Alternating Current — Question 4

Question 4

An asymmetric two-level waveform can have zero mean without equal amplitudes

In every period, a source is +A for fraction d and -B for fraction $1-d$, with positive A and B. Require zero mean. Let A=12.0 V and d=1/4.

1. Find B and the rms voltage.

2. Find the ratio of rms voltage to the positive amplitude for general d.

Question 4 - Solution

An asymmetric two-level waveform can have zero mean without equal amplitudes

(a) Method and calculation

$$dA-(1-d)B=0\Rightarrow B=dA/(1-d)=4.00\,V.$$$$V_{\mathrm{rms}}^2=dA^2+(1-d)B^2\Rightarrow\boxed{V_{\mathrm{rms}}=\sqrt{48}=6.92820\,V}.$$

(b) Interpretation and further calculation

$$V_{\mathrm{rms}}^2=A^2\left[d+\frac{d^2}{1-d}\right]=A^2\frac d{1-d}.$$$$\boxed{V_{\mathrm{rms}}/A=\sqrt{d/(1-d)}}.$$

Physical check

Zero mean constrains areas under the waveform, not equal time above and below zero.