Alternating Current — Question 1

Question 1

A zero-crossing slope and rms reading determine an AC source

A sinusoidal source has rms voltage 24.0 V. At a rising zero crossing its measured slope is $1.20\times10^4$ V/s. Take that crossing as t=0.

1. Find peak voltage, angular frequency, and ordinary frequency.

2. Write v(t) and find the first time it reaches half its positive peak.

Question 1 - Solution

A zero-crossing slope and rms reading determine an AC source

(a) Method and calculation

$$V_p=\sqrt2V_{\mathrm{rms}}=33.9411\,V,\quad\dot v(0)=\omega V_p.$$$$\boxed{\omega=353.553\,s^{-1},\quad f=\omega/(2\pi)=56.2698\,Hz}.$$

(b) Interpretation and further calculation

$$\boxed{v(t)=33.9411\sin(353.553t)\,V}.$$$$\sin\omega t=1/2\Rightarrow\boxed{t=\pi/(6\omega)=1.48096\,ms}.$$

Physical check

The slope determines angular frequency only after the peak, rather than rms, voltage is used.