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Internal Resistance Calculator

Find terminal voltage and current for a battery with internal resistance r

Parameters

Vⓘ
Ωⓘ
Ωⓘ
Show Trail

Controls

xⓘ

Calculated Values

Current:
1.14;A1.14;A
Terminal Voltage:
11.43;V11.43;V
Power in Load:
13.06;W13.06;W

Examples

12 V cell, r=0.5 Ω, R=10 Ω

Typical small battery load.

  • Current: 1.141.14

9 V, r=1 Ω, R=8 Ω

Transistor radio load.

  • Terminal Voltage: 8.008.00

Visualization

EMF, Internal Resistance, and Terminal Voltage

Real voltage sources (cells, batteries) have internal resistance r in series with ideal EMF ℰ. Terminal voltage V = ℰ − Ir when current I flows outward.

Open circuit (I = 0): V_terminal = ℰ — voltmeter across terminals reads full EMF. Short circuit (R_load ≈ 0): I = ℰ/r — very large if r is small (dangerous).

Loaded circuit: I = ℰ/(r + R). Terminal V = IR = ℰR/(r+R). As R decreases, I increases and terminal voltage drops.

Maximum power transfer: load R = r delivers max power to load (P_max = ℰ²/4r). Efficiency is only 50% at this point — used when matching matters more than efficiency.

Plotting V vs I gives straight line: intercept ℰ, slope −r. Aging batteries: ℰ may drop and r increases — sag under load.

Series cells: ℰ add, r add. Parallel identical cells: ℰ same, r divides (if matched).

Key Concepts

  • ℰ — electromotive force (open-circuit V)
  • V_terminal = ℰ − Ir
  • I = ℰ/(r + R_load)
  • Max power when R = r
  • V–I graph: intercept ℰ, slope −r
  • Aging increases r

Real-World Applications

  • Battery state-of-health testing
  • Power supply regulation design
  • Automotive starter current sag
  • Class 12 cells and combinations
  • Maximum power transfer matching

Explore Further

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  • Power Calculator

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  • Series Circuit

    Analyze series circuits with total resistance, current, and voltage drops.

  • Parallel Circuit

    Analyze parallel circuits with current division and total resistance.

Physics Equations

Current:
I=Er+RI = \frac{\mathcal{E}}{r + R}
Terminal Voltage:
V=E−IrV = \mathcal{E} - Ir

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Real Battery Model

Ideal EMF ℰ in series with internal resistance r, then load R.

Equation:

E→r→R\mathcal{E} \rightarrow r \rightarrow R

Explanation:

Terminal voltage is measured across the load, between battery terminals.

2

Step 2: Total Resistance and Current

Equation:

I=Er+RI = \frac{\mathcal{E}}{r + R}

Calculation:

I=120.5+10=1210.5=1.142857 AI = \frac{12}{0.5 + 10} = \frac{12}{10.5} = 1.142857 \text{ A}

Result:

I=1.142857AI = 1.142857 A

Explanation:

Series resistances add. Smaller load R → larger current.

3

Step 3: Voltage Drop Across Internal Resistance

Equation:

Vr=IrV_r = I r

Calculation:

Vr=1.142857×0.5=0.571429 VV_r = 1.142857 \times 0.5 = 0.571429 \text{ V}

Result:

Lostvoltage=0.571429VLost voltage = 0.571429 V

Explanation:

This drop is wasted as heat inside the battery — not available at terminals.

4

Step 4: Terminal Voltage

Equation:

Vterminal=E−Ir=IRV_{terminal} = \mathcal{E} - Ir = IR

Calculation:

Vterminal=12−0.571429=11.428571 VV_{terminal} = 12 - 0.571429 = 11.428571 \text{ V}

Result:

Vterminal=11.428571VV_terminal = 11.428571 V

Explanation:

Also equals IR — voltage across load equals terminal voltage when r is in series.

5

Step 5: Power Distribution

Split power between useful load and internal loss.

Calculation:

Pload=I2R=13.0612 W,Pinternal=I2r=0.6531 WP_{load} = I^2 R = 13.0612 \text{ W}, \quad P_{internal} = I^2 r = 0.6531 \text{ W}

Explanation:

Useful fraction ≈ 95.2% of total power delivered by source.

6

Step 6: Limiting Cases

Open circuit and short circuit behavior.

Explanation:

Open (R→∞): I→0, V_terminal→ℰ=12 V. Short (R→0): I→ℰ/r=24.0000 A — limited by internal r.

Frequently Asked Questions (FAQ)

EMF vs terminal voltage?

EMF is open-circuit potential; terminal voltage is what you get under load (lower by Ir).

Why do car lights dim when starting?

Starter draws large I; voltage drop across battery internal r reduces voltage to other loads.

How to measure r?

Measure ℰ (open) and V, I with known load; r = (ℰ−V)/I.

Is maximum power transfer efficient?

No — 50% wasted in r at R=r. Use when signal power matters, not energy efficiency.

Ideal battery?

r = 0 — terminal V always ℰ regardless of I (idealization).

Practice MCQs

  1. Open-circuit terminal voltage equals:
  2. As load resistance increases, terminal voltage:
  3. Maximum power to load occurs when R_load =
  4. Internal resistance r causes:
  5. Two identical cells in series give total ℰ and r:
  6. Short-circuit current is approximately:

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