# AC Circuits - Power vs. Voltage and Current

In an AC circuit - alternating current is generated from a sinusoidal voltage source

### Voltage

Currents in circuits with ** pure ** resistive , capacitive ** or ** inductive loads.

The momentary voltage in an sinusoidal AC circuit can be expressed on the ** time-domain form ** as

* u(t) = U _{ max } cos(ω t + θ) (1) *

* where *

* u(t) = voltage in the circuit at time t (V) *

* U _{ max } = maximal voltage at the amplitude of the sinusoidal wave (V) *

* t = time (s) *

* ω = 2 π f *

* = angular frequency of sinusoidal wave ( rad /s) *

* f = frequency (Hz, 1/s) *

* θ = phase shift of the sinusoidal wave (rad) *

The momentary voltage can alternatively be expressed in the ** frequency-domain (or phasor) form ** as

* U = U(jω) = U _{ max } e ^{ jθ } (1a) *

* where *

* U(jω) = U = complex voltage (V) *

A phasor is a complex number expressed in polar form consisting of a magnitude equal to the peak amplitude of the sinusoidal signal and a phase angle equal to the phase shift of the sinusoidal signal with reference to a cosine signal.

Note that the specific angular frequency - * ω * - is not explicitly used in the phasor expression.

### Current

The momentary current can be expressed can be expressed in the ** time-domain form ** as

* i(t) = I _{m} cos(ω t + θ) (2) *

* where *

* i(t) = current at time t (A) *

* I _{ max } = maximal current at the amplitude of the sinusoidal wave (A) *

Currents in circuits with ** pure ** resistive, capacitive ** or ** inductive loads are indicated in the figure above. The current in a "real" circuit with resistive, inductive ** and ** capacitive loads are indicated in the figure below.

The momentary current in an AC circuit can alternatively be expressed in the ** frequency-domain (or phasor) form ** as

* I = I(jω) = I _{ max } e^{jθ } (2a) *

* where *

* I = I(jω) = complex current (A) *

### Frequency

Note that the frequency of most AC systems are fixed - like * 60 Hz * in North America and * 50 Hz * in most of the rest of the world.

The angular frequency for North America is

* ω = 2 π 60 *

* = 377 rad/s *

The angular frequency for most of the rest of the world is

* ω = 2 π 50 *

* = 314 rad/s *

### Resistive Load

The voltage over a resistive load in an AC system can be expressed as

* U = R I (4) *

* where *

* R = resistance (ohm) *

For a resistance load in an AC circuit the voltage is ** in phase ** with the current.

### Inductive Load

The voltage over an inductive load in an AC system can be expressed as

* U = j ω L I (5) *

* where *

* L = inductance (henry) *

For an inductive load the current in an AC circuit is * π/2 (90^{o}) *

**phase after**the voltage (or voltage before the current).

### Capacitive Load

The voltage over an inductive load in an AC system can be expressed as

* U = 1 / (j ω C) I (6) *

* where *

* C = capacitance (farad) *

For a capacitive load the current in an AC circuit leads the voltage by * π/2 (90^{o}) *

**phase**.

In a real electrical circuit there is a mix of resistive, capacitive and inductive loads with a voltage/current phase shift in the range - * π/2 * <= * φ <= π/2 * as illustrated in the figure below.

The current in a "real" circuit with a mix of resistive, inductive ** and ** capacitive loads. φ is the phase angle between the current and the voltage.

### Impedance

Ohm's law for complex alternating current can be expressed as

* U _{ z } = I _{ z } Z (7) *

* where *

* U _{ z } = voltage drop over the load (volts, V) *

* I _{ z } = current through the load (ampere, A) *

* Z = impedance of the load (ohms, Ω) *

The impedance in an AC circuit can be regarded as complex resistance. The impedance acts as a frequency dependent resistor where the resistance is a function of the frequency of the sinusoidal excitation.

#### Impedances in Serie

The resulting impedance for impedances in series can be expressed as

* Z = Z _{1} + Z_{2} (7b) *

#### Impedances in Parallel

The resulting impedance for impedances in parallel can be expressed as

* 1 / Z = 1 / Z _{1} + 1 / Z_{2} (7c) *

#### Admittance

Admittance is the inverted impedance

* Y = 1 / Z (8) *

* where *

* Y = admittance (1/ohm) *

### RMS or Effective Voltage

The RMS value is the effective value of a sinusoidal voltage or current.

RMS - Root Mean Square - or effective voltage can be expressed as

* U _{ rms } = U _{ eff } *

* = U _{ max } / (2)^{1/2 } *

* = 0.707 U _{ max } (9) *

* where *

* U _{ rms } = U _{ eff } *

* = RMS voltage (V) *

* U _{ max } = maximum voltage (amplitude) of sinusoidal voltage source (V) *

RMS - Root Mean Square - or effective current can be expressed as

* I _{ rms } = I _{ eff } *

* = I _{ max } / (2)^{1/2 } *

* = 0.707 I _{ max }(10) *

* where *

* I _{ rms } = I _{ eff } *

* = RMS current (A) *

* I _{ max } = maximum current (amplitude) of sinusoidal voltage source (A) *

AC voltmeters and ammeters shows the RMS value of the voltage or current - or 0.707 times the max peak values. The max peak values are 1.41 times the voltmeter values.

Example

- for a
*230V*system the*U*and_{ rms }= 230V*U*_{ max }= 324 V - for a
*120V*system the*U*and_{ rms }= 120V*U*_{ max }= 169 V

### AC Three Phase Voltage - Line to Line and Line to Neutral

In an AC three phase system the voltage can be delivered between the lines and the neutral (phase potential), or between the lines (line potential). The resulting voltages for two common systems - the European 400/230V and the North American 208/120V system are indicated for one period in the figures below.

#### 400V/230V AC

print 400/230V Three Phase Diagram

*L1, L2 and L3*are the three phases line to to neutral potentials -*phase potentials**L1 to L2, L1 to L3 and L2 to L3*are the three phases line to line potentials -*line potentials**L2, L2 and L3*is the resulting potential of the tree phases in a balanced circuit -*resulting potential = 0*

The magnitude of the line potentials is equal to * 3 ^{ 1/2 } (1.73) * the magnitude of the phase potential.

* U _{ rms, line } = 1.73 U _{ rms, phase } (11) *

#### 208V/120V AC

print 208/120V Three Phase Diagram

### Power

Active - or real or true - power that do the actual work in the circuit - can be calculated as

* P = U _{ rms } I _{ rms } cos φ (12) *

* where *

* P = active real power (W) *

* φ = the phase angle between the current and the voltage (rad, degrees) *

Cos φ is also called the Power Factor.

Reactive power in the circuit can be calculated as

* Q = U _{ rms } I _{ rms } sin φ (13) *

* Q = reactive power (VAR) *

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