Chapter 4, Problem 7P

To determine

The missing values.

Explanation of Solution

The given values are shown in below table:

Primary		Secondary 1		Secondary 2		Secondary 3
${\text{E}}_{\text{P}}$	208	${\text{E}}_{\text{S1}}$	320	${\text{E}}_{\text{S1}}$	120	${\text{E}}_{\text{S1}}$	24
${\text{I}}_{\text{P}}$		${\text{I}}_{\text{S1}}$		${\text{I}}_{\text{S1}}$		${\text{I}}_{\text{S1}}$
${\text{N}}_{\text{P}}$	800	${\text{N}}_{\text{S1}}$		${\text{N}}_{\text{S1}}$		${\text{N}}_{\text{S1}}$
		Turns Ratio 1		Turns Ratio 2		Turns Ratio 3
		${\text{R}}_1$	$12\text{k}\Omega$	${\text{R}}_2$	$6\Omega$	${\text{R}}_3$	$8\Omega$

Primary – Secondary 1:

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

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S1}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S1}}}\\ =\frac{208}{320}\\ =\frac{1}{1.5385}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $1:1.5385$.

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

  $\begin{array}{c}{\text{I}}_{\text{S1}}=\frac{{\text{E}}_{\text{S1}}}{{\text{R}}_1}\\ =\frac{320}{12\text{k}\Omega \left[\frac{1000\Omega }{1\text{k}\Omega }\right]}\\ =0.0267\end{array}$

Calculate the no. of turns in secondary winding 1.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S1}}}=\frac{1}{1.5385}\\ {\text{N}}_{\text{S1}}=1.5385\times {\text{N}}_{\text{P}}\\ =1.5385\times 800\\ =1230.8\\ \approx 1231\end{array}$

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

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S1}\right)}}{{\text{I}}_{\text{S1}}}=\frac{{\text{N}}_{\text{S1}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S1}\right)}\text{=}\frac{{\text{N}}_{\text{S1}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S1}}\\ =\frac{1.5385}{1}\times 0.0267\\ =0.0411\end{array}$

Primary – Secondary 2:

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

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S2}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S2}}}\\ =\frac{208}{120}\\ =\frac{1.7333}{1}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $1.7333:1$.

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

  $\begin{array}{c}{\text{I}}_{\text{S2}}=\frac{{\text{E}}_{\text{S2}}}{{\text{R}}_2}\\ =\frac{120}{6\Omega }\\ =20\end{array}$

Calculate the no. of turns in secondary winding 2.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S2}}}=\frac{1.7333}{1}\\ {\text{N}}_{\text{S2}}=\frac{1}{1.7333}\times {\text{N}}_{\text{P}}\\ =\frac{1}{1.7333}\times 800\\ =461.5473\\ \approx 462\end{array}$

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

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S2}\right)}}{{\text{I}}_{\text{S2}}}=\frac{{\text{N}}_{\text{S2}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S2}\right)}\text{=}\frac{{\text{N}}_{\text{S2}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S2}}\\ =\frac{1}{1.7333}\times 20\\ =11.5386\end{array}$

Primary – Secondary 3:

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

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S3}}}=\frac{{\text{E}}_{\text{P}}}{{\text{E}}_{\text{S3}}}\\ =\frac{208}{24}\\ =\frac{8.6667}{1}\end{array}$

The turns ratio 1$\left({\text{N}}_{\text{P}}:{\text{N}}_{\text{S1}}\right)$ is $8.6667:1$.

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

  $\begin{array}{c}{\text{I}}_{\text{S3}}=\frac{{\text{E}}_{\text{S3}}}{{\text{R}}_3}\\ =\frac{24}{8\Omega }\\ =3\end{array}$

Calculate the no. of turns in secondary winding 3.

  $\begin{array}{c}\frac{{\text{N}}_{\text{P}}}{{\text{N}}_{\text{S3}}}=\frac{8.6667}{1}\\ {\text{N}}_{\text{S3}}=\frac{1}{8.6667}\times {\text{N}}_{\text{P}}\\ =\frac{1}{8.6667}\times 800\\ =92.3073\\ \approx 92\end{array}$

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

  $\begin{array}{c}\frac{{\text{I}}_{\text{P}\left(\text{S3}\right)}}{{\text{I}}_{\text{S3}}}=\frac{{\text{N}}_{\text{S3}}}{{\text{N}}_{\text{P}}}\\ {\text{I}}_{\text{P}\left(\text{S3}\right)}\text{=}\frac{{\text{N}}_{\text{S3}}}{{\text{N}}_{\text{P}}}\times {\text{I}}_{\text{S3}}\\ =\frac{1}{8.6667}\times 3\\ =0.3462\end{array}$

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

  $\begin{array}{c}{\text{I}}_{\text{P}}={\text{I}}_{\text{P}\left(\text{S1}\right)}+{\text{I}}_{\text{P}\left(\text{S2}\right)}+{\text{I}}_{\text{P}\left(\text{S3}\right)}\\ =0.0411+11.5386+0.3462\\ =11.9259\\ \approx 11.93\end{array}$

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

Primary		Secondary 1		Secondary 2		Secondary 3
${\text{E}}_{\text{P}}$	208	${\text{E}}_{\text{S1}}$	320	${\text{E}}_{\text{S2}}$	120	${\text{E}}_{\text{S3}}$	24
${\text{I}}_{\text{P}}$	11.93	${\text{I}}_{\text{S1}}$	0.0267	${\text{I}}_{\text{S2}}$	20	${\text{I}}_{\text{S3}}$	3
${\text{N}}_{\text{P}}$	800	${\text{N}}_{\text{S1}}$	1231	${\text{N}}_{\text{S2}}$	462	${\text{N}}_{\text{S3}}$	92
		Turns Ratio 1	1:1.5385	Turns Ratio 2	1.7333:1	Turns Ratio 3	8.6667:1
		${\text{R}}_1$	$12\text{k}\Omega$	${\text{R}}_2$	$6\Omega$	${\text{R}}_3$	$8\Omega$