Solve node voltages and audit delivered power
Current paths must be defined before writing a matrix. A 6 V source connects to node 1 through Rs. Node 1 connects to ground through 2 Ω and to node 2 through 1 Ω. Node 2 connects to ground through R2. All resistances are positive, constant and lumped; ground is an explicitly chosen zero potential.
Learning outcome: Change R2 while holding Rs fixed. Predict which node and source currents change, then retain both node residuals and the independently calculated power residual. Explain why a zero residual checks arithmetic under this topology but does not verify the topology experimentally.
Starting recap
Review charge continuity, definite integrals, first-order differential equations, initial conditions and signed source work. Distinguish displayed time from physical time.
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Try
For Rs=2 Ω and R2=3 Ω, the equations are 2V1−V2=3 and −V1+(4/3)V2=0. Thus V1=2.4 V and V2=1.8 V. Source current is 1.8 A, bridge current 0.6 A and source power 10.8 W. The four resistor powers are 6.48, 2.88, 0.36 and 1.08 W, also totaling 10.8 W.
Open the experimentPredict
For Rs=2 Ω and R2=3 Ω, what is the bridge current from node 1 to node 2?
Experiment
Change R2 while holding Rs fixed. Predict which node and source currents change, then retain both node residuals and the independently calculated power residual. Explain why a zero residual checks arithmetic under this topology but does not verify the topology experimentally.
Solve coupled current balances and power
Worked teaching model · synthetic values · Ideal 6 V source to node 1 through Rsource; node 1 to ground through 2 ohm and to node 2 through 1 ohm; node 2 to ground through R2. Constant positive resistances, steady lumped circuit and explicit ground. Virtual only; no source or resistor handling.
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Bridge current for alternative second loads
Unlike graph quantities rescale independently; read units and values, not screen angles. 1: same source-side network (solid). Joined curves show the supplied model, not additional observations.
Equivalent numerical coordinates for Bridge current for alternative second loads
same source-side network
| second ground resistance (ohm) | bridge current (A) |
|---|---|
| 1 | 1 |
| 1.05 | 0.9836066 |
| 1.1 | 0.9677419 |
| 1.15 | 0.952381 |
| 1.2 | 0.9375 |
| 1.25 | 0.9230769 |
| 1.3 | 0.9090909 |
| 1.35 | 0.8955224 |
| 1.4 | 0.8823529 |
| 1.45 | 0.8695652 |
| 1.5 | 0.8571429 |
| 1.55 | 0.8450704 |
| 1.6 | 0.8333333 |
| 1.65 | 0.8219178 |
| 1.7 | 0.8108108 |
| 1.75 | 0.8 |
| 1.8 | 0.7894737 |
| 1.85 | 0.7792208 |
| 1.9 | 0.7692308 |
| 1.95 | 0.7594937 |
| 2 | 0.75 |
| 2.05 | 0.7407407 |
| 2.1 | 0.7317073 |
| 2.15 | 0.7228916 |
| 2.2 | 0.7142857 |
| 2.25 | 0.7058824 |
| 2.3 | 0.6976744 |
| 2.35 | 0.6896552 |
| 2.4 | 0.6818182 |
| 2.45 | 0.6741573 |
| 2.5 | 0.6666667 |
| 2.55 | 0.6593407 |
| 2.6 | 0.6521739 |
| 2.65 | 0.6451613 |
| 2.7 | 0.6382979 |
| 2.75 | 0.6315789 |
| 2.8 | 0.625 |
| 2.85 | 0.6185567 |
| 2.9 | 0.6122449 |
| 2.95 | 0.6060606 |
| 3 | 0.6 |
| 3.05 | 0.5940594 |
| 3.1 | 0.5882353 |
| 3.15 | 0.5825243 |
| 3.2 | 0.5769231 |
| 3.25 | 0.5714286 |
| 3.3 | 0.5660377 |
| 3.35 | 0.5607477 |
| 3.4 | 0.5555556 |
| 3.45 | 0.5504587 |
| 3.5 | 0.5454545 |
| 3.55 | 0.5405405 |
| 3.6 | 0.5357143 |
| 3.65 | 0.5309735 |
| 3.7 | 0.5263158 |
| 3.75 | 0.5217391 |
| 3.8 | 0.5172414 |
| 3.85 | 0.5128205 |
| 3.9 | 0.5084746 |
| 3.95 | 0.5042017 |
| 4 | 0.5 |
| 4.05 | 0.4958678 |
| 4.1 | 0.4918033 |
| 4.15 | 0.4878049 |
| 4.2 | 0.483871 |
| 4.25 | 0.48 |
| 4.3 | 0.4761905 |
| 4.35 | 0.4724409 |
| 4.4 | 0.46875 |
| 4.45 | 0.4651163 |
| 4.5 | 0.4615385 |
| 4.55 | 0.4580153 |
| 4.6 | 0.4545455 |
| 4.65 | 0.4511278 |
| 4.7 | 0.4477612 |
| 4.75 | 0.4444444 |
| 4.8 | 0.4411765 |
| 4.85 | 0.4379562 |
| 4.9 | 0.4347826 |
| 4.95 | 0.4316547 |
| 5 | 0.4285714 |
At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.
| Quantity | Value | Unit |
|---|---|---|
| First node potential | 2.4 | V |
| Second node potential | 1.8 | V |
| Source current | 1.8 | A |
| Bridge current from node 1 | 0.6 | A |
| First-node current residual | -4.44089e-16 | A |
| Second-node current residual | 0 | A |
| Source power delivered | 10.8 | W |
| Total resistive conversion | 10.8 | W |
| Power residual | 0 | W |
Node equations enforce local charge continuity and the separate power sum checks the signs.
The conductance matrix is [(1/Rs+1/2+1),−1;−1,(1+1/R2)]. Multiplying by the node-voltage vector gives [6/Rs,0]. Currents use potential differences divided by their own resistances. The voltage-source delivered power equals the sum I²R over all four resistors. No accumulation occurs in this stipulated steady state; it does not justify ignoring capacitive charge during a switch.
Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.
Node equations enforce local charge continuity and the separate power sum checks the signs.
The conductance matrix is [(1/Rs+1/2+1),−1;−1,(1+1/R2)]. Multiplying by the node-voltage vector gives [6/Rs,0]. Currents use potential differences divided by their own resistances. The voltage-source delivered power equals the sum I²R over all four resistors. No accumulation occurs in this stipulated steady state; it does not justify ignoring capacitive charge during a switch.
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