Transformer Turns Ratio⚠ unverified

Understanding the Parameters

• $V_p$, $V_s$ — the RMS voltages on the two windings. Their ratio equals the turns ratio because both windings link the same core flux.

• $n$ — $n>1$ is a step‑down transformer (fewer secondary turns, lower secondary voltage); $n<1$ steps up. Current and impedance transform oppositely (see Transformer Current Ratio).

Derivation (Approaching a Proof)

Both windings are wound on a common magnetic core carrying a shared time‑varying flux $\Phi(t)$. By Faraday's law, the EMF induced in a winding of $N$ turns is $N\,d\Phi/dt$. Applied to each winding (same $\Phi$):

$$V_p = N_p\frac{d\Phi}{dt}, \qquad V_s = N_s\frac{d\Phi}{dt}.$$

Dividing, the common $d\Phi/dt$ cancels:

$$\frac{V_p}{V_s} = \frac{N_p}{N_s} \equiv n.$$

So the voltage ratio is the turns ratio — the defining property of a transformer, exact for an ideal (loss‑free, fully coupled) device and very nearly so for real ones.

History

The transformer emerged in the 1880s (Gaulard & Gibbs; Zipernowsky, Bláthy & Déri's closed‑core "ZBD"; Stanley for Westinghouse). Its ability to change AC voltage cheaply — via the turns ratio — is precisely why alternating current won the "war of currents": power can be generated, transmitted at high voltage (low loss), and stepped back down for use.

Related Concepts: Transformer Current Ratio, Transformer Efficiency, Watt's Law

Notes: Registry calculator transformer-turns-ratio (unverified). Ideal transformer; real units have leakage flux, winding resistance, and magnetizing current that make the ratio slightly load‑dependent.