• Tags: transformer, turns-ratio, efficiency
• Difficulty: Moderate

A transformer works on mutual induction: an AC current in the primary coil creates a changing magnetic flux that induces an EMF in the secondary coil. For an ideal transformer: $V_s$/$V_p$ = $\frac{N_s}{N_p}$ (turns ratio k), and $V_p$$I_p$ = $V_s$$I_s$ (power conservation), so $I_s$/$I_p$ = $\frac{N_p}{N_s}$ = 1/k. Step-up (k > 1) increases voltage but decreases current. Step-down (k < 1) does the opposite. Impedance transformation: $Z_s$ = $k^2$ * $Z_p$ (used in impedance matching). Real transformers have losses: (1) Copper losses: $I^{2R}$ heating in windings — minimized by using thick, low-resistance wire. (2) Eddy current losses: circulating currents in the core — minimized by using laminated core. (3) Hysteresis losses: energy spent magnetizing/demagnetizing core — minimized by using soft iron (narrow hysteresis loop). (4) Flux leakage: not all flux links both coils. Efficiency = $\frac{V_s*I_s}{(V_p*I_p)}$ x 100%. Transformers only work with AC because DC produces constant flux (no induced EMF).