Say v v v v = V m sin ( ω t + ϕ ) v=V_{m}\sin(\omega{t}+\phi) v = V m sin ( ω t + ϕ )
Production of AC is less expensive
AC devices are efficient and require less maintenance
Maximum instantaneous value. V m V_m V m
Maximum variation between maximum positive and negative instantaneous values.
For a sinusoidal waveform, this is twice the peak value. 2 V m 2V_m 2 V m
v mean = 1 T ∫ T 0 T 0 + T v ( t ) d t v_{\text{mean}}=
\frac{1}{T}
\int_{T_0}^{T_0+T}{v(t)\text{d}t}
v mean = T 1 ∫ T 0 T 0 + T v ( t ) d t Here:
T 0 T_0 T 0 T T T
For any symmetric waveform, mean value is 0 0 0
Mean value of the rectified version of a waveform.
For symmetric waveforms, half-cycle mean value is taken as the average value.
v average = 2 T ∫ T 0 T 0 + T 2 v ( t ) d t v_{\text{average}}=
\frac{2}{T}
\int_{T_0}^{T_0+\frac{T}{2}}{v(t)\,\text{d}t}
v average = T 2 ∫ T 0 T 0 + 2 T v ( t ) d t For sinusoidal waveforms, from the example:
v average = 2 T ∫ T 0 T 0 + T 2 V m sin ( ω t + ϕ ) d t v_{\text{average}}
=
\frac{2}{T}
\int_{T_0}^{T_0+\frac{T}{2}}{V_{m}\sin(\omega{t}+\phi)\,\text{d}t}
v average = T 2 ∫ T 0 T 0 + 2 T V m sin ( ω t + ϕ ) d t = 2 π V m = 0.637 V m =
\frac{2}{\pi}V_m
=
0.637V_m
= π 2 V m = 0.637 V m Aka. effective value. rms value is always used to express the magnitude of a
time varying quantity.
v rms = 1 T ∫ T 0 T 0 + T v ( t ) 2 d t v_{\text{rms}}=
\sqrt{
\frac{1}{T}
\int_{T_0}^{T_0+T}{v(t)^2\,\text{d}t}
}
v rms = T 1 ∫ T 0 T 0 + T v ( t ) 2 d t For sinusoidal waveforms:
v rms = V m 1 T ∫ T 0 T 0 + T sin 2 ( ω t + ϕ ) d t = V m 2 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}}
v rms = V m T 1 ∫ T 0 T 0 + T sin 2 ( ω t + ϕ ) d t = 2 V m P = v i = i 2 R P=vi=i^2R
P = v i = i 2 R Form factor = rms value average value = V m 2 × π 2 V m = 1.111 \text{Form factor}
=
\frac{\text{rms value}}{\text{average value}}
=
{\frac{V_m}{\sqrt{2}}}\times{\frac{\pi}{2{V_m}}}
=1.111
Form factor = average value rms value = 2 V m × 2 V m π = 1.111 Peak factor = peak value rms value = V m × 2 V m = 1.412 \text{Peak factor}
=\frac{\text{peak value}}{\text{rms value}}
={V_m}\times{\frac{\sqrt{2}}{V_m}}
=1.412
Peak factor = rms value peak value = V m × V m 2 = 1.412