Power dissipation

Electrical power is the rate at which energy crosses an element. A resistor converts electrical energy into modeled internal energy or heat, but a voltage source can supply or absorb energy depending on its operating point. The sign is meaningful only after voltage polarity and current reference direction are defined together.

Think like a programmer

Use one passive-sign-convention record for every branch: voltage is positive at the terminal where the reference current enters. Calculate p = vi from those same signed values, never from separately displayed magnitudes. Emit a per-element power table and a whole-network residual at every solved state. Integrate power over time with timestamps when a transient model needs energy rather than instantaneous transfer rate.

Model checklist

Inputs: Branch voltage polarity, current reference direction, component parameters, source schedule, sample times, integration rule, and power-balance tolerance.
State: Signed branch voltages and currents, instantaneous element powers, cumulative energy by element, total source/sink power, and residual history.
Rule: Multiply compatible signed voltage and current for each branch; sum all element powers; integrate selected powers across recorded time intervals.
Output: Power in watts, energy in joules, source/absorber classification under the chosen convention, and network balance residual.
Check: A positive-resistance element has nonnegative I²R dissipation; reversing both voltage and current leaves p unchanged; reversing only one reference flips p; a closed ideal solved network has summed instantaneous power near zero; transient stored-energy change plus transfers balances within integration tolerance.

\[P=VI=I^2R=\frac{V^2}{R}\]

The three resistor expressions are equivalent only for an ideal resistor with compatible values: P = VI = I²R = V²/R. Use each as a targeted check. With fixed resistance, doubling current must quadruple dissipation. With fixed voltage, doubling resistance halves dissipation. A negative result from I²R for positive R is not “a source mode”; it is an inconsistent input or bug. A negative VI can be valid under the passive convention and means the element is delivering power rather than absorbing it.

\[p_k(t)=v_k(t)i_k(t),\qquad \sum_k p_k(t)\approx0,\qquad \Delta U\approx\int_{t_0}^{t_1}p(t)\,dt\]

In a static circuit, the residual is the sum of every branch’s signed power. A resistor may absorb +12 W while a source reports −12 W; both signs are correct if their voltage/current references follow the same convention. In a transient circuit, add stored capacitor and inductor energy changes and boundary/source transfer over the same interval. Sampling at different timestamps or mixing one branch’s RMS quantity with another branch’s instantaneous quantity invalidates the ledger.

Numerical power balance is a debugging instrument, not a proof that every component law is physical. A solver can satisfy an algebraic balance while using the wrong resistance or an unrealistic source. Pair the residual with component-law checks, units, source metadata, and an analytic case. If a model intentionally includes an untracked loss term, name it; do not hide it in a residual.

Try this experiment

Prediction: Doubling current quadruples ideal resistor power at fixed resistance, while a source and resistor have opposite signed roles in a passive-sign-convention ledger.

Choose a resistor current reference and voltage polarity. Calculate its power at I and 2I. Add an ideal source that supplies the same magnitude, assign its reference directions, and verify the two signed powers sum to zero. For a transient trace, integrate source and resistor power over one named interval and compare with the change in stored energy.

Where this model breaks

Heating changes resistance and physical temperature distribution. Ideal power accounting does not model thermal safety, electromagnetic radiation, chemical source dynamics, switching loss, parasitic elements, frequency-dependent impedance, or measurement bandwidth. A numerical energy integral also inherits the time resolution and interpolation policy of its samples.

Summary

Use signed power from the solver’s own voltage/current references, then maintain instantaneous and time-integrated energy ledgers. Balance residuals expose sign, topology, timestamp, and transfer-accounting bugs, but must be paired with component-law checks.

Glossary

Power: energy transfer per time.
Dissipation: conversion to internal energy.
Passive sign convention: orientation convention for element power.
Power residual: summed signed element power that should vanish for a closed ideal instantaneous network.
Energy ledger: accumulated transfers and stored-energy changes over a stated interval.
Self-check
1. How does ideal resistor power scale with I at fixed R?
2. Why are voltage and current reference directions part of a power input?
3. What does a negative VI mean under the passive sign convention?
4. Which additional term belongs in a transient energy balance?

Sources

OpenStax, University Physics Volume 2, Electric Power.
C. K. Alexander and M. N. O. Sadiku, Fundamentals of Electric Circuits.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Power Dissipation, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Currents and Circuits, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Power Dissipation as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Power Dissipation into a test

Account for signed element power and verify whole-network electrical energy balance.

Name the inputs and units that the currents and circuits model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

Share to: