Electricity and Circuit Foundations
Resistance, Nonlinear Loads and Heat
Understand why Ohm's law is a model and why real loads may respond differently.
You are applying “Resistance, Nonlinear Loads and Heat” to a real system and need the concept, assumptions and verification boundary to stay clear.
Learn the difference between resistance at one operating point and genuinely ohmic behaviour, calculate resistive heating, and screen a real resistor against power, voltage, temperature and pulse limits.
- Select or check first
- One quotient is not a straight line. Static resistance R = V/I describes the line from the origin to one point on an current–voltage graph. Ohmic behaviour requires voltage to remain proportional to current under stated, substantially constant physical conditions.
- Controlling assumption
- Circuit state, waveform, source impedance, ratings, temperature and protection topology match the equipment being analysed.
- Top risk
- Low current does not make high voltage safe. A high-value resistor can carry a small operating current while its terminals remain at hazardous voltage. The wider source may also deliver far more fault current or stored energy than the branch calculation suggests.
- Proof
- Check the exact manufacturer data, units and limits, then verify the result by calculation, measurement or a controlled acceptance test.
Start with the basics Optional foundation before “Resistance, Nonlinear Loads and Heat”
Electrical-engineering basics
Electrical behaviour becomes easier to reason about when every voltage has a reference and every current has a complete return path.
- Voltage
- Voltage is potential difference between two points. A voltage statement is incomplete unless both points or its reference are known.
- Current
- Current is charge flow through a path. It does not disappear after a component; in a complete circuit it returns to the source.
- Resistance and impedance
- Resistance opposes steady current. Impedance extends that idea to changing signals and can depend on frequency through capacitance and inductance.
- Power and heat
- Electrical power is the rate of energy transfer. Some becomes useful output and some becomes heat that must stay within component and enclosure limits.
- Rating versus operating point
- A rating is a limit or qualified condition, not a target. The operating point must remain inside all relevant voltage, current, power, temperature and timing limits.
How to use this article: Draw the source, load, reference and return path before calculating. Treat exposed or stored electrical energy as hazardous until isolation is proved.
On this page
Dividing voltage by current can produce a resistance value at one operating point. It does not, by itself, prove that the same value will predict the device at another voltage, current or temperature. A fixed resistor may behave nearly linearly within stated conditions; a diode, lamp filament, thermistor, motor or semiconductor load may not. Good circuit work therefore asks two questions: what relationship is the device following now, and which limits apply while it follows it?
Resistance and Ohm's law are related, but not identical
Diagram description
A straight current-voltage line labelled ohmic and a curved line labelled nonlinear meet at one operating point. Construction lines emphasize that static resistance at one point cannot establish linearity.
Static or chord resistance at an operating point
R_static = V/I, for I ≠ 0; 1 Ω = 1 V/A
This quotient is useful when voltage and current refer to the same terminals, directions and instant or steady condition. It is a description of that point. Treating it as a constant that predicts other points is an additional model assumption.
- R_static
- static or chord resistance at the selected point, Ω
- V
- voltage across the selected terminals, V
- I
- current through the selected branch, A
If a device has 5.0 V across it and 50 mA through it, V/I = 100 Ω at that operating point. A second measurement or a known device model is needed before concluding that 10 V would produce 100 mA.
For an ideal ohmic element with constant resistance, V = IR is a predictive linear model: doubling V doubles I. Real metallic resistors approximate that model only within specified conditions because resistance also changes with temperature, manufacturing tolerance, frequency, voltage coefficient, ageing and mechanical or environmental stress. “Constant resistance” always has an implied accuracy and operating range.
Local slope resistance for a nonlinear device
r_d = dV/dI at the operating point
The local or small-signal slope can differ greatly from V/I. It predicts sufficiently small changes around the selected bias point, not the entire curve. When a graph plots current vertically and voltage horizontally, its visible slope is dI/dV, the reciprocal of r_d where the derivative exists.
- r_d
- differential or small-signal resistance, Ω
- dV
- small change in terminal voltage, V
- dI
- corresponding small change in current, A
A diode may have a large V/I quotient yet a much smaller dV/dI near its biased operating point. Neither value should be substituted for a complete large-signal diode model.
| Question | Useful quantity or model | Important condition |
|---|---|---|
| What is V/I at this point? | Static resistance R_static = V/I | Voltage and current must describe the same operating point |
| How will a small change behave near this bias? | Differential resistance r_d = dV/dI | Valid locally; frequency and temperature may matter |
| Can one constant predict a range of points? | Ohmic model V = IR with constant R | Check linearity and keep stated physical conditions substantially constant |
| How much resistor should be purchased? | Nominal resistance plus tolerance and temperature coefficient | Also check power, voltage, pulse, package and environment ratings |
Why common loads become nonlinear
A filament lamp heats as current rises, which changes its resistance. A thermistor is designed for resistance to vary strongly with temperature. A diode's carrier transport produces a curved current–voltage relationship. Motors add back electromotive force, winding resistance, inductance and mechanical speed or load. Electronic power supplies may draw switched current. These are not violations of Ohm's law; they are devices for which a single constant R is not a complete model.
Resistive electrical power becomes thermal power
Diagram description
An illustrative DC resistor example labels resistance, voltage and current, then evaluates P = VI, P = I squared R and P = V squared divided by R. Heat arrows and a caution about thermal conditions surround the component.
Equivalent power expressions for an ohmic resistor
P = VI = I²R = V²/R
The last two forms follow from V = IR and therefore require a resistance value appropriate to the operating condition. Use RMS voltage or current for average heating in an ideal linear resistor under periodic AC. For nonlinear, reactive or switched loads, calculate instantaneous p = vi or use the correct device and waveform model.
- P
- average dissipated power under the applicable condition, W
- V
- DC value or appropriate RMS voltage, V
- I
- DC value or appropriate RMS current, A
- R
- resistance applicable at that condition, Ω
For 100 Ω at 5.0 V, I = 5/100 = 0.050 A and P = 5 × 0.050 = 0.25 W. The three power forms agree because the example assumes a constant 100 Ω resistance.
The dissipated energy does not remain an abstract number. It raises the resistor and nearby materials above ambient until heat flow to leads, copper, board, air and surrounding structures balances the incoming average power—or until the operating interval ends. Component temperature therefore depends on power, duration, mounting, copper area, airflow, enclosure, nearby heat sources and ambient temperature.
A first thermal screen, not a temperature guarantee
ΔT ≈ P × θ
This linear steady-state screen is useful only when θ is defined for the same heat path, mounting and boundary conditions. A datasheet derating curve or terminal-temperature rating may be more appropriate. Transient pulses require thermal-impedance or pulse-load information rather than a steady θ value.
- ΔT
- estimated temperature rise across the defined thermal path, K or °C difference
- P
- average dissipated power for the model, W
- θ
- thermal resistance for the specified path and conditions, K/W
A generic θ value copied from a different package or PCB cannot reliably predict this resistor's body or film temperature. Use the selected manufacturer's test conditions.
A resistor has several independent limits
Diagram description
A high-resistance chip resistor example compares low current and moderate power with a separate 100-volt operating limit. A checklist calls for continuous dissipation, maximum voltage and pulse or overload verification against the current datasheet.
| Constraint | What it controls | What to verify |
|---|---|---|
| Nominal resistance and tolerance | Initial resistance range | Tolerance class, preferred value and production variation |
| Temperature coefficient and temperature range | Resistance change and permitted environment | TCR, specified temperature span and self-heating |
| Rated dissipation and derating | Allowable continuous heat generation | Reference ambient or terminal temperature, mounting assumptions and derating curve |
| Maximum operating or working voltage | Electrical stress across the resistive element | AC RMS/DC wording, applicable resistance range and any lower calculated power limit |
| Pulse or overload capability | Short-duration energy and peak stress | Amplitude, duration, repetition, waveform, resistance value and lifetime criteria |
| Insulation or dielectric voltage | Isolation between defined parts | Do not substitute it for maximum continuous working voltage |
| Package and assembly | Heat path and physical reliability | Land pattern, PCB material, solder process, coating, altitude and environment as applicable |
Two-limit voltage screen for a linear resistor
V_screen ≤ min(U_working, √(P_allow × R))
This is only an initial continuous-operation screen. P_allow must already reflect the correct ambient or terminal temperature and mounting condition. Tolerance, margin, transient voltage, pulse energy, waveform, insulation, altitude and safety requirements can impose lower limits.
