Copper saving in auto transformer

An autotransformer has a single winding, part of which is common to both primary and secondary. Power is transferred partly by conduction and partly by induction, so it needs less copper than a two-winding transformer of the same rating.

Assumptions

1. Same rating (same $V_1$, $V_2$, $I_1$, $I_2$) for both transformers.
2. Same current density in all sections, so copper weight ∝ (current × number of turns) of each section (length ∝ turns, cross-section ∝ current).
3. No-load current and losses are neglected, so $N_1I_1 = N_2I_2$.
4. Step-down case with $K = \frac{N_2}{N_1} = \frac{V_2}{V_1} = \frac{I_1}{I_2} < 1$.

          I1
  o------>----+ A
              S  section AB: (N1-N2) turns,
  V1          S  current I1
              +---------+---> I2  o
              S  B      |
              S section BC: N2 turns,     V2
              S current (I2 - I1)
              S         |
  o-----------+ C ------+---------o

Two-winding transformer

Primary: $N_1$ turns carrying $I_1$; secondary: $N_2$ turns carrying $I_2$.

$$W_{tw} \propto N_1I_1 + N_2I_2$$

Autotransformer

• Section AB has $(N_1 - N_2)$ turns carrying $I_1$.
• Section BC has $N_2$ turns carrying $(I_2 - I_1)$.

$$W_{auto} \propto (N_1 - N_2)I_1 + N_2(I_2 - I_1)$$

Ratio of copper weights

$$\begin{aligned} \frac{W_{auto}}{W_{tw}} &= \frac{N_1I_1 - N_2I_1 + N_2I_2 - N_2I_1}{N_1I_1 + N_2I_2} \end{aligned}$$

Using $N_2I_2 = N_1I_1$ and $N_2 = KN_1$:

$$\begin{aligned} \frac{W_{auto}}{W_{tw}} &= \frac{2N_1I_1 - 2KN_1I_1}{2N_1I_1} = 1 - K \\ W_{auto} &= (1 - K)\,W_{tw} \end{aligned}$$

Therefore

$$\text{Saving of copper} = W_{tw} - W_{auto} = K\,W_{tw}$$

Significance when K is near unity

The saving is $K$ times the copper of the two-winding transformer, so it grows as $K \to 1$:

Physically, when $K \approx 1$ the common section BC carries only the small difference $(I_2 - I_1)$, and section AB has very few turns, so very little copper is needed. Only the fraction $(1-K)$ of the power is transformed inductively; the rest is conducted directly.

Hence autotransformers are economical when the voltage ratio is close to 1 (e.g. 400/440 V, 220/132 kV interconnection, variacs, starting of induction motors), but not for large ratios, where saving is small and the lack of electrical isolation is a safety risk.