Chapter 4, Problem 8P

To determine

The missing values.

Explanation of Solution

The given values are shown in below table:

Primary		Secondary 1		Secondary 2		Secondary 3
${\text{E}}_{\text{P}}$	277	${\text{E}}_{\text{S1}}$	480	${\text{E}}_{\text{S1}}$	208	${\text{E}}_{\text{S1}}$	120
${\text{I}}_{\text{P}}$		${\text{I}}_{\text{S1}}$		${\text{I}}_{\text{S1}}$		${\text{I}}_{\text{S1}}$
${\text{N}}_{\text{P}}$	350	${\text{N}}_{\text{S1}}$		${\text{N}}_{\text{S1}}$		${\text{N}}_{\text{S1}}$
		Turns Ratio 1		Turns Ratio 2		Turns Ratio 3
		${\text{R}}_1$	$200\Omega$	${\text{R}}_2$	$60\Omega$	${\text{R}}_3$	$24\Omega$

Primary – Secondary 1:

Calculate the turns ratio 1$\left({\text{N}}_{\text{P}}/{\text{N}}_{\text{S1}}\right)$.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S1}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S1}}}\\ =\frac{277}{480}\\ =\frac{1}{1.7328}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $1:1.7328$.

Calculate the current in secondary winding 1$\left({\text{I}}_{\text{S1}}\right)$.

  $\begin{array}{c}{\text{I}}_{\text{S1}}=\frac{{\text{E}}_{\text{S1}}}{{\text{R}}_1}\\ =\frac{480}{200\Omega }\\ =2.4\end{array}$

Calculate the no. of turns in secondary winding 1.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S1}}}=\frac{1}{1.7328}\\ {\text{N}}_{\text{S1}}=1.7328\times {\text{N}}_{\text{P}}\\ =1.7328\times 350\\ =606.48\\ \approx 606\end{array}$

Calculate the current in primary winding due to secondary winding 1$\left({\text{I}}_{\text{P}\left(\text{S1}\right)}\right)$.

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S1}\right)}}{{\text{I}}_{\text{S1}}}=\frac{{\text{N}}_{\text{S1}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S1}\right)}\text{=}\frac{{\text{N}}_{\text{S1}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S1}}\\ =\frac{1.7328}{1}\times 2.4\\ =4.1587\end{array}$

Primary – Secondary 2:

Calculate the turns ratio 2$\left({\text{N}}_{\text{P}}/{\text{N}}_{\text{S2}}\right)$.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S2}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S2}}}\\ =\frac{277}{208}\\ =\frac{1.3317}{1}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $1.3317:1$.

Calculate the current in secondary winding 2$\left({\text{I}}_{\text{S2}}\right)$.

  $\begin{array}{c}{\text{I}}_{\text{S2}}=\frac{{\text{E}}_{\text{S2}}}{{\text{R}}_2}\\ =\frac{208}{60\Omega }\\ =3.4667\end{array}$

Calculate the no. of turns in secondary winding 2.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S2}}}=\frac{1.3317}{1}\\ {\text{N}}_{\text{S2}}=\frac{1}{1.3317}\times {\text{N}}_{\text{P}}\\ =\frac{1}{1.3317}\times 350\\ =262.8219\\ \approx 263\end{array}$

Calculate the current in primary winding due to secondary winding 2$\left({\text{I}}_{\text{P}\left(\text{S2}\right)}\right)$.

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S2}\right)}}{{\text{I}}_{\text{S2}}}=\frac{{\text{N}}_{\text{S2}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S2}\right)}\text{=}\frac{{\text{N}}_{\text{S2}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S2}}\\ =\frac{1}{1.3317}\times 3.4667\\ =2.6032\end{array}$

Primary – Secondary 3:

Calculate the turns ratio 3$\left({\text{N}}_{\text{P}}/{\text{N}}_{\text{S3}}\right)$.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S3}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S3}}}\\ =\frac{277}{120}\\ =\frac{2.3083}{1}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $2.3083:1$.

Calculate the current in secondary winding 3$\left({\text{I}}_{\text{S3}}\right)$.

  $\begin{array}{c}{\text{I}}_{\text{S3}}=\frac{{\text{E}}_{\text{S3}}}{{\text{R}}_3}\\ =\frac{120}{24\Omega }\\ =5\end{array}$

Calculate the no. of turns in secondary winding 3.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S3}}}=\frac{2.3083}{1}\\ {\text{N}}_{\text{S3}}=\frac{1}{2.3083}\times \text{350}\\ =\frac{1}{2.3083}\times 350\\ =151.6267\\ \approx 152\end{array}$

Calculate the current in primary winding due to secondary winding 3$\left({\text{I}}_{\text{P}\left(\text{S3}\right)}\right)$.

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S3}\right)}}{{\text{I}}_{\text{S3}}}=\frac{{\text{N}}_{\text{S3}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S3}\right)}\text{=}\frac{{\text{N}}_{\text{S3}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S3}}\\ =\frac{1}{2.3083}\times 5\\ =2.166\end{array}$

Calculate the total current in primary winding $\left({\text{I}}_{\text{P}}\right)$.

  $\begin{array}{c}{\text{I}}_{\text{P}}={\text{I}}_{\text{P}\left(\text{S1}\right)}+{\text{I}}_{\text{P}\left(\text{S2}\right)}+{\text{I}}_{\text{P}\left(\text{S3}\right)}\\ =4.1587+2.6032+2.166\\ =8.9279\\ \approx 8.93\end{array}$

Thus, the all missing values are calculated and shown in below table:

Primary		Secondary 1		Secondary 2		Secondary 3
${\text{E}}_{\text{P}}$	277	${\text{E}}_{\text{S1}}$	480	${\text{E}}_{\text{S2}}$	208	${\text{E}}_{\text{S3}}$	120
${\text{I}}_{\text{P}}$	8.93	${\text{I}}_{\text{S1}}$	2.4	${\text{I}}_{\text{S2}}$	3.4667	${\text{I}}_{\text{S3}}$	5
${\text{N}}_{\text{P}}$	350	${\text{N}}_{\text{S1}}$	606	${\text{N}}_{\text{S2}}$	263	${\text{N}}_{\text{S3}}$	152
		Turns Ratio 1	1:1.7328	Turns Ratio 2	1.3317:1	Turns Ratio 3	2.3083:1
		${\text{R}}_1$	$200\Omega$	${\text{R}}_2$	$60\Omega$	${\text{R}}_3$	$24\Omega$