Figure shows an “autotransformer.” It consists of a single coil (with an iron core). Three taps Ti are provided. Between taps T1 and T2 there are 200 turns, and between taps T2 and t3 there are800turns. Any two taps can be chosen as the primary terminals, and any two taps can be chosen as the secondary terminals. For choices producing a step-up transformer, (a)What is the smallest? (b)What is the second smallest? (c) What is the largest value of the ratio Vs/Vp? (d) For a step-down transformer, what is the smallest?(e)For a step-down transformer, what is the second smallest? (f) For a step-down transformer, what are the largest values of Vs/Vp?

Number of turns between taps is $N_{T1T2}=200$.

Number of turns between taps is $N_{T2T3}=800$.

Figure 31-37 is the transformer.

We use the concept of transformer. Using the equation of ratio of voltages related to ratio of number of turns, we can find the ratio of voltages.

a.

The smallest ratio when primary turns are larger than secondary; take $T_2{T}_3$as primary and $T_1{T}_3$as secondary and write the following formula.

$$\begin{gathered} \frac{V_s}{V_p}=\frac{V_{13}}{V_{23}} \\ =\frac{N_s}{N_p} \\ =\frac{\left(800+200\right)}{800} \\ =1.25 \end{gathered}$$

Hence, the smallest value of ratio $\frac{V_s}{V_p}$ for step-up transformer is 1.25.

b.

For the second smallest ratio, take $T_1{T}_2$as primary and $T_2{T}_3$as secondary, and define the ratio as follow.

$$\begin{gathered} \frac{V_{23}}{V_{12}}=\frac{800}{200} \\ =4.00 \end{gathered}$$

Hence, the second smallest value of ratio $\frac{V_s}{V_p}$for step-up transformer is 4.00.

c.

Largest value $\frac{V_s}{V_p}$for step-up transformer:

For largest ratio, take$T_1{T}_3$ as primary and $T_1{T}_3$as secondary, and define the ratio as below.

$$\begin{gathered} \frac{V_{13}}{V_{12}}=\frac{\left(800+200\right)}{200} \\ =5.00 \end{gathered}$$

Largest value of $\frac{V_s}{V_p}$ for step-up transformer 5.0.

d.

Exchange the primary and secondary values, so you obtain

$$\begin{gathered} \frac{V_{12}}{V_{13}}=\frac{1}{5.00} \\ =0.200 \end{gathered}$$

Thus, the smallest value of ratio $\frac{V_s}{V_p}$ for step-down transformer is 0.20.

e.

Second smallest value of $\frac{V_s}{V_p}$for step-down transformer:

$$\begin{gathered} \frac{V_{12}}{V_{23}}=\frac{1}{4.00} \\ =0.250 \end{gathered}$$

Second smallest value of ratio $\frac{V_s}{V_p}$ for step-down transformer is 0.250.

f.

Largest value of $\frac{V_s}{V_p}$for step-down transformer.

$$\begin{gathered} \frac{V_{23}}{V_{13}}=\frac{1}{1.25} \\ =0.800 \end{gathered}$$

Hence, the largest value of $\frac{V_s}{V_p}$ for step-up transformer is 0.800.