Answer:
a. 127 V b 311 V
Explanation:
a. The RMS value of E(t) = 1/T∫[E(t)]². E(t) = 180 sin(200πt). Since the frequency f = 100 cycles per second, the period, T = 1/f = 1/100 = 0.01 s.
So, 1/T∫[E(t)]² = 1/T∫[180 sin(200πt)]² = 180²/0.01∫sin(200πt)]²
Using trigonometric identity sin²Ф = (1 - cos2Ф)/2 where Ф = ωt
1/T∫[E(t)]² = 180²/0.01∫(1 - cos2Ф)/2. We integrate from 0 to T, , we have
1/T∫[E(t)]² = 180²/(0.01 × 2)(t - sin2ωt/2ω)
1/T∫[E(t)]² = 180²/(0.02[(T - (sin2π)/(2 × 200π) ) - (0 - [sin(2 × 0)]/(2 × 200π))
1/T∫[E(t)]² = 180²/(0.02)[(0.01 - 0)
1/T∫[E(t)]² = 180²/2
E(t)RMS = √1/T∫[E(t)]²
= √180²/2
= 180/√2
= 127.28 V
= 127 V to the nearest whole number
b. Since E(t)(RMS) = A/√2 where A = voltage amplitude and E(t)(RMS) = 220 V,
A = √2E(t)(RMS) =
√2 × 220 V
= 311.13 V
= 311 V to the nearest whole number
