A transformer has N1=350turns and NOW=2000turns. If the input voltage is 170cos(omega time)v, what rms voltage is developed across the secondary
To find the rms voltage developed across the secondary of the transformer, we can use the turns ratio formula:
V2/V1 = N2/N1
Where V1 is the input voltage (170cos(omega time)v), N1 is the number of turns on the primary coil (350 turns), N2 is the number of turns on the secondary coil (2000 turns), and V2 is the voltage developed across the secondary.
Rearranging the formula, we can solve for V2:
V2 = (V1*N2) / N1
Substituting the values:
V2 = (170cos(omega time)v * 2000) / 350
Since we are interested in the rms voltage, we need to take the average of the squared values over a full cycle:
V2rms = sqrt[(1/T) * integral(V2^2, 0 to T)]
Where T is the period, which in this case is 2π/omega.
Let's assume omega = 1 rad/s for simplicity. Thus, the period T = 2π/1 = 2π.
V2rms = sqrt[(1/2π) * integral((170cos(t) * 2000 / 350)^2, 0 to 2π)]
V2rms = sqrt[(1/2π) * integral((170/350)^2 * cos(t)^2, 0 to 2π)]
V2rms = sqrt[(1/2π) * integral((170/350)^2 * (1 + cos(2t))/2, 0 to 2π)]
V2rms = sqrt[(1/2π) * [(170/350)^2 * [(t + sin(2t))/2] from 0 to 2π]
V2rms = sqrt[(1/2π) * [(170/350)^2 * [(2π + sin(4π) - 0 - sin(0))/2]]
V2rms = sqrt[(1/2π) * [(170/350)^2 * 2π]]
V2rms = sqrt[(17/35)^2 * π]
Using a calculator, we can evaluate the expression:
V2rms ≈ 104.348 V
Therefore, the rms voltage developed across the secondary of the transformer is approximately 104.348 V.