- V_screen
- screened continuous voltage magnitude, V
- U_working
- manufacturer's applicable maximum operating or working voltage, V
- P_allow
- allowable dissipation after applicable derating, W
- R
- resistance at the assessed condition, Ω
A very high resistance can keep V²/R below the wattage limit while voltage itself reaches the element's working-voltage limit. A low current is therefore not proof of low electrical stress.
Worked example: the selected Vishay TNPW0603 e3 family
The April 2026 Vishay datasheet reviewed for the figure lists 0.210 W rated dissipation at 70 °C and 100 V AC RMS/DC maximum operating voltage for TNPW0603 e3. It also presents a higher terminal-temperature-based dissipation condition that depends on customer thermal management and PCB design. These are family-specific published conditions, not generic 0603 values and not permission to mix the most favourable numbers from different test bases.
| Check | Calculation or evidence | Result at the stated point |
|---|---|---|
| Current | I = V/R = 100 V / 100,000 Ω | 1.00 mA |
| Power | P = V²/R = 100²/100,000 | 0.10 W; below 0.210 W before any required derating |
| Operating voltage | Compare 100 V with listed Umax = 100 V AC RMS/DC | At the stated maximum, leaving no voltage margin |
| Temperature | Use the applicable ambient- or terminal-based rating and PCB condition | Not established by the electrical arithmetic |
| Pulse or transient | Check peak, duration, repetition and relevant pulse curve | Not established by the DC check |
| Design decision | Apply tolerance, supply variation, transient and reliability margins | Cannot be approved from 0.10 W alone |
A practical selection and verification sequence
| Step | Question |
|---|---|
| 1. Define | What resistance, accuracy, waveform, duty, ambient, lifetime and fault conditions are required? |
| 2. Calculate | What are nominal and worst-case voltage, current, continuous power and transient energy? |
| 3. Select | Which exact manufacturer, family, case size, resistance and tolerance meet those conditions? |
| 4. Derate | What allowable power remains at the real ambient or terminal temperature and mounting condition? |
| 5. Check independent limits | Are working voltage, pulse, overload, temperature and insulation requirements each satisfied with margin? |
| 6. Verify thermally | Does the intended PCB and enclosure keep the component within specified temperature limits? |
| 7. Control | Are the manufacturer part number, revision-sensitive requirements and approved substitutions documented? |
If voltage, current, power and energy still feel interchangeable, revisit Voltage, Current, Power and Energy Are Different. That vocabulary makes the distinction here precise: resistance relates voltage and current under a model, while power sets the rate of energy conversion into heat.
One-minute questions
Questions to answer before you apply this lesson
What should I select or check first for Resistance, Nonlinear Loads and Heat?
One quotient is not a straight line. Static resistance R = V/I describes the line from the origin to one point on an current–voltage graph. Ohmic behaviour requires voltage to remain proportional to current under stated, substantially constant physical conditions.
Which assumption can change the conclusion?
Circuit state, waveform, source impedance, ratings, temperature and protection topology match the equipment being analysed.
What can go wrong if I copy the example directly?
Low current does not make high voltage safe. A high-value resistor can carry a small operating current while its terminals remain at hazardous voltage. The wider source may also deliver far more fault current or stored energy than the branch calculation suggests.
How do I verify the result in the real system?
Check the exact manufacturer data, units and limits, then verify the result by calculation, measurement or a controlled acceptance test.
Sources and further reading
- SI Units — Electric CurrentNational Institute of Standards and Technology · 2026-10-13Authoritative metrology source reviewed for the ampere and the ohm relationship 1 Ω = 1 V/A.
- Ohm's Law — University Physics Volume 2OpenStax, Rice University · 2026-10-13University physics source reviewed for ohmic proportionality under constant physical conditions, nonlinear devices and temperature-dependent resistance.
- TNPW e3 — High Stability Thin Film Flat Chip ResistorsVishay Intertechnology · 2026-10-13Current manufacturer datasheet, document 28758 revision 10-Apr-2026, reviewed for TNPW0603 e3 resistance range, rated dissipation, terminal-temperature condition, operating voltage, temperature range, derating and pulse information.
- TNPW e3 Product InformationVishay Intertechnology · 2026-10-13Manufacturer product page reviewed to confirm the selected family, current datasheet and linked technical resources.