Ac Circuit

Reference data and engineering information about ac circuit for electrical applications.

Overview

Engineering reference data for Ac Circuit in electrical engineering.

Key Formulas

Power

$P = VI = I^2R = V^2/R$

Electrical power.

Variables

Symbol	Description	Unit
$V$	Voltage	V
$I$	Current	A
$R$	Resistance	Ω
$P$	Power	W

Phasor Representation

In AC circuit analysis, sinusoidal signals can be represented in phasor form for simplified calculation:

Voltage Phasor: $\underline{U} = U_{max} e^{j\theta}$

Current Phasor: $\underline{I} = I_{max} e^{j\theta}$

where $j$ is the imaginary unit, and the phasor magnitude represents the peak amplitude while the angle represents the phase shift relative to a cosine reference.

Impedance

Impedance $Z$ extends Ohm's law to AC circuits, acting as a complex, frequency-dependent resistance:

Ohm's Law for AC: $\underline{U} = \underline{I} Z$

Impedance of Components:

Resistor: $Z_R = R$
Inductor: $Z_L = j\omega L$
Capacitor: $Z_C = \frac{1}{j\omega C}$

Series Impedance: $Z_{total} = Z_1 + Z_2$

Parallel Impedance: $\frac{1}{Z_{total}} = \frac{1}{Z_1} + \frac{1}{Z_2}$

Admittance ( $Y$ ) is the inverse of impedance: $Y = \frac{1}{Z}$ (Units: siemens, S, or 1/Ω)

RMS (Effective) Values

Standard AC instruments measure the RMS (Root Mean Square) value, which is the equivalent DC value that delivers the same power.

RMS Voltage: $U_{rms} = \frac{U_{max}}{\sqrt{2}} \approx 0.707 \, U_{max}$

RMS Current: $I_{rms} = \frac{I_{max}}{\sqrt{2}} \approx 0.707 \, I_{max}$

The peak value is $\sqrt{2}$ (≈1.41) times the RMS value. For example, a 230 V AC supply has a peak voltage of about 325 V.

Phase Relationships in Loads

The phase difference $\phi$ between voltage and current defines the load character:

Purely Resistive Load: Voltage and current are in phase ( $\phi = 0$ ).
Purely Inductive Load: Current lags voltage by 90° ( $\phi = +90°$ ).
Purely Capacitive Load: Current leads voltage by 90° ( $\phi = -90°$ ).

For a real circuit with mixed loads, the phase angle $\phi$ lies between -90° and +90°.

Three-Phase Systems

Three-phase systems deliver power via three sinusoidal voltages, phase-shifted by 120°.

Common Voltage Levels:

Europe / IEC: 400 V (line-to-line) / 230 V (line-to-neutral)
North America: 208 V (line-to-line) / 120 V (line-to-neutral)

Key Terms:

L1, L2, L3: The three individual phase lines, each providing a phase-to-neutral (phase) potential.
Line-to-Line: The voltage between any two phase lines (e.g., L1-L2). This is $\sqrt{3}$ times the line-to-neutral voltage in a balanced system.
Resultant Neutral Point: In a balanced three-phase system, the vector sum of the three phase voltages is zero, establishing the neutral point.

Phasor representation uses complex numbers in polar form where the magnitude equals the peak amplitude and the phase angle equals the phase shift relative to a cosine reference. The specific angular frequency ω is not explicitly included in the phasor expression since the system operates at a single frequency.

Common AC System Voltages

The following table provides reference values for common residential and commercial AC voltage systems: