PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

226 Practical MATLAB® Applications for Engineers

3.3 Background

R.3.1 The general form of a sinusoid AC voltage is given by

v(t) = Vm sin(t ± 1 )

or

v(t) = Vm cos(t ± 2 )

where Vm is the maximum or peak value, given in volts, ω the angular velocity or

frequency, given in radians/second, and θ 1 or θ 2 are the phase angles, expressed in

radians or in degrees. (Conversions are sometimes required between radians and

degrees depending on the nature of the problem.)

R.3.2 Sinusoidal waves are periodic functions, which means that the wave repeats itself
indefi nitely. Obviously, it can safely be stated that by knowing one cycle of a sinusoid
wave the nature of the waveform for all and any time is then known.
Recall that one period or cycle of a sinusoid waveform is defi ned by the following
formula:

T

2 

 (expressed in seconds)

R.3.3 The frequency of a sinusoid is the reciprocal of its period, which means that if the
value of the period T is known, then f is defi ned by

f

T

1

(expressed in cycles per second (cps))

where f is the frequency, given in Hertz or cps, and it is related to ω by ω = 2 πf, with

units in radians per second.

R.3.4 A sinusoid AC signal can be expressed either by a sine or a cosine function because

sin()t cos t






2









or

cos()t sin t






2









Note t h at whe n c h a ng i ng a si ne wave i nto a co si ne wave or v ice ver s a, it i s ne ce s s a r y

to change the phase angle by subtracting or adding π/2 (radians) or 90° (degrees).

R.3.5 Recall that sinusoidal waves are related to exponential functions by the Euler’s
identities and also be expressed by a Taylor’s series as follows:

ej

j (^)
cos( ) sin( ) (Euler’s identity)