The voltage 'V' and current 'A' across a load are as follows. V(t) = 100 sin(ωt) i(t) = 10 sin(ωt - 60°) + 2 sin(3ωt) + 5 sin(5ωt) The average power consumed by the load, in W, is___________.
Instantaneous voltage, v(t) = 100 sin (ωt)
Instantaneous current, i(t) = 10 sin (ω t – 60°) + 2 sin (3 ωt) + 5 sin (5 ωt)
\(Average\;Power = \frac{1}{T}\mathop \smallint \nolimits_0^T v\left( t \right).i\left( t \right)dt\)
\(= \frac{1}{T}\left[ {\mathop \smallint \nolimits_0^T \left\{ {\left( {100\sin \left( {\omega t} \right)} \right).\left( {10\sin \left( {\omega t} \right) - 60^\circ } \right) + \left( {2\sin \left( {3\omega t} \right) + 5\sin \left( {5\omega t} \right)} \right)} \right\}dt} \right]\)
\(= \frac{1}{T}\left[ {\mathop \smallint \nolimits_0^T \left( {100\sin \left( {\omega t} \right)} \right).\left[ {10\sin \left( {\omega t - 60^\circ } \right)} \right]dt + 0 + 0} \right]\)
(∵ In case of average power multiplication of different frequency component is zero)
\(= \frac{1}{{2T}}\left[ {\mathop \smallint \nolimits_0^T 1000\left( {\cos \left( {\omega t - \omega t} \right) + 60^\circ } \right) - \cos \left( {\omega t + \omega t - 60^\circ } \right)dt} \right]\)
[∵ 2 sin A. sin B = cos (A - B) – cos (A + B)]
\(= \frac{1}{{2T}}\left[ {\mathop \smallint \nolimits_0^T 1000\left[ {\cos 60^\circ - \cos \left( {2\omega t - 60^\circ } \right)} \right]dt} \right]\)
\(= \frac{1}{{2T}}\left[ {\mathop \smallint \nolimits_0^T 1000\left( {\frac{1}{2} - 0} \right)dt} \right]\)
[∵ integration of cosine function over a complete cycle is zero]
∴ Average power = 250 watt
Common mistake:
Generally students use formulae Pavg = Vrms.Irms which is not correct when the phase difference is given. The correct formulae for calculating active power is Pavg = Vrms.Irms cos ϕ
Whereas apparent power, S = Vrms.Irms
Important Questions from RMS Value of Time Varying Waveforms
RMS value is defined based on which of the following?
Which of the following factor have value of 1.1 for sinusoidal alternating current only?
xR and xA are, respectively, the rms and average values of x(t) = x(t - T), and similarly, yR and yA are, respectively, the rms and average values of y(t) = kx(t), k, T are independent of t. Which of the following is true?
The rms value of a sinusoidal ac current is numerically equal to its value at an angle of _______ degrees
A resistor connected to a DC supply of 20 V produces the same heating effect as an AC supply connected across the same resistor. What is the RMS value of the AC voltage?
