# Thevenin Equivalent Problem

So to get started on part a I am finding the REQ of the circuit by shorting it. Now to get from A to B in this circuit that is having R4 and R5 in series with R3 in parallel so REQ = (1/900 + 1/300)^-1 = 225 ohms. That is assuming R2 is skipped because it goes directly to ground after R3. To finish up the Thevenin Equivalent I need to find a voltage, but I have no clue where to go from this.

• Welcome to EE.SE. Find the voltage by the resistor ratios. How is this 100 volts divided up? HINT: R2 is parallel to the voltage so it has no effect. R3 is in series so it counts as part of a divider. – user105652 Feb 6 '18 at 3:41

I see in a comment that you are looking for nodal analysis. Here's my own approach to it:

Equation rules:

• Keep all outgoing currents on the left side
• Keep all incoming currents on the right side
• If not already chosen by the problem, pick a convenient node and call it "ground." This allows you to assign $0\:\text{V}$ to that node and this will simplify some of the equations. Which node to select is up to you. But I will pick one that either helps make the schematic easier to understand, or else will simplify the equations more than other choices.

I like to imagine a 3D chessboard -- those funny chessboards with tiny sub-chessboard areas that sit at different heights. Each node in the circuit is represented as one of these sub-areas (or "platforms"), sitting at some unknown elevation and held there on a long pole beneath it. This unknown elevation is the voltage you must find. One of these can be placed at "0 height." But that's just arbitrary, since then there still may be platforms above and/or below. So it's just a way of conveniently locating the "ground floor," so to speak.

I treat the currents as water. Water flows onto and off of each platform. The left side of the equation, the outgoing currents, is represented by the water that is flowing off of the perimeter of the platform. The right side of the equation, the incoming currents, is represented by the water that is flowing onto the platform from other platforms above it. The amount of water falling onto such a platform must equal the water flowing off of it via the perimeter edge. (Of course, otherwise water would magically build up on the platform -- or the reverse, get created out of thin air. Both are impossible.)

(No water is ever lost and all of it remains in play and the water cannot accumulate on the floors -- the net flow onto a floor must equal the net flow off of it -- KCL! -- and hence the justification for setting the left side equal to the right side.)

simulate this circuit – Schematic created using CircuitLab

$N_3$ is free, since it has a voltage source holding its value. So $V_{N_3}=V_1=100\:\text{V}$. This only leaves two other nodes to worry over.

I'm going to provide the general solution here, because the approach can be used broadly. Your circuit could actually be "cut short" by the fact that you already know that $R_1$ is to be removed and/or shorted out. But this would be a special-case solution. I'd rather teach you the general solution, out of which all other questions can be answered.

The nodal equations are:

\begin{align*} \frac{V_{N_1}}{R_4}+\frac{V_{N_1}}{R_5}&=10\:\text{mA}+\frac{V_{N_2}}{R_4}+\frac{0\:\text{V}}{R_5}\\\\ \frac{V_{N_2}}{R_1}+\frac{V_{N_2}}{R_3}+\frac{V_{N_2}}{R_4}&=\frac{0\:\text{V}}{R_1}+\frac{V_{N_3}=100\:\text{V}}{R_3}+\frac{V_{N_1}}{R_4} \end{align*}

There are two unknowns and two variables to solve for, here.

\begin{align*} V_{N_1}&=R_5\frac{V_1\:R_1+I_1\left(R_1\:R_3+R_1\:R_4+R_3\:R_4\right)}{R_1\:R_3+R_1\:R_4+R_1\:R_5+R_3\:R_4+R_3\:R_5}\tag{R_1\ne\infty}\\\\ &=R_5\frac{V_1+I_1\left(R_3+R_4\right)}{R_3+R_4+R_5}\tag{R_1=\infty}\\\\ V_{N_2}&=R_1\frac{V_1\left(R_4+R_5\right)+I_1\:R_3\:R_5}{R_1\:R_3+R_1\:R_4+R_1\:R_5+R_3\:R_4+R_3\:R_5}\tag{R_1\ne\infty}\\\\ &=\frac{V_1\left(R_4+R_5\right)+I_1\:R_3\:R_5}{R_3+R_4+R_5}\tag{R_1=\infty} \end{align*}

Note that if you plug in the actual value for $R_1=100\:\Omega$ in the above equations (the ones where $R_1\ne\infty$), then you will get the voltages you already worked out:

\begin{align*} V_{N_1}\approx 15.25641\:\text{V}\\\\ V_{N_2}\approx 23.46154\:\text{V} \end{align*}

Once you solve these equations, you can work out the fact that if you short $R_1$ (just set $R_1=0\:\Omega$ in the above equations) then the current through that short is $\frac{61}{180}\:\text{A}$ (the sum of the currents coming in from the left and the right.) And if you open $R_1$ (just set $R_1=\infty\:\Omega$ in the above equations) then the voltage across the vacant location for $R_1$ is $V_{N_2}=76\frac{1}{4}\:\text{V}$.

