 2\pi f L$
  • Phase Angle — phase difference between voltage and current; for series RLC, $\tan\phi = \frac{X_L - X_C}{R}$
    • Positive when inductive ($X_L > X_C$): voltage leads current
    • Negative when capacitive ($X_C > X_L$): current leads voltage
  • Resonance — when $X_L = X_C$, minimum impedance
  • Power Factor — $\cos\phi = \frac{R}{Z}$, ratio of real to apparent power
  • Power Triangle — $S^2 = P^2 + Q^2$, geometric relationship between real, reactive, and apparent power
  • Pure Reactive Elements — inductors and capacitors dissipate zero average power; they only exchange reactive power with the source
  • Power Factor Correction — adding capacitors to partially cancel inductive reactance and raise the power factor toward unity
  • Quality Factor — $Q = \frac{\omega_0 L}{R}$

RLC Series Circuit

In an RLC series circuit, the inductor and capacitor voltages are $180°$ out of phase and partially cancel:

  • Net Reactance: $X = X_L - X_C$
  • Total Voltage: $V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$
  • Circuit Behavior:
    • Inductive dominance ($X_L > X_C$): circuit behaves like RL, voltage leads current
    • Capacitive dominance ($X_C > X_L$): circuit behaves like RC, current leads voltage
    • Resonance ($X_L = X_C$): $Z = R$, purely resistive, $\phi = 0$
  • RL Series Circuit — $Z = \sqrt{R^2 + X_L^2}$, voltage leads current by $\phi = \tan^{-1}(X_L/R)$
  • RC Series Circuit — current leads voltage by negative phase angle
    • Total voltage: $V_T = \sqrt{V_R^2 + V_C^2}$
    • Impedance: $Z = \sqrt{R^2 + X_C^2}$
    • Phase angle: $\phi = \tan^{-1}\left(\frac{-X_C}{R}\right)$ (negative because current leads)
stateDiagram
    [*] --> Compare: Apply AC Source
    state "Compare Reactances<br/>X = X_L - X_C" as Compare
    Compare --> Inductive: X_L > X_C
    Compare --> Capacitive: X_C > X_L
    Compare --> Resonant: X_L = X_C

    state "Inductive Dominance" as Inductive
    Inductive : Voltage leads Current
    Inductive : Circuit behaves like RL

    state "Capacitive Dominance" as Capacitive
    Capacitive : Current leads Voltage
    Capacitive : Circuit behaves like RC

    state "Resonance" as Resonant
    Resonant : Purely resistive
    Resonant : Z = R and phi = 0

Pure Circuits

Analysis of circuits containing only a single type of element (R, C, or L).

Pure Resistive Circuit (PRC)

A pure resistor has no capacitance and no self-inductance.

  • $I = I_0 \sin(\omega t)$ and $V_R = V_0 \sin(\omega t)$
  • Current is in phase with voltage: $\Delta\phi = 0$
  • Impedance: $Z = R$

Pure Capacitive Circuit (PCC)

A pure capacitor has no resistance and no self-inductance.

  • $V_C = V_0 \sin(\omega t)$ and $I = I_0 \sin\left(\omega t + \frac{\pi}{2}\right)$
  • Current leads voltage by $\pi/2$ (or $90°$); equivalently, voltage lags current by $\pi/2$
  • Capacitive reactance: $X_C = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0} = \frac{1}{2\pi f C}$
  • $X_C$ is a scalar quantity with unit ohm ($\Omega$)
  • $X_C \propto \frac{1}{f}$ — inversely proportional to frequency

Pure Inductive Circuit (PLC)

A pure inductor has no resistance and no capacitance.

