Chapter 5, Problem 6RQ

To determine

The voltages between the points $\text{A}-\text{B}$, $\text{A}-\text{C}$, $\text{A}-\text{D}$, $\text{A}-\text{E}$, $\text{B}-\text{C}$, $\text{B}-\text{D}$, $\text{B}-\text{E}$, $\text{C}-\text{D}$, $\text{C}-\text{E}$ and $\text{D}-\text{E}$.

Explanation of Solution

Given:

The number of turns between the windings $\text{A}-\text{B}$ $\left({\text{N}}_{\text{A}-\text{B}}\right)$ is $120$.

The number of turns between the windings $\text{B}-\text{C}$ $\left({\text{N}}_{\text{B}-\text{C}}\right)$ is $180$.

The number of turns between the windings $\text{C}-\text{D}$ $\left({\text{N}}_{\text{C}-\text{D}}\right)$ is $250$.

The number of turns between the windings $\text{D}-\text{E}$ $\left({\text{N}}_{\text{D}-\text{E}}\right)$ is $300$.

The voltage connected across windings $\text{B}-\text{E}$ $\left({\text{E}}_{\text{B}-\text{E}}\right)$ is $240\text{volts}$.

The below figure represents the autotransformer windings.

Calculation:

Calculate the voltage per unit turns $\left(\text{volts}\text{per}\text{turn}\right)$ using the relation.

  $\text{volts}\text{per}\text{turn}=\frac{{\text{E}}_{\text{B}-\text{E}}}{{\text{N}}_{\text{B}-\text{E}}}$

  $\begin{array}{l}\text{volts}\text{per}\text{turn}=\frac{\text{240}\text{volts}}{\left(180+250+300\right)}\\ \text{volts}\text{per}\text{turn}=\frac{24}{73}\end{array}$

Calculate the voltage between the windings $\text{A}-\text{B}$ $\left({\text{E}}_{\text{A}-\text{B}}\right)$ using the relation.

  ${\text{E}}_{\text{A}-\text{B}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{A}-\text{B}}$

  $\begin{array}{c}{\text{E}}_{\text{A}-\text{B}}=\frac{24}{73}\times \text{120}\\ {\text{E}}_{\text{A}-\text{B}}=\text{39}\text{.452}\text{volts}\end{array}$

Calculate the voltage between the windings $\text{A}-\text{C}$, $\left({\text{E}}_{\text{A}-\text{C}}\right)$ using the relation.

  ${\text{E}}_{\text{A}-\text{C}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{A}-\text{C}}$

  $\begin{array}{c}{\text{E}}_{\text{A}-\text{C}}=\frac{24}{73}\times \left(\text{120}+\text{180}\right)\\ {\text{E}}_{\text{A}-\text{C}}=98.63\text{volts}\end{array}$

Calculate the voltage between the windings $\text{A}-\text{D}$ $\left({\text{E}}_{\text{A}-\text{D}}\right)$ using the relation.

  ${\text{E}}_{\text{A}-\text{D}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{A-D}}$

  $\begin{array}{c}{\text{E}}_{\text{A}-\text{D}}=\frac{24}{73}\times \left(\text{120}+\text{180}+\text{250}\right)\\ {\text{E}}_{\text{A}-\text{D}}=180.82\text{volts}\end{array}$

Calculate the voltage between the windings $\text{A}-\text{E}$ $\left({\text{E}}_{\text{A}-\text{E}}\right)$ using the relation.

  ${\text{E}}_{\text{A}-\text{E}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{A}-\text{E}}$

  $\begin{array}{c}{\text{E}}_{\text{A}-\text{E}}=\frac{24}{73}\times \left(\text{120}+\text{180}+\text{250}+\text{300}\right)\\ {\text{E}}_{\text{A}-\text{E}}=279.452\text{volts}\end{array}$

Calculate the voltage between the windings $\text{B}-\text{C}$ $\left({\text{E}}_{\text{B}-\text{C}}\right)$ using the relation.