The above equations allow you to set $R_1=0$. Here, the voltage at $V_{N_2}$ is going to obviously be zero volts (since you just shorted it to ground.) However, the voltage at $V_{N_1}$ will now be very important. It will cause a current through $R_4$! So compute $V_{N_1}$ using the above equation with $R_1=0$ and then work out the current in $R_4$ from that. Add to that, the current that must also be created now in $R_3$. Since you know that $V_{N_2}=0$ these two currents should be trivial to work out. Sum them. This is the short circuit current.

These equations also support $R_1=\infty$. In this case, you just solve for $V_{N_2}$ and you have the open circuit voltage.

P.S. If you are not already using Sage and/or Sympy, you should be. These programs help you rapidly solve equations, symbolically. It took me mere seconds of time to develop the equations I showed above from the two nodal equations.

 var('vn1 vn2 vn3 r1 r2 r3 r4 r5 i1')
ans=solve([Eq(vn1/r4+vn1/r5,i1+vn2/r4),Eq(vn2/r1+vn2/r3+vn2/r4,vn3/r3+vn1/r4)],[vn1,vn2])


From that, the short circuit current was easy:

 (ans[vn1].subs({r1:0,r2:200,r3:300,r4:400,r5:500,i1:.01,vn3:100})/400+100/300).n()


And so was the open circuit node voltage for $V_{N_2}$:

 y1[vn2].subs({r1:1E20,r2:200,r3:300,r4:400,r5:500,i1:.01,vn3:100}).n()


If you were to take this as a nodal analysis problem, but wanted to solve it more specifically than the above general approach, then you have the following two circuits:

simulate this circuit

The nodal equations for the top schematic is:

\begin{align*} \frac{V_{N_1}}{R_4}+\frac{V_{N_1}}{R_5}&=10\:\text{mA}+\frac{V_{N_2}}{R_4}+\frac{0\:\text{V}}{R_5}\\\\ \frac{V_{N_2}}{R_3}+\frac{V_{N_2}}{R_4}&=\frac{V_{N_3}=100\:\text{V}}{R_3}+\frac{V_{N_1}}{R_4} \end{align*}

This solves out as $V_{N_2}=76\frac{1}{4}\:\text{V}$.

The nodal equation for the bottom schematic is trivially:\begin{align*} \frac{V_{N_1}}{R_4}+\frac{V_{N_1}}{R_5}&=10\:\text{mA}+\frac{0\:\text{V}}{R_4}+\frac{0\:\text{V}}{R_5}\\\\ \end{align*}

This solves out as $V_{N_1}=2\frac{2}{9}\:\text{V}$. So it is now easy to figure out the current through $V_2$ as the sum, $I_{V_2}=\frac{2\frac{2}{9}\:\text{V}}{400\:\Omega}+\frac{100\:\text{V}}{300\:\Omega}=\frac{61}{180}\:\text{A}$. And once again you are at the same place and able to work out the Thevenin resistance by dividing the open circuit voltage by the short circuit current.

• Thanks for the detailed answer, but what am I supposed to do with those voltages once solved? I have only seen single nodal problems like this before. – user177121 Feb 6 '18 at 5:39
• @krstr96 Those solved voltages, using $R_1=100\:\omega$ are pretty much useless. That's why I provided full equation solutions in my answer. Those equations let you apply any value you want for $R_1$. They are very flexible. I'll add a note to my answer to help you on this point. – jonk Feb 6 '18 at 6:00
• @krstr96 Added. Hope that helps. – jonk Feb 6 '18 at 6:05

Yes, since R2 is shorted it does not contribute to the Thevenin resistance. Your calculated resistance looks correct to me.

For finding the voltage at node A, you will want to use either mesh analysis, nodal analysis, or superposition, take your pick. I'll get you started on superposition here.

simulate this circuit – Schematic created using CircuitLab

First write the equation for $V_{A1}$, should be pretty straightforward. You can pretty much disregard R2, as it doesn't change the voltage across the resistor divider formed by R3, R4, and R5.

simulate this circuit

Next you'll want to write the equation for $V_{A2}$. It's a little bit trickier, you're going to want to find the current through R3+R4 by using the current divider rule: $$I=10mA*\frac{(R3+R4)//R5}{(R3+R4)+(R3+R4)//R5}$$

Then use Ohm's law to find the voltage at $V_{A2}$. Finally, just add $V_{A1}$ and $V_{A2}$ to get your thevenin voltage

• Is there any way you could set me up for nodal? I am still a bit confused and have never seen superposition before. – user177121 Feb 6 '18 at 3:18
• Oh, I figured you'd want superposition because it's very similar to what you did to calculate the resistance. For nodal, you need to figure out the voltages at the node between R5 and R4, and between R4 and R3. Write the equation summing the currents going out of the node (and set it equal to 0), and solve for the voltages – C_Elegans Feb 6 '18 at 3:23
• Is that current divider equation giving the current of the R3R4 branch? – Nedd Feb 6 '18 at 7:34
• Yes it is. Are you using superposition now? – C_Elegans Feb 6 '18 at 13:45
• Your current divider equations seems to have a problem. Try calculating both branch currents, they don't total to 10ma. – Nedd Feb 11 '18 at 19:46

simulate this circuit – Schematic created using CircuitLab

simulate this circuit

simulate this circuit

simulate this circuit