Question

Question (10 points) Find the complex power supplied by the source and the power factor State whether the power factori ding o n t: RMS voltages are not shown with cosine 0.5 H 50 cos(101 + 30°) (+ 10 F

Answer 1

Complex power:-

According to laws of basic circuits, for the given R, L, C circuit, if any voltage source or current source is present, the source will act as an active element and all other circuit element (R, L,C) will act as a passive elements. And also, the total power delivered by all the sources is equal to the total power received by all the passive elements in the circuit.

For a given above circuit, if $V = |V| \angle \alpha$ is the rms voltage and $I = |I| \angle \beta$ is the rms current in the circuit, then the complex power of the voltage source supplied is defined by the formula $S =V I ^* = \frac{1} {2} |V_m| |I_m| \angle( \alpha - \beta) ^o$ ​​​​​​ ​​​​​​, where I*  is the complex conjugate of the current flowing in the circuit, also the relation between rms and peak terms is given by $V = \frac {V_m} { \sqrt {2} }, I = \frac{I_m} { \sqrt{2}}$ .

Given voltage source Vs =50 cos(10t+30°), where the circuit has frequency $\omega = 10 rad/sec$

Procedure: So, the given circuit contains the voltage source. We first convert the time domain circuit into phasor domain and calculate the overall impedance across the voltage source. Then after, determine the current through the circuit by the ohm's law. Then, we determine the complex power delivered by the voltage source by the above mentioned formula. The below steps of simplifications determine the complex power:-

0.5H 10-12 + lor © jour Socesot+30° So Lest tott 30°) , T v Y50430° 50/30 om 0.5H Real Zeyna (sj 10) + (-2) = lotsj) -24 103 -44+2 2th - 10 toj 2764 2-5 24 h " 2-j. *- = (2+21)2 Zero 2 (1+ y)2 = 2.82 _45° 2 From Ohmis law, » Nm = I'm 50/30 Zed Riable hear * Im = m . Lea Equivalent Circuit 50 Lov 282 245²2 Im = 17.08 L-15° ^] Complese power s= 2 Yes In sa ! Vendit.Imd 2$, Vm = 11ml L*, Im = 1 Im 12B, (0=1-B)

- Inne 17-68 215° A S = von In* c* * 50x17.68 %(30° +15°) = S = 442 L450VA

Power factor:

Power factor is defined as the ratio between real power and apparent power of the circuit, where the powet factor can be calculated using the phase angle calculated while determining the complex power S for the given voltage source. So, the power factor $p. f = \cos \phi , \phi = \alpha - \beta$ ​​​​​

The power factor is calculated as determined in the below image:-

If s= NA Loo is complex power then. power factor = cos o From the above result, s= VA Loo E 442 1450 VA comparing with the formula, 0 = 45° => cos $ = cos 45° = = 0.707. ... Power factor = cos $ = 0.7077

The power factor is said to be lagging because the phase angle is positive and the lagging power factor means the load is inductive and current lags voltage by a phase of 45°.