Internal Resistance

Internal Resistance ($r$) is the resistance to the flow of electric current that exists within the source itself (battery, generator, or any EMF source). It acts as if it were a resistor connected in series with the ideal EMF source.

Definition

When there is current in a circuit:

  • The potential difference across the terminals is always less than the EMF of the source
  • This is due to the resistance to current flow within the source itself
  • This resistance is called the internal resistance ($r$)

Physical Origin

Internal resistance arises from:

  1. In batteries: Resistance of the electrolyte and electrodes
  2. In generators: Resistance of the windings and brushes
  3. In solar cells: Resistance of the semiconductor material and contacts

Circuit Model

A real source can be modeled as:

flowchart LR
    A["Ideal EMF<br/>ε"] --- B["Internal<br/>Resistance r"]
    B --- C["Terminal +"]
    C --- D["External Load R"]
    D --- E["Terminal -"]
    E --- A
    
    style A fill:#90EE90
    style B fill:#FFB6C1
  • An ideal EMF source ($\varepsilon$) in series with internal resistance ($r$)
  • The combination is connected to an external load ($R$)

Key Equations

Terminal Voltage

When delivering current (discharging):

$$V = \varepsilon - Ir$$

where:

  • $V$ = terminal voltage (voltage across external circuit)
  • $\varepsilon$ = EMF of the source
  • $I$ = current
  • $r$ = internal resistance

Internal Resistance Formula

$$r = \frac{\varepsilon - V}{I}$$

Complete Circuit Equation

$$\varepsilon = IR + Ir = V + Ir$$

where $IR = V$ is the voltage drop across the external resistance.

Three Operating Conditions

1. Delivering Current (Discharging)

When the battery supplies current to an external circuit:

$$V = \varepsilon - Ir$$

  • Terminal voltage decreases as current increases
  • Terminal voltage is always less than EMF
  • This is the normal operating condition for a battery powering a device

2. Receiving Current (Charging)

When an external source charges the battery:

$$V = \varepsilon + Ir$$

  • Terminal voltage exceeds EMF
  • Higher voltage is needed to overcome both the EMF and internal resistance
  • This is why car battery chargers output ~14-15 V for a 12 V battery

3. No Current (Open Circuit)

When the circuit is open ($I = 0$):

$$V = \varepsilon$$

  • Terminal voltage equals EMF
  • No voltage drop across internal resistance
  • Used to measure the true EMF of a source

Power Dissipation

Power Lost in Internal Resistance

When current flows, power is dissipated as heat in the internal resistance:

$$P_{r} = I^{2}r$$

This represents energy lost within the source rather than being delivered to the external circuit.

Power Delivered to Load

$$P_{R} = I^{2}R = V \cdot I$$

Total Power Generated

$$P_{\text{total}} = \varepsilon \cdot I = P_{R} + P_{r}$$

Efficiency

$$\eta = \frac{P_{R}}{P_{\text{total}}} = \frac{R}{R + r}$$

  • Efficiency approaches 100% when $R \gg r$
  • Efficiency is 50% when $R = r$ (maximum power transfer condition)

Typical Values

Source Typical EMF Typical Internal Resistance
AA Alkaline (fresh) 1.5 V 0.1 - 0.5 Ω
AA Alkaline (used) 1.5 V 0.5 - 5 Ω
Lead-acid Car Battery 12 V 0.005 - 0.02 Ω
Lithium-ion Cell 3.7 V 0.05 - 0.2 Ω
Lab Power Supply Variable 0.001 - 0.1 Ω

Effects of Internal Resistance

1. Voltage Drop Under Load

As current increases, terminal voltage decreases:

$$\Delta V = \varepsilon - V = Ir$$

2. Heating in the Source

Internal resistance causes the source to heat up during operation:

$$P_{\text{heat}} = I^{2}r$$

3. Limited Short-Circuit Current

Maximum current occurs when $R = 0$ (short circuit):

$$I_{\text{max}} = \frac{\varepsilon}{r}$$

A small internal resistance limits the short-circuit current (safety feature).

Measurement of Internal Resistance

Method 1: Using Terminal Voltage and Current

  1. Measure EMF with open circuit: $\varepsilon = V_{\text{open}}$
  2. Connect known load $R$ and measure terminal voltage $V$
  3. Calculate current: $I = V/R$
  4. Calculate internal resistance: $r = \frac{\varepsilon - V}{I}$

Method 2: Using Two Different Loads

With two different loads $R_1$ and $R_2$:

$$r = \frac{V_2 - V_1}{I_1 - I_2}$$

where $I_1 = V_1/R_1$ and $I_2 = V_2/R_2$.

Worked Examples

Example 1: Finding Terminal Voltage

Problem: A battery has EMF 9.0 V and internal resistance 0.5 Ω. What is its terminal voltage when delivering 2.0 A?

Solution:

$$V = \varepsilon - Ir = 9.0 - (2.0)(0.5) = 9.0 - 1.0 = 8.0 \text{ V}$$

Example 2: Finding Internal Resistance

Problem: A cell has EMF 1.5 V. When connected to a 5 Ω resistor, the terminal voltage is 1.4 V. Find the internal resistance.

Solution:

First find current: $$I = \frac{V}{R} = \frac{1.4}{5} = 0.28 \text{ A}$$

Then: $$r = \frac{\varepsilon - V}{I} = \frac{1.5 - 1.4}{0.28} = \frac{0.1}{0.28} = 0.36 \text{ Ω}$$

Example 3: Charging a Battery

Problem: A 12 V car battery with internal resistance 0.05 Ω is being charged at 60 A. What is the terminal voltage?

Solution:

Since charging (receiving current):

$$V = \varepsilon + Ir = 12 + (60)(0.05) = 12 + 3 = 15 \text{ V}$$

Example 4: Power Dissipation

Problem: For the car battery in Example 3, calculate: (a) Power dissipated as heat in the battery (b) Rate of chemical energy storage (c) Total power input

Solution:

(a) Power dissipated: $$P_{r} = I^{2}r = (60)^{2}(0.05) = 180 \text{ W}$$

(b) Chemical energy storage rate: $$P_{\text{chem}} = \varepsilon I = (12)(60) = 720 \text{ W}$$

(c) Total power input: $$P_{\text{total}} = P_{r} + P_{\text{chem}} = 180 + 720 = 900 \text{ W}$$

Or: $$P_{\text{total}} = VI = (15)(60) = 900 \text{ W}$$

Related Concepts

Key Takeaways

  1. Internal resistance is unavoidable in all real sources
  2. Terminal voltage < EMF when delivering current
  3. Terminal voltage > EMF when charging
  4. Power is dissipated as heat in the internal resistance ($I^{2}r$)
  5. Smaller internal resistance means better performance under load
  6. Internal resistance typically increases as batteries age or deplete