Explain what are meant by phase and phase difference of sinusoidal waves..
Phase:
Phase of an alternating quantity at any instant is the time that has elapsed since the instantaneous value of the quantity assumed zero value was going from negative to positive direction. This time interval is measured in times of the fraction of the time period of the waveform. The phase \(\mathrm{\Phi}=nT\), where n represents the fraction.
Fig. 7 Representation phase of a sinusoidal current source
The phase is also measured in terms of angle \(\mathrm{\Phi}=\omega t=\frac{2\pi}{T}t\). For example the phase of current at point A as shown in Fig. 7 is T/4 in terms of fraction of time period T or \(\mathrm{\Phi}=\omega t=\frac{2\pi}{T}\times\frac{T}{4}=\pi/2\)
Phase difference:
The phase difference between two waves of same frequency is the difference in their phases at any instant of time. When two waves having the same frequency attain their zeros and peak values simultaneously, i.e., at the same instant of time, the waves are said to be in same phase or their phase difference is zero. If the waves attain the zeros and maximum values at different instants of time then the waves are said to have a phase difference. If either the zero or the peak of first wave occurs earlier in time with respect to the peak of the second wave then the first wave is said to lead in phase with respect to second wave and the second wave is said to lag behind in phase with respect to the first wave. The phase lead and lag performance of a sinusoidal waveform is given in Fig. 8. Here four waves namely A, B, C and D respectively is in phase lead and lag condition. The waveform C and D are phase lagging by \(\theta\) and Φ with respect to the wave B. Similarly the wave A is lead by phase angle \(\alpha\) with respect to the wave B.
Let us consider the sinusoidal current wave B is
\(i_B=I_m\sin{\omega t}\)
The waveform A, C and D can be represented as
\(i_C=I_m\sin{(\omega t-\theta)}\)
\(i_D=I_m\sin{(\omega t-\mathrm{\Phi})}\)
And
\(i_A=I_m\sin{(\omega t+\alpha)}\)
Fig. 8 three sinusoidal wave in phase lag and lead condition
Draw a mathematical expression for RMS value of a sinusoidal voltage \(v=V_{m}\sin{{\omega t}}\).
Explain
Find the current in each branch of the network using Kirchhoff’s law
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Fig. 15(a)