Transformers — Question 3

Question 3

The applied volts per turn determine sinusoidal core flux density

A transformer primary has N turns around uniform core area A. Its flux density is sinusoidal with peak Bmax. Neglect winding voltage drop and leakage. Applied rms voltage is V at frequency f.

1. Derive the relation among V, f, N, A, and Bmax from Faraday induction.

2. Find the minimum integer turns for V=230 V, f=50.0 Hz, $A=4.00\times10^{-4}$ square meters, and allowed peak Bmax=1.20 T.

Question 3 - Solution

The applied volts per turn determine sinusoidal core flux density

(a) Method and calculation

$$\Phi=AB_{\max}\sin\omega t,\quad |v|_{\max}=N\omega AB_{\max}.$$$$\boxed{V=\frac{2\pi}{\sqrt2}fNAB_{\max}\approx4.44288fNAB_{\max}}.$$

(b) Interpretation and further calculation

$$N\ge\frac{230\sqrt2}{2\pi(50)(4.00\times10^{-4})(1.20)}=2157.01.$$$$\boxed{N_{\min}=2158\ \text{turns}}.$$

Physical check

The integer choice must round upward because it enforces an upper bound on peak flux density.