As you know, if you put two DC voltage sources in series, you can provide a larg

Question

As you know, if you put two DC voltage sources in series, you can provide a larger voltage to an application. For example, two 1.5V AA Batteries in series provides 3V. You can do the same thing with AC voltage, though the situation is more complicated. Suppose you have two AC voltage sources that have the same amplitude and frequency, but are out of phase with one another. In mathematical terms, one of the voltage functions looks like V_0 middot sin (omega t + delta). delta is referred to as a phase offset and is in radians. If you add those sources together, what's the rms voltage of the resulting signal? Let V_0 = 12 V, omega = 124 rad/s, and delta = 0.95 rad. You'll probably need to work from the definition of root-mean-square. You can get a short, tidy algebraic answer by using trig identities, or you may use a computer to grind out any difficult Integrals. Whatever you prefer.

Explanation / Answer

given V1 = Vosin(wt)
V2 = Vosin(wt + del)
so, V1 + V2 = Vo(sin(wt) + sin(wt+del))
but sinA + sinB = 2sin([A+B]/2)cos([A-B]/2)
so, V1 + V2 = Vo(sin(wt) + sin(wt+del)) = 2Vo(sin([2wt + del]/2)cos(-del/2)) = 2Vo(sin(wt + del/2)cos(del/2))
so amplitude oif this voltage is 2Vocos(del/2)
now given VO = 12 V
del = 0.95 rad = 54.43 deg
so, Voltage amplitudfer, V = 2Vocos(del/2) = 2*12*cos(54.43/2) = 21.34 VRMS value of this voltage = Peak voltage/sqroot(2) = 15.09 V

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