CT secondary open; auto transformer economy

Why the secondary of a CT must not be left open

The primary of a CT carries the line current, which is independent of the secondary. With the secondary closed, the mmf $N_2I_2$ nearly cancels $N_1I_1$, leaving only a small exciting mmf. If the secondary is open, $I_2 = 0$ and the whole primary mmf magnetises the core. The flux rises far into saturation, causing very high induced voltage in the many-turn secondary (insulation breakdown and shock hazard), excessive core heating, and residual magnetism that destroys accuracy.

Autotransformer is economical when the ratio is near unity

An autotransformer has a single winding, part of which is common to the primary and the secondary. Let the primary have $N_1$ turns (terminals A–C), the secondary $N_2$ turns (taps B–C), and take $N_2 < N_1$. Define $K = \dfrac{N_2}{N_1} = \dfrac{V_2}{V_1} = \dfrac{I_1}{I_2}$.

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

The amount of copper is proportional to (turns × current) of each winding portion.

Two-winding transformer (primary $N_1$ at $I_1$, secondary $N_2$ at $I_2$):

$$W_{2w} \propto N_1I_1 + N_2I_2 = 2N_1I_1$$

Autotransformer:

$$\begin{aligned} W_{auto} &\propto (N_1 - N_2)I_1 + N_2(I_2 - I_1) \\ &= N_1I_1(1 - K) + N_1I_1(1 - K)\quad(\text{using } N_2I_2 = N_1I_1,\ N_2I_1 = KN_1I_1) \\ &= 2N_1I_1(1 - K) \end{aligned}$$ $$\frac{W_{auto}}{W_{2w}} = 1 - K,\qquad \text{saving} = W_{2w} - W_{auto} = K\,W_{2w}$$

The saving in copper is the fraction $K$ of the copper of a two-winding transformer. When $K$ is close to 1 (for example 0.9), the autotransformer needs only $(1 - K)$ = 10% of the copper, a saving of 90%. When $K$ is small (say 0.2), the saving is only 20%, so it is not economical. Also, only a small part of the power, $(1 - K)$, is transferred by induction, while the rest is passed directly by conduction, so the core, losses and cost fall in the same way.

Applications: starting of induction motors (autotransformer starter), voltage regulators (variacs), boosting or reducing line voltage by 10 to 20%, and interconnecting 220 kV and 132 kV systems.