Vrms of offset AC voltage sources

Question

As you know, if you put two DC voltage sources in series, you can provide a larger voltage to an apillication. For example, two 1.5V A batteries in series provide 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 Vo*sin(wt) and the other looks like Vo*sin(wt + s). s is referred to as a phase offset and is in radians. If you add those sources together, what is the rms voltage of the resulting signal?

Let Vo = 23 V, w = 141 rad/s, and s = 1.15 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.

Answer 1

The equation for the voltage of the first AC source is Vosin(wt), and the equation for the voltage of the second AC source is Vosin(wt + s). Adding these together, we get:

V = Vosin(wt) + Vosin(wt + s)

Using the trigonometric identity for the sum of two sines, we can simplify this expression:

V = 2Vosin((wt + wt + s)/2)*cos((wt - wt - s)/2)

V = 2Vosin((2wt + s)/2)*cos(s/2)

V = 2Vosin(wt + s/2)*cos(s/2)

To find the rms voltage of this signal, we need to take the square root of the average of the squared voltage over one period:

Vrms = sqrt((1/T)integral(V^2dt))

The period T of the signal is 2*pi/w, so we have:

Vrms = sqrt((1/(2*pi/w))integral(V^2dt))

Vrms = sqrt((w/2pi)integral((2Vosin(wt + s/2)cos(s/2))^2dt))

Vrms = sqrt((w/2pi)integral(4Vo^2sin^2(wt + s/2)*cos^2(s/2)*dt))

Vrms = sqrt((w/2pi)integral(2Vo^2(1 - cos(2wt + s))*cos^2(s/2)*dt))

Vrms = sqrt((w/2pi)(2Vo^2(t - (1/(2*w))*sin(2wt + s)*cos^2(s/2))|0 to T))

Vrms = sqrt((w/2pi)(2Vo^2T - 2Vo^2(1/(2*w))sin(2pi + s)*cos^2(s/2)))

Since sin(2*pi + s) = sin(s), this simplifies to:

Vrms = sqrt(Vo^2*cos^2(s/2))

Plugging in the given values for Vo and s, we have:

Vrms = sqrt((23 V)^2*cos^2(1.15/2))

Vrms = 20.97 V (rounded to two decimal places)

Therefore, the rms voltage of the resulting signal is approximately 20.97 V.