Nodal Analysis

Nodal Analysis is a method to calculate unknown parameters in a circuit design. It relies heavily on Kirchoff's Current Law, which say that the sum of the currents entering any node is equal to the sum of the currents exiting that same node.

In this experiment we are asked to analyze the following circuit:

Some information that we might like to have is the voltage between R1 and R2 (Which we will call V3), and the Voltage between R2 and R3 (V4). 

 

The appropriate method to determine these unknown voltages is to write KCL equations at each node, and then solve the resulting system of equations. If the bottom rail is assigned the value of 0V or Ground, the expression governing the node between R1 and R2 is:

 

(V3-V1)/100+V3/1000+(V3-V4)/220 = 0

 

for the second node the equation is:

 

(V4-V3) /220+V4/1000+(V4-V2)/220 = 0

At this point it may concern you that all of the currents are directed out of each node, but the voltages will take care of the direction of the current. For instance, the term (V4-V2)/220 if V4 is greater than V2 then the current will flow from left to right, or out of the node. If V2 is greater than V4 the current will be negative, indicating the opposite direction.

At this point you can solve for the node voltages by noting that V1 is 12 V and that V2 = 9V. Alternately, you could run this simulation in pSpice and arrive at the following values.

Using these results it is possible to calculate the current through each branch. The result should look something like this.

Based on this analysis, the 12V power supply is delivering 17.45 mA of current, and the 9V power supply is delivering 1.484 mA of current.

The power from V1 is (12V*.01745A) = .2094 W

Power from V2 is (9V *.001484A) = .01336 W

Now that the analysis is complete, it is time to build the circuit. In place of the two voltage supplies we are using 2 lab volt unregulated power supplies. Both are capable of providing the voltage and current necessary for the experiment.

Below is a picture of the circuit constructed on a bread board. Note that the orientation is upside down in reference to the earlier schematic.

Using this circuit, we took measurements of voltage and current to compare with the simulation.

The results are summarized in the following table:

The largest error can be found in the current from the second supply, however a small error in the output voltage of the 12V supply can be magnified significantly in the 9V supply, as its total current is an order of magnitude smaller.

The actual power delivered by each of the power supplies was found to be:

P1 = .2244 W
P2 = .01197 W

P1 error = 7.1%
P2 error = -10.4%

In the second step of the experiment we are asked to determine the  necessary power supply voltage such that V3=V4=9V.

Below is a diagram which describes the necessary source voltages to achieve this:

We constructed this circuit and then measured the actual voltages at each node, in addition to the currents leaving each voltage source.

I1 = 8.4mA
I2 = 9.5mA
V3 = 8.89V
V4 = 8.98V

The difference between the measured values and the calculated values can be mostly attributed to the non-ideal resistance of the resistors in the circuit, in addition to additional circuit loads such as the resistance of connecting cables, and the resistance of the ammeter when it was connected.