Math Bridge: LDR Power-law Compression

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Why equal light ratios make shrinking voltage steps

One thread from the LDR power law to the chapter's 10, 100, and 1,000 lux divider outputs.

Phoebe, the physics guidePhoebe guides
The one targetUse γ to connect light, resistance, voltage, and behaviour.
The chapter case10 kΩ at 10 lux, γ=0.7, 10 kΩ divider, 3.3 V.
What it buys youPredict when steering sensitivity will flatten.

A field team faces an unresolved physical question: Why equal light ratios make shrinking voltage steps They must answer it before changing ldr gamma exponent 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 ldr gamma exponent. The middle card applies this page's relationship. The green card is r at 1,000 lux. 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.

LDR gamma exponent changes r at 1,000 lux An input card leads through the page relationship to the r at 1,000 lux result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The resistance readouts use the chapter's power law. The voltage readouts put those same resistances into its 3.3 V divider equation.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for ldr gamma exponent is 0.7.

  2. 2

    Name the relationship. R(E)=R_ref(E/E_ref)^(-γ); V_out=V_s R_fixed/(R_LDR+R_fixed)

  3. 3

    Substitute the chapter fixture. Set ldr gamma exponent to 0.7. The page ledger gives r at 1,000 lux as 0.398 kohm.

  4. 4

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

Predict, then change ldr gamma exponent

Try Predict the direction of r at 1,000 lux. Move one control, calculate, then check your prediction.

0.7
Chapter baseline
R at 1,000 lux

Observe The resistance readouts use the chapter's power law. The voltage readouts put those same resistances into its 3.3 V divider equation. Reset the control to 0.7 and compare r at 1,000 lux.

Explain Only ldr gamma exponent 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 ldr gamma exponent moves. Field effects named in the page's technical boundary stay fixed.

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

1. Read a power law in words

R=R_ref(E/E_ref)^(−γ)

E is light level. R is LDR resistance. The minus sign means resistance falls as light rises. γ controls how steeply it falls.

Phoebe: A tenfold light increase always multiplies resistance by 10^(−γ). It does not subtract a fixed number of ohms.

2. Bound the exponent with recombination

At steady state, carrier generation G balances recombination R(n): dn/dt=G−R(n)=0. If recombination is proportional to n, then n∝E and γ=1. If it is proportional to n², then n∝√E and γ=0.5. The chapter's γ=0.7 sits between those ideal limits.

3. Work the resistance points

1

At 10 luxR=10 kΩ.

2

At 100 luxR=10×10^(−0.7)=1.995 kΩ, about 2.0 kΩ.

3

At 1,000 luxR=10×100^(−0.7)=0.398 kΩ, about 0.4 kΩ.

4. Try the exponent γ

R(E)=R_ref(E/E_ref)^(−γ); V_out=V_s R_fixed/(R_LDR+R_fixed)

TryMove γ between the 0.5 and 1.0 physical limits and watch the same three light points.

R at 100 lux
R at 1,000 lux
V at 10 lux
V at 100 lux
V at 1,000 lux
First voltage step
Second voltage step

ObserveAt γ=0.7, equal 10× light jumps give 1.101 V and then only 0.423 V.

ExplainThe resistance readouts use the chapter's power law. The voltage readouts put those same resistances into its 3.3 V divider equation.

Technical boundaries.

One γ is an approximate fit over a limited light and temperature range.

Real LDRs vary by unit, remember prior illumination, respond slowly, and may saturate with the ADC or motor driver
Needs separate evidence

Use field evidence or a deeper model before release.

5. Turn resistance into divider voltage

V_out=3.3×10/(R_LDR+10)

At 10 lux this is 1.650 V. At 100 lux it is 2.751 V. At 1,000 lux it is 3.174 V. The voltage rises because the top LDR resistance falls.

6. Connect the curve to behaviour

The first light decade moves the motor command much more than the second. Near bright light, a left-right lux difference can make only a small voltage difference. The vehicle becomes less sensitive even though the physical light still changes.

7. Check yourself

What does the minus sign in E^(−γ) mean?
Answer: Resistance decreases as illuminance increases.
Why does 100 lux give about 2.0 kΩ?
Answer: 100/10=10, so R=10×10^(−0.7)=1.995 kΩ.
Why are the two voltage steps unequal?
Answer: Both the LDR power law and the divider are nonlinear, so equal light ratios do not create equal voltage additions.
Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

10 kΩ at 10 lux
Resistance or impedance value
γ=0.7
Named physical or model constant
10 kΩ fixed resistor
Resistance or impedance value
3.3 V supply
Voltage or voltage-step value
2.0/0.4 kΩ
Resistance or impedance value
1.65/2.75/3.17 V
Voltage or voltage-step value
1.995
Chapter input or worked result
0.398
Chapter input or worked result
2.751
Chapter input or worked result
3.174
Chapter input or worked result
1.101
Chapter input or worked result
0.423 results keep extra digits only to show the calculation
Chapter input or worked result

The simple power law is not a production calibration.

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