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I am studying the superposition principle using Engineering Circuit Analysis by Hayt, Kemmerly, and Durbin. The book gives the following exercise:

PRACTICE 5.2: For the circuit of Fig. 5.7, use superposition to obtain the voltage across each current source. enter image description here

My Attempt:

Using the superposition, I first deactivate the \$2\ \text{A}\$ current source.

[![strong text][2]][2]

Next, I apply KCL (Kirchhoff’s Current Law) at the red highlighted node:

$$\frac{v_1`-v_2`}{15\ \Omega}-i`+4\cdot i` =0\ \text{A} \label{1} \tag{I}$$

and I also use

$$i`=\frac{v_2`}{5\ \Omega} \label{2} \tag{II}$$

Substituting \eqref{1} into \eqref{2} I obtain

$$\frac{1}{15\ \Omega} \cdot v_1` + \frac{8}{15\ \Omega} \cdot v_2` = 0\ \text{A}$$

Where I’m stuck:

At this point, I end up with one equation involving two unknowns \$v_1`\$ and \$v_2`\$. I’m not sure how to obtain a second independent equation. A hint in the right direction would be greatly appreciated.


Thanks to the answers:

so first only open \$2\ \text{A}\$ current source. apply KCL on \$v_1`\$ and KCL on \$v_2`\$:

$$\begin{align} -\frac{22}{105\ \Omega}\cdot v_1` \ + \ \frac{1}{15\ \Omega}\cdot v_2` \ &= - \frac{3}{7}\ \text{A} \\ \frac{1}{15\ \Omega}\cdot v_1` \ + \ \frac{8}{15\ \Omega}\cdot v_2` \ &= 0\ \text{A} \\ \end{align}$$

Solution: \$v_1` \ =\ \frac{120}{61}\ \text{V} \$, \$v_2` \ =\ -\frac{15}{61}\ \text{V} \$

then only short \$3\ \text{V}\$ voltage source. apply KCL on \$v_1``\$ and KCL on \$v_2``\$:

$$\begin{align} -\frac{22}{105\ \Omega}\cdot v_1`` \ + \ \frac{1}{15\ \Omega}\cdot v_2`` \ &= - 2\ \text{A} \\ \frac{1}{15\ \Omega}\cdot v_1`` \ + \ \frac{8}{15\ \Omega}\cdot v_2`` \ &= 0\ \text{A} \\ \end{align}$$

Solution: \$v_1`` \ =\ \frac{560}{61}\ \text{V} \$, \$v_2`` \ =\ -\frac{70}{61}\ \text{V} \$

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  • \$\begingroup\$ You don't appear to have set the 3V source to zero at any point, superposition means you superpose several partial solutions \$\endgroup\$ Commented yesterday
  • \$\begingroup\$ I want to first calculate \$v_1` \$ and \$v_2` \$ for the case of an open \$3\ \text{A}\$ current source. and then zero the \$3\ \text{V}\$ voltage source and calculate \$v_1`` \$ and \$v_2`` \$. and then calculate \$v_1=v_1` +v_1`` \$ and \$v_2=v_2` +v_2`` \$ \$\endgroup\$ Commented yesterday
  • \$\begingroup\$ Strange that the authors spread the misconception about the superposition principle and dependent sources. And they incorrectly repeat that you cannot remove dependent sources when applying the superposition principle, which is not true. \$\endgroup\$ Commented yesterday
  • \$\begingroup\$ @G36 so you mean it is possible to remove dependent current sources when applying superposition theorem? \$\endgroup\$ Commented yesterday
  • \$\begingroup\$ @MarcoMoldenhauer Did you see/read my answer? \$\endgroup\$ Commented yesterday

3 Answers 3

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You cannot remove dependent sources in applying superposition theorem. For removing voltage sources, you have to short them. For removing current sources, open circuit them .Do nodal analysis for each step and see if you get the result.

Small hint:

Use superposition theorem to calculate \$V_1\$ and \$V_2\$ by adding \$V_1\$ and \$V_2\$ when current source is open circuit and when voltage source is short circuited while keeping the other source and the dependent source active. With that, you can calculate currents.

Big hint:

For example, removing the current source, I get these equations: $$\frac{V_1 - 3}{7} + \frac{V_1 - V_2}{15} = 0 \tag 1$$ Substituting current i, $$\frac{V_2}{5} + \frac{V_2 - V_1}{15} = 4 \frac{V_2}{5} \tag 2$$ From these,\$V_1\$ and \$V_2\$ can be calculated. Now, shorting voltage source and writing nodal equations. $$\frac{{V_1}^`}{7} + \frac{{V_1}^` - {V_2}^`}{15} = 2 \tag 3$$ $$\frac{{V_2}^`}{5} + \frac{{V_2}^` - {V_1}^`}{15} = 4 \frac{{V_2}^`}{5} \tag 4$$ From these, \$V_1\$ and \$V_2\$ and \${V_1}^`\$ and \${V_2}^`\$ can be calculated.

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We can use superposition with the dependent sources. But the controlling parameter cannot be removed in any of the individual cases. See the example:

Case 1: Independent current source acting alone

Independent current source

$$i_1 = 2A \times \frac{7\Omega}{7\Omega + 15\Omega + 5\Omega} = \frac{14}{27}\:[A] $$

Case 2:Independent voltage source acting alone:

Independent voltage

$$i_2 = \frac{3V}{7\Omega + 15\Omega + 5\Omega} = \frac{3}{27}\:[A]$$

Case 3: Dependent source acting alone:

Dependent source

$$i_3 = 4i \times \frac{22\Omega}{5\Omega + 22\Omega} = \frac{88}{27}i \:[A]$$

Now we have $$i = i_1 + i_2 + i_3 = \frac{14}{27}+ \frac{3}{27} + \frac{88}{27}i$$

And we solve for \$i\$

$$i = - \frac{17}{61}\:[A] \approx -278.689mA$$

Therefore $$V_2 = 5\Omega \times (- \frac{17}{61}) = - 1.393V$$

$$V_1 = 15\Omega \times -3i + V_2 = 12.541V + (-1.393V) = 11.1475V$$

And we are done.

More examples here:

Why are dependent sources not disabled with the superposition principle?

https://www.youtube.com/watch?v=P-9o75WZ0rM

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  • \$\begingroup\$ Insightful answer!!! \$\endgroup\$ Commented yesterday
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When you deactivate the \$2A\$ current source, the \$7Ω\$ and \$15Ω\$ resistors both are in series.

Both carry the same current. This is what you perhaps overlooked.

So, the equations are:

  1. \$\frac{3-v_2}{22}\$ \$+\$ \$\frac{4v_2}{5}\$ \$=\$ \$\frac{v_2}{5}\$

  2. \$i\$ = \$\frac{v_2}{5}\$

And, to determine \$v_1\$:

  1. \$\frac{3-v_1}{7}\$ \$=\$ \$\frac{v_1-v_2}{15}\$

Now you have 3 equations and 3 variables.

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