Alternating Current — Question 7

Question 7

Equal-frequency sources combine through phase as well as amplitude

Two ideal series sources are $v_1=10\cos\omega t$ V and $v_2=10\cos(\omega t+2\pi/3)$ V, with additive voltage orientation.

1. Express their sum as a single sinusoid and find its rms value.

2. Find the rms sum for a general phase separation phi and state the cancellation condition.

Question 7 - Solution

Equal-frequency sources combine through phase as well as amplitude

(a) Method and calculation

$$\cos a+\cos b=2\cos[(a-b)/2]\cos[(a+b)/2].$$$$\boxed{v_1+v_2=10\cos(\omega t+\pi/3)\,V,\quad V_{\mathrm{rms}}=5\sqrt2\,V}.$$

(b) Interpretation and further calculation

$$V_{{\mathrm{sum}},p}=2V_p|\cos(\phi/2)|.$$$$\boxed{V_{{\mathrm{sum}},\mathrm{rms}}=\sqrt2V_p|\cos(\phi/2)|;\quad\phi=\pi\Rightarrow0}.$$

Physical check

Adding rms magnitudes arithmetically would miss destructive interference between sources.