AC Theory | Sahithyan's Notes

Say $v$ is alternating as in $v=V_{m}\sin(\omega{t}+\phi)$ .

Why AC Instead of DC?

Production of AC is less expensive
AC devices are efficient and require less maintenance

Peak Value

Maximum instantaneous value. $V_m$ in the example.

Peak-to-Peak Value

Maximum variation between maximum positive and negative instantaneous values. For a sinusoidal waveform, this is twice the peak value. $2V_m$ in the example.

Mean Value

v_{\text{mean}}= \frac{1}{T} \int_{T_0}^{T_0+T}{v(t)\text{d}t}

Here:

$T_0$: the starting time of a cycle
$T$: the periodic time

For any symmetric waveform, mean value is $0$ .

Average Value

Mean value of the rectified version of a waveform.

For symmetric waveforms, half-cycle mean value is taken as the average value.

v_{\text{average}}= \frac{2}{T} \int_{T_0}^{T_0+\frac{T}{2}}{v(t)\,\text{d}t}

For sinusoidal waveforms, from the example:

v_{\text{average}} = \frac{2}{T} \int_{T_0}^{T_0+\frac{T}{2}}{V_{m}\sin(\omega{t}+\phi)\,\text{d}t}

= \frac{2}{\pi}V_m = 0.637V_m

rms Value

Also known as effective value. rms value is always used to express the magnitude of a time varying quantity.

v_{\text{rms}}= \sqrt{ \frac{1}{T} \int_{T_0}^{T_0+T}{v(t)^2\,\text{d}t} }

For sinusoidal waveforms:

v_{\text{rms}}= V_m \sqrt{ \frac{1}{T} \int_{T_0}^{T_0+T}{\sin^2{(\omega{t}+\phi)}\,\text{d}t} } = \frac{V_m}{\sqrt{2}}

Instantaneous Power

P=vi=i^2R

Form Factor

\text{Form factor} = \frac{\text{rms value}}{\text{average value}} = {\frac{V_m}{\sqrt{2}}}\times{\frac{\pi}{2{V_m}}} =1.111

Peak Factor

\text{Peak factor} =\frac{\text{peak value}}{\text{rms value}} ={V_m}\times{\frac{\sqrt{2}}{V_m}} =1.412