Alternating Circuit (AC Circuit)

varying emf by mechanically rotating a loop of wire in an external magnetic field, a sinusoidally varying electromagnetic force could be obtained in the rotating loop of wire.

Ԑ = ωAB sin ωt

Ԑ = Ԑmax sin ωt

where

Ԑmax = ωAB

ω is the angular speed of rotating coil, A is the cross-sectional area of the coil, B is the magnetic field that the coil is rotated through.

the resulting alternating current in the coil was found to be

i = (Ԑmax /R) sin ωt

ω related to the frequency of the alternating current by

ω = 2∏f

The effective current and voltage in an AC circuit

current, i start from 0 increase in one direction until reaches max value i(max)
decrease to 0 and increase again in the opposite direction
increase negatively to -i(max)

alternating current i as

i = imax sin(2∏ft)

effective current ieff is defined as a constant current that generates heat in a resistor R at the same rate as an alternating current

i =

= 0.707imax

voltage,

V = imax Rsin(2∏ft), imax R = Vmax

hence,

V = Vmax (2∏ft)

Veff =

Alternating current (AC) circuits explained using time and phasor animations. Impedance, phase relations, resonance and RMS quantities.

The first alternator to produce alternating current was a dynamo electric generator based on Michael Faraday's principles constructed by the French instrument maker Hippolyte Pixii in 1832. Pixii later added a commutator to his device to produce the (then) more commonly used direct current. The earliest recorded practical application of alternating current is by Guillaume Duchenne, inventor and developer of electrotherapy. In 1855, he announced that AC was superior to direct current for electrotherapeutic triggering of muscle contractions.

Mathematics of AC voltages

A sine wave, over one cycle (360°). The dashed line represents the root mean square (RMS) value at about 0.707

Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

$v(t)=V_\mathrm{peak}\cdot\sin(\omega t)$ ,

where

$\displaystyle V_{\rm peak}$ is the peak voltage (unit: volt),
$\displaystyle\omega$ is the angular frequency (unit: radians per second)
- The angular frequency is related to the physical frequency, $\displaystyle f$ (unit = hertz), which represents the number of cycles per second, by the equation $\displaystyle\omega = 2\pi f$ .
$\displaystyle t$ is the time (unit: second).

The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of $\sin(x)$ is +1 and the minimum value is −1, an AC voltage swings between $+V_{\rm peak}$ and $-V_{\rm peak}$ . The peak-to-peak voltage, usually written as $V_{\rm pp}$ or $V_{\rm P-P}$ , is therefore $V_{\rm peak} - (-V_{\rm peak}) = 2 V_{\rm peak}$ .

[edit]Power and root mean square

The relationship between voltage and the power delivered is

$p(t) = \frac{v^2(t)}{R}$ where $R$ represents a load resistance.

Rather than using instantaneous power, $p(t)$ , it is more practical to use a time averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a root mean square(RMS) value, written as $V_{\rm rms}$ , because