Phase difference is also used to compare two different currents

Phase difference is also used to compare two different currents or a current and a voltage. If the
phase difference between two currents, two voltages, or a voltage and a current is zero degrees,
they are said to be "in-phase." If the phase difference is an amount other than zero, they are said
to be "out-of-phase."
Voltage Calculations
Equation (7-9) is a mathematical representation of the voltage associated with any particular
orientation of a coil (inductor).
e E (7-9) max sinq
where
e = induced EMF

Emax = maximum induced EMF
q = angle from reference (degrees or radians)
Example 1: What is the induced EMF in a coil producing a maximum EMF of 120 V when
the angle from reference is 45°?
e = Emax sin q
e = 120 V (sin 45°)
e = 84.84 V
The maximum induced voltage can also be called peak voltage Ep. If (t) is the time in which the
coil turns through the angle (q), then the angular velocity (w) of the coil is equal to q/t and is
expressed in units of radians/sec. Equation (7-10) is the mathematical representation of the
angular velocity.
q wt (7-10)
where
w = angular velocity (radians/sec)
t = time to turn through the angle from reference (sec)
q = angle from reference (radians)
Using substitution laws, a relationship between the voltage induced, the maximum induced
voltage, and the angular velocity can be expressed. Equation (7-11) is the mathematical
representation of the relationship between the voltage induced, the maximum voltage, and the
angular velocity, and is equal to the output of an AC Generator.
e E (7-11) max sin (wt)
ES-07 Page 8 Rev. 0
Basic AC Theory AC GENERATION ANALYSIS
where
e = induced EMF (volts)
Emax = maximum induced EMF (volts)
w = angular velocity (radians/sec)
t = time to turn through the angle from reference (sec)
Current Calculations
Maximum induced current is calculated in a similar fashion. Equation (7-12) is a mathematical
representation of the relationship between the maximum induced current and the angular velocity.
i I (7-12) max sin (wt)
where
i = induced current (amps)
Imax = maximum induced current (amps)
w = angular velocity (radians/sec)
t = time to turn through the angle from reference (sec)
Frequency Calculations
The frequency of an alternating voltage or current can be related directly to the angular velocity
of a rotating coil. The units of angular velocity are radians per second, and 2p radians is a full
revolution. A radian is an angle that subtends an arc equal to the radius of a circle. One radian
equals 57.3 degrees. One cycle of the sine wave is generated when the coil rotates 2p radians.
Equation (7-13) is the mathematical relationship between frequency (f) and the angular velocity
(w) in an AC circuit.
w 2pf (7-13)
where
w = angular velocity (radians/sec)
f = frequency (HZ)