Theme 5 Alternating Current and Voltage

Alternating Voltage Acts in alternate directions periodically. Alternating voltage generated usually by a rotating machine called an alternator – in a sinusoidal manner. Otherwise Alternating voltages can be of rectangular type, triangular…… Main interest is of a sinusoidal waveform- utility supply

Can be easily stepped up or down using a transformer. Numerous devices in industry use alternatin voltage/current to operate. E.g. an induction motor.

V max - V max time 2π2π 0.5ππ 1.5 π T

Mathematically, a sinusoid that’s integrated or differentiated- results in a sinusoid of the same frequency: Summing different sinusoids of the same frequency and different amplitudes results in a sinusoid of the same frequency. A cosine wave is just a sine wave with shifted phase

Phase shifted sinusoidal waveforms: B Lags A by angle θ Or, A Leads B by angle θ θ A B

Phase shifted sinusoidal waveforms: B Leads A by angle θ Or, A Lags B by angle θ θ A B

Average Value of a Sinusoid Wave t 1= 0 π=t 2 True Average value=0 Finite average value for half the sinusoid wave can be found= average value of a sinusoid

Average Value of a Sinusoid Wave t 1= 0 π=t 2

RMS Value of a Sinusoidal Waveform If a resistor is connected across a sinusoidal voltage source, a sinusoidal current will flow in the resistor. The RMS value/effective Value is the current that produces the same heating effect as a direct current flowing…i.e.

Average power dissipated by the resistor over a time T is;

F RMS is the RMS of any function f(t)

Obtaining I RMS for a sinusoidal current waveform t 1= 0 t2=2πt2=2π I time

t 1= 0 t2=2πt2=2π I time

Similarly, V RMS is;

Crest/Peak Factor for a sinusoidal waveform

Form Factor of a Sinusoid

OPERATOR j Alternating current or voltage is a vector quantity But instantaneous values are constantly changing with time Thus it can be represented by a ‘rotating’ phasor- which rotates are a constant angular velocity

t 1= 0 t2=2πt2=2π I time Phasor Rotation Angular momentum =ώt j r ImIm ImIm ImIm 0 deg

j-operator 90 degrees 270 degrees 180 degrees0 degrees An operator which turns a phasor by 90 degrees

Polar and Rectangular form Polar Form Rectangular Form

Phasor Algebra Algebraic operations same as complex number manipulations Rectangular Form

Converting from rectangular to polar

Polar Algebraic Manipulations

Assignment Example 7.7 Example 7.8 Problems 7.3, 7.4, 7.7, 7.8, 7.19, 7.20 (page )