Alternating Current — Question 3

Question 3

A DC offset adds in quadrature to the sinusoidal rms contribution

A voltage is $v(t)=6.00+8.00\cos\omega t$ volts and is applied to a 10.0-ohm resistor.

1. Find its mean, total rms, and AC-only rms values.

2. Find mean resistor power and determine whether the current reverses.

Question 3 - Solution

A DC offset adds in quadrature to the sinusoidal rms contribution

(a) Method and calculation

$$\langle v\rangle=6.00\,V,\quad\langle v^2\rangle=6^2+8^2/2=68\,V^2.$$$$\boxed{V_{\mathrm{rms}}=\sqrt{68}=8.24621\,V,\quad V_{{\mathrm{ac}},\mathrm{rms}}=8/\sqrt2=5.65685\,V}.$$

(b) Interpretation and further calculation

$$\boxed{\langle P\rangle=68/10=6.80\,W}.$$$$v_{\min}=-2.00\,V<0<v_{\max}=14.0\,V\Rightarrow\boxed{\text{current reverses}}.$$

Physical check

An AC-coupled meter excludes the DC contribution and would understate total resistor heating.