  ${\text{E}}_{\text{B}-\text{C}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{B}-\text{C}}$

  $\begin{array}{c}{\text{E}}_{\text{B}-\text{C}}=\frac{24}{73}\times \left(\text{180}\right)\\ {\text{E}}_{\text{B}-\text{C}}=59.178\text{volts}\end{array}$

Calculate the voltage between the windings $\text{B}-\text{D}$ $\left({\text{E}}_{\text{B}-\text{D}}\right)$ using the relation.

  ${\text{E}}_{\text{B}-\text{D}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{B}-\text{D}}$

  $\begin{array}{c}{\text{E}}_{\text{B}-\text{D}}=\frac{24}{73}\times \left(\text{180}+\text{250}\right)\\ {\text{E}}_{\text{B}-\text{D}}=141.37\text{volts}\end{array}$

Calculate the voltage between the windings $\text{B}-\text{E}$ $\left({\text{E}}_{\text{B}-\text{E}}\right)$ using the relation.

  ${\text{E}}_{\text{B}-\text{E}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{B}-\text{E}}$

  $\begin{array}{c}{\text{E}}_{\text{B}-\text{E}}=\frac{24}{73}\times \left(\text{180}+\text{250}+\text{300}\right)\\ {\text{E}}_{\text{B}-\text{E}}=240\text{volts}\end{array}$

Calculate the voltage between the windings $\text{C}-\text{D}$ $\left({\text{E}}_{\text{C}-\text{D}}\right)$ using the relation.

  ${\text{E}}_{\text{C}-\text{D}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{C}-\text{D}}$

  $\begin{array}{c}{\text{E}}_{\text{C}-\text{D}}=\frac{24}{73}\times \left(\text{250}\right)\\ {\text{E}}_{\text{C}-\text{D}}=82.19\text{volts}\end{array}$

Calculate the voltage between the windings $\text{C}-\text{E}$ $\left({\text{E}}_{\text{C}-\text{E}}\right)$ using the relation.

  ${\text{E}}_{\text{C}-\text{E}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{C}-\text{E}}$

  $\begin{array}{c}{\text{E}}_{\text{C}-\text{E}}=\frac{24}{73}\times \left(\text{250}+\text{300}\right)\\ {\text{E}}_{\text{C}-\text{E}}=180.82\text{volts}\end{array}$

Calculate the voltage between the windings $\text{D}-\text{E}$ $\left({\text{E}}_{\text{D}-\text{E}}\right)$ using the relation.

  ${\text{E}}_{\text{D}-\text{E}}=\left(\text{volts}\text{per}\text{turn}\right)\times {\text{N}}_{\text{D}-\text{E}}$

  $\begin{array}{c}{\text{E}}_{\text{D}-\text{E}}=\frac{24}{73}\times \left(\text{300}\right)\\ {\text{E}}_{\text{D}-\text{E}}=98.63\text{volts}\end{array}$

Thus, the voltages between the windings $\text{A}-\text{B}$ is $\underset{\_}{39.452\text{volts}}$, between windings $\text{A}-\text{C}$ is $\underset{\_}{98.63\text{volts}}$, between windings $\text{A}-\text{D}$ is $\underset{\_}{180.82\text{volts}}$, between windings $\text{A}-\text{E}$ is $\underset{\_}{279.452\text{volts}}$, between windings $\text{B}-\text{C}$ is $\underset{\_}{59.178\text{volts}}$, between windings $\text{B}-\text{D}$ is $\underset{\_}{141.37\text{volts}}$, between windings $\text{B}-\text{E}$ is $\underset{\_}{240\text{volts}}$, between windings $\text{C}-\text{D}$ is $\underset{\_}{82.19\text{volts}}$, between windings $\text{C}-\text{E}$ is $\underset{\_}{180.82\text{volts}}$ and between windings $\text{D}-\text{E}$ is $\underset{\_}{98.63\text{volts}}$.