Math Bridge: Transistor Switch Loss

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Math BridgeElectronicsStruggle-friendly runway

Why can the same motor make one transistor hot and another merely warm?

Connect drive current and channel resistance to loss, package heating, and a safer switch choice.

Eddie, the electronics guideEddie guides
The one targetTurn load current into comparable BJT and MOSFET heat ledgers.
The chapter caseA 2.00 A motor, forced beta 25, and a 28 milliohm logic-level MOSFET.
What it buys youA switch choice tied to drive limits and temperature, not a part label.

A field team faces an unresolved physical question: Why can the same motor make one transistor hot and another merely warm? They must answer it before changing load current on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is load current. The middle card applies this page's relationship. The green card is bjt base current. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Load current changes bjt base current An input card leads through the page relationship to the bjt base current result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. Doubling current doubles the simple BJT ledger but quadruples I-squared-R loss, so neither device stays cool by label alone.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for load current is 2.

  2. 2

    Name the relationship. IB=2.00/25=0.080 A PBJT=0.5(2.00)+(5.00-0.700)(0.080)=1.344 W PMOS=(2.00)^2(0.0280)=0.112 W loss ratio=1.344/0.112=12.0 delta TJ,MOS=0.112(62)=6.94 °C

  3. 3

    Substitute the chapter fixture. Set load current to 2. The page ledger gives bjt base current as 80.0 mA.

  4. 4

    Read the result. Keep mA beside the value. Use it only inside the technical boundary on this page.

Predict, then change load current

Try Predict the direction of bjt base current. Move one control, calculate, then check your prediction.

2
Chapter baseline
BJT base current

Observe Doubling current doubles the simple BJT ledger but quadruples I-squared-R loss, so neither device stays cool by label alone. Reset the control to 2 and compare bjt base current.

Explain Only load current moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only load current moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only load current. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A BJT needs continuing base current and keeps a saturation voltage across its main path. A fully enhanced MOSFET instead looks mainly like a small resistance. Both losses become heat.

Eddie: Compare what each device must spend while carrying exactly the same load current.

2. Name every algebra move

1

Find base currentDivide load current by forced beta: IB=I/beta.

2

Price BJT conductionMultiply saturation voltage by current.

3

Price base driveMultiply the driver voltage left after VBE by IB.

4

Price the MOSFETSquare current and multiply by RDS(on).

5

Turn loss into heatMultiply watts by package thermal resistance.

3. Reproduce the chapter case

IB=2.00/25=0.080 A
PBJT=0.5(2.00)+(5.00−0.700)(0.080)=1.344 W
PMOS=(2.00)^2(0.0280)=0.112 W
loss ratio=1.344/0.112=12.0
delta TJ,MOS=0.112(62)=6.94 °C

The smaller MOSFET loss is useful only if 28 milliohms is guaranteed at the real gate voltage.

4. Try one real input

TryMove the motor current, predict which loss rises fastest, then compare the full ledger.

Load current
BJT base current
BJT path loss
Base-drive loss
BJT total loss
MOSFET loss
Loss advantage
MOSFET rise
BJT rise

ObserveBJT terms rise with current, while MOSFET channel loss rises with current squared.

ExplainDoubling current doubles the simple BJT ledger but quadruples I-squared-R loss, so neither device stays cool by label alone.

Technical boundaries.

This transparent steady-on ledger uses the chapter constants.

Drive
Forced beta and RDS(on) must be verified at the real GPIO voltage and temperature.
Switching
Gate charge, transition loss, frequency, and inductive turn-off are outside this calculation.
Thermal
The quoted theta JA is an illustrative package-to-ambient path, not a board guarantee.

Correct, not complete: this ledger does not select, derate, protect, or thermally qualify a transistor stage.

5. Use the result in the design

Reject a BJT drive current the GPIO cannot supply, then verify the MOSFET's on-resistance at the actual gate voltage and worst-case junction temperature.

6. Record the evidence state

Record load current including stall, gate or base voltage, measured device drop, switching frequency, ambient temperature, package, copper area, and protection path.

7. Check yourself

Why is VGS(th) not an on-resistance guarantee?
Answer: Threshold only marks the start of channel formation at a tiny test current; low RDS(on) needs a stronger specified gate drive.
Why does MOSFET conduction loss accelerate with current?
Answer: The same current creates the voltage drop IR, so power is I(IR)=I²R.
Does a low steady loss prove the switch is safe?
Answer: No. Switching loss, stall current, avalanche, package limits, board cooling, and ambient conditions still matter.
Honesty boundary.

The arithmetic reproduces the named chapter case; it is an inspectable comparison, not a component approval.

Drive
Forced beta and RDS(on) must be verified at the real GPIO voltage and temperature.
Switching
Gate charge, transition loss, frequency, and inductive turn-off are outside this calculation.
Thermal
The quoted theta JA is an illustrative package-to-ambient path, not a board guarantee.

Correct, not complete: this ledger does not select, derate, protect, or thermally qualify a transistor stage.

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