Transformers — Question 6

Question 6

Maximum transformer efficiency occurs when variable and fixed losses balance

A transformer model has fixed core loss P0, copper loss $kI_s^2$, and approximately constant secondary rms voltage Vs feeding a resistive load. Ignore all other losses.

1. Maximize $\eta=V_sI_s/(V_sI_s+P_0+kI_s^2)$ over positive current.

2. Find optimal current and maximum efficiency for P0=20.0 W, k=.400 ohm, and Vs=24.0 V.

Question 6 - Solution

Maximum transformer efficiency occurs when variable and fixed losses balance

(a) Method and calculation

$$\eta^{-1}=1+\frac{P_0}{V_sI_s}+\frac{kI_s}{V_s},\quad\frac d{dI_s}\eta^{-1}=\frac1{V_s}\left(k-\frac{P_0}{I_s^2}\right).$$$$\boxed{I_{s,*}=\sqrt{P_0/k},\quad kI_{s,*}^2=P_0}.$$

(b) Interpretation and further calculation

$$I_{s,*}=\sqrt{20/.400}=7.07107\,A,\quad P_{L,*}=24I_{s,*}=169.706\,W.$$$$\boxed{\eta_{\max}=\frac{169.706}{169.706+40}=.809256}.$$

Physical check

At very low current fixed loss dominates; at high current copper loss grows faster than useful power.