# AC Circuit - Voltage, Current and Power

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

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 }(3)*

*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 }(3)*

*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.

### 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 (9)*

*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 (9b)*

*Q = reactive power (VAR)*

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