The maximum values of the alternating voltages and current are `400 V` and `20 A` respectively in a circuit connected to `50 Hz` supply and these quantities are sinusoidal. The instantaneous values of the voltage and current are `200sqrt2V` and `10 A`, respectively. At `t = 0`, both are increasing positively.
(a) Write down the expression for voltage and current at time `t`.
(b) Determine the power consumed in the circuit.

Explanation

`V_(0) = 400 V. i_(0) =20A, f = 50 Hz`
`omega =2pi f = 100 pi rad//sec, V =283 V, I = 10 A`
`V = V_(0) sin (omegat + phi 1)`
`283 =400 sin (100 phi t + phi_(1))`
At `t = 0`
`200sqrt2 =400 sin (0 + phi_1) implies sin phi_(1) = (1)/sqrt2`
`phi_(1) (pi)/(4)`
`i = i_(0) sin (omega t + phi_(2))`
`10 = 20 sin (0 + phi_(2)) implies sin phi_(2) = (1)/(2) implies phi_(2) = (pi)/(2)`
`V =400 sin (100 pit + pi/4)`
`i =20 sin (100 pit + pi//6)`
`P = (1)/(2) V_(0)i_(0) cos phi`
where phi phase difference between voltage and current
`phi = (pi)/(4) -(pi)/(6) = (pi)/(12) = 15^(@)`
`P = (1)/ (2) (400) (20) cos 15^(@) = 4000 cos 15^(@)`
` = 3864 W`
Note: `cos 15^(@) = cos (45^(@) cos 30^(@) + sin 45^(@) sin 30^(@)` .