  • $V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$ and $I = I_0 \sin(\omega t)$
  • Voltage leads current by $\pi/2$ (or $90°$); equivalently, current lags voltage by $\pi/2$
  • Inductive reactance: $X_L = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0} = 2\pi f L$
  • $X_L$ is a scalar quantity with unit ohm ($\Omega$)
  • $X_L \propto f$ — directly proportional to frequency

CIVIL Mnemonic

A memory aid for remembering which quantity leads:

  • C (Capacitor): I leads V
  • L (Inductor): V leads I (or I lags V)
graph TB
    subgraph elements["Pure AC Circuit Elements"]
        R["Pure Resistor"]
        C["Pure Capacitor"]
        L["Pure Inductor"]
    end

    R --> RV["Current and Voltage are in phase"]
    C --> CV["Current leads Voltage by 90°"]
    L --> LV["Voltage leads Current by 90°"]

    subgraph mnemonic["CIVIL Mnemonic"]
        CM["Capacitor: Current leads Voltage"]
        LM["Inductor: Voltage leads Current"]
    end

    C -.-> CM
    L -.-> LM

    classDef element fill:#e7f5ff,stroke:#1971c2,stroke-width:2px
    classDef phase fill:#fff4e6,stroke:#e67700,stroke-width:2px
    classDef memory fill:#e5dbff,stroke:#5f3dc4,stroke-width:2px
    class R,C,L element
    class RV,CV,LV phase
    class CM,LM memory

Phasor Diagrams

A phasor diagram represents an AC quantity as a rotating vector. The phasor rotates anticlockwise at angular velocity $\omega$ from the positive x-axis. The vertical projection of the phasor onto the time axis traces the corresponding sinusoidal waveform.

Phase Angle & Phase Shift

The phase angle $\phi$ describes the horizontal shift of a sine wave relative to a reference:

Condition Equation Sign
In-phase $A(t) = A_m \sin(\omega t)$ $\phi = 0^\circ$
Lead (left shift) $A(t) = A_m \sin(\omega t + \phi)$ positive $\phi$
Lag (right shift) $A(t) = A_m \sin(\omega t - \phi)$ negative $\phi$

[!note] Sign Convention LEFT is positive (+$\phi$) → LEAD RIGHT is negative (-$\phi$) → LAG

Leading & Lagging

  • A signal leads when it reaches its peak or zero-crossing earlier than the reference signal.
  • A signal lags when it reaches its peak or zero-crossing later than the reference signal.
  • In AC circuit analysis, the phase angle is often defined as the phase difference between voltage and current.

Key Formulas

Formula Description
$V_{rms} = \frac{V_m}{\sqrt{2}}$ RMS voltage
$I_{rms} = \frac{I_m}{\sqrt{2}}$ RMS current
$Z = \frac{V_{rms}}{I_{rms}} = \frac{V_0}{I_0}$ Impedance (general definition)
$Z = R$ Impedance in pure resistor
$Z = X_C$ Impedance in pure capacitor
$Z = X_L$ Impedance in pure inductor
$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$ Capacitive reactance
$X_L = \omega L = 2\pi f L$ Inductive reactance
$X = X_L - X_C$ Net reactance (RLC series)
$Z = \sqrt{R^2 + (X_L - X_C)^2}$ Series RLC impedance
$\tan\phi = \frac{X_L - X_C}{R}$ Phase angle (positive = inductive)
$Z_{RL} = \sqrt{R^2 + X_L^2}$ Series RL impedance
$\tan\phi = \frac{X_L}{R}$ RL phase angle (voltage leads)
$V_T = \sqrt{V_R^2 + V_L^2}$ Total voltage in RL series
$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$ Total voltage in RLC series
$f_0 = \frac{1}{2\pi\sqrt{LC}}$ Resonant frequency
$P_{avg} = V_{rms}I_{rms}\cos\phi$ Average power
$S = V_{rms}I_{rms}$ Apparent power
$Q = V_{rms}I_{rms}\sin\phi$ Reactive power
$S = \sqrt{P^2 + Q^2}$ Apparent power (power triangle)
$PF = \frac{R}{Z} = \cos\phi$ Power factor

Related Concepts

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