Answer to Question #113247 in Electric Circuits for James

Question

The current in an a.c. circuit at any time t seconds is given by:
i = 5 sin(100πt − 0.432) A
Determine (a) the amplitude, frequency, periodic time and phase angle (in
degrees), (b) the value of current at t = 0, (c) the value of current at t =8ms,
(d) the time when the current is first a maximum, (e) the time when the current
first reaches 3A. Sketch one cycle of the waveform showing relevant points.

Answer 1

$i = I_0\sin(\omega t-\phi_0) = 5\sin(100\pi t-0.435)$ .

Comparing left and right side, find:

a) The amplitude is $I_0 = 5 A$ , frequency is $f = \omega/2\pi = 50 Hz$ , periodic time is $T = 1/f = 0.02s$ and phase angle is $\phi_0 = 0.435 rad = 0.435\cdot 180/\pi = 24.9 \degree.$

b) The value of current at t = 0: $i(0) = 5\sin(100\pi \cdot 0-0.435) = 5\sin(-0.435) = -2.107A$

c) The value of current at t =8ms: $i(8ms) = 5\sin(100\pi \cdot 8\cdot10^{-3}-0.435) =4.37 A$

d) The time when the current is first a maximum is a time, when $\frac{di}{dt} = 0$ . Thus, $\frac{di}{dt} = 5\cdot 100\pi \cos(100\pi t-0.435) = 0 \Rightarrow 100\pi t-0.435 = \pi/2 \Rightarrow t = 6.38ms.$

e) The time when the current first reaches 3A: $i(t) = 5\sin(100\pi \cdot t-0.435) = 3\Rightarrow100\pi \cdot t-0.435 = \arcsin\frac{3}{5}\Rightarrow t = 3.43ms$

Sketch one cycle of the waveform showing relevant points: