Nodal Analysis

PROBLEM
We want to construct a "reliable" power system - a system that has the ability to survive a casualty or cascade of damage. We model a circuit consisting of 2 power supplies, 2 loads, and cables with cable resistances.

Given the circuit above, we use nodal analysis to find the two unknown voltages V2, V3.


CALCULATIONS
  Nodal equations:
  (V2-Vbatt1)/Rc1 + (V2-V3)/Rc2 + (V2)/RL1 = 0
  (V3-V2)/Rc2 + (V3-V4)/Rc3 + V3/RL2 = 0
where: Rc1 = 100 ohms, Rc2=Rc3= 220 ohms, RL1=RL2=1 kohm, Vbatt1 = 12V, Vbatt2= 9V.
Solving the two equations:  V2 = 10.2 V,  V3 = 8.7 V

Next, we find the current leaving each battery & the power supplied by each battery.

Ibatt1= (Vbatt1-V2)/Rc1 = 0.018 A = 18mA
Ibatt2 = (Vbatt2-V3)/Rc3= 0.014 A = 14mA

Psupp = Vi
Pbatt1= 0.216 W
Pbatt2= 0.126 W

Data
Components Data Table

Component	Nominal Val	Measured Val	Power/Current Rating
Rc1	100 ohms	98.2 ohms	0.25 W
Rc2	220 ohms	217 ohms	.0125 W
Rc3	220 ohms	217 ohms	.0125 W
RL1	1 kohm	976 ohms	.0125 W
RL2	1 kohm	979 ohms	.0125 W
Vbatt1	12 V	12.07 V	2A
Vbatt2	9 V	9.08 V	2A

Experimental Values Data Table

Variable	Theoretical Val	Measured Val	Percent Error
Ibatt1	0.017 A	17.4 mA
Ibatt2	0.0014 A	12 mA
V2	10.2 V	10.32
V3	8.7 V	8.7

Next, we calculate the power delivered by the two batteries:
Pbatt1 - 17.4mA*(12.07V) = 0.210 W
Pbatt2 - 12mA*(9.08V) = 0.109 W

Finally, assume we change the problem: suppose V2=V3=9V. Use the nodal equations to find the required battery voltages Vbatt1, Vbatt2.

Vbatt1 = 9.9 V
Vbatt2 = 10.8 V

Implement the circuit using the new battery voltages and record the achieved node voltages and battery currents.

V2 = 8.96 V
V3 = 8.79 V
Ibatt1 = 9.48 mA
Ibatt2 = 8.24 mA