Question

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?

Short answer

a. The smallest value of ratio $\frac{V_s}{V_p}$for step-up transformer is 1.25.

b. The second smallest value of ratio $\frac{V_s}{V_p}$for step-up transformer is 4.0.

c. The largest value of $\frac{V_s}{V_p}$for step-up transformer is 5.00.

d. The smallest value of ratio $\frac{V_s}{V_p}$for step-down transformer is 0.20.

e. The second smallest value of the ratio $\frac{V_s}{V_p}$for step-down transformer is 0.25.

f. The largest value of $\frac{V_s}{V_p}$ for step-up transformer is 0.80.

Step-by-step solution

Listing the given quantities

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

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

Figure 31-37 is the transformer.

Understanding the concepts of 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.

Calculations of the smallest value of ratio VsVp for step-up transformer

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.

Calculations of the second smallest value of ratio  for step-up transformer

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.

Calculations of the largest value of ratio VsVp for step-up transformer

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.

Calculations of the smallest value of ratio VsVp for step-down transformer

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.

Calculations of the second smallest value of ratio VsVp for step-down transformer

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.

Calculations of the largest value of ratio VsVp for step-down transformer

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.