Terminal Voltage

Terminal Voltage ($V$) is the operating potential difference measured across the terminals of a source (battery, generator, etc.) when connected to a circuit. It represents the actual voltage available to drive current through the external circuit.

Definition

Terminal voltage is the potential difference between the positive and negative terminals of a source under operating conditions:

$$V = V_{+} - V_{-}$$

Relationship to EMF and Internal Resistance

Terminal voltage depends on:

The EMF of the source ($\varepsilon$)
The current flowing ($I$)
The internal resistance of the source ($r$)

Fundamental Equation

$$V = \varepsilon - Ir$$

This is the most common form, applicable when the source is delivering current (discharging).

Three Operating Conditions

1. Discharging (Delivering Current)

When the source supplies current to an external load:

$$V = \varepsilon - Ir$$

Terminal voltage is less than EMF
The $Ir$ term represents the voltage drop across internal resistance
As current increases, terminal voltage decreases
This is the normal operating mode for batteries powering devices

Physical interpretation: Some energy is lost overcoming the internal resistance, leaving less voltage for the external circuit.

2. Charging (Receiving Current)

When an external source forces current into the battery:

$$V = \varepsilon + Ir$$

Terminal voltage is greater than EMF
Higher voltage is needed to overcome both the chemical EMF and internal resistance
Used when charging rechargeable batteries

Example: A 12 V car battery being charged at high current might have a terminal voltage of 14-15 V.

3. Open Circuit (No Current)

When no current flows ($I = 0$):

$$V = \varepsilon$$

Terminal voltage equals EMF
No voltage drop across internal resistance
Used to measure the true EMF of a source with a high-resistance voltmeter

Visual Representation

flowchart LR
    subgraph Source["EMF Source with Internal Resistance"]
        direction TB
        E["EMF ε<br/>↑<br/>Ideal Source"] --- Rint["Internal<br/>Resistance r"]
    end
    
    A["Terminal +<br/>V = ?"] --- Load["External<br/>Load R"]
    Load --- B["Terminal -"]
    
    Rint --- A
    E --- B
    
    style E fill:#90EE90
    style Rint fill:#FFB6C1
    style Load fill:#87CEEB

Complete Circuit Analysis

Voltage Drops in Series

$$\varepsilon = V_{\text{external}} + V_{\text{internal}}$$

where:

$V_{\text{external}} = IR$ (voltage across external load)
$V_{\text{internal}} = Ir$ (voltage drop across internal resistance)

Therefore: $$\varepsilon = IR + Ir = I(R + r)$$

Current in the Circuit

$$I = \frac{\varepsilon}{R + r}$$

Substituting back to find terminal voltage:

$$V = IR = \varepsilon \cdot \frac{R}{R + r}$$

This shows terminal voltage is a fraction of EMF determined by the ratio of external to total resistance.

Factors Affecting Terminal Voltage

1. Load Resistance

As load resistance $R$ decreases:

Current $I$ increases
Terminal voltage $V$ decreases
When $R \to 0$ (short circuit): $V \to 0$ and $I \to \varepsilon/r$ (maximum current)

2. Internal Resistance

Higher internal resistance causes:

Greater voltage drop for the same current
Lower terminal voltage under load
More power dissipated as heat within the source

3. Current Drawn

The relationship between terminal voltage and current is linear:

$$V = \varepsilon - Ir$$

Plotting $V$ vs $I$ gives a straight line with:

Intercept = EMF (at $I = 0$)
Slope = $-r$ (negative internal resistance)

Terminal Voltage vs Current Graph

Terminal
Voltage V
    ↑
    │
 ε ─┼────────────●─────
    │           /│
    │          / │
    │         /  │
    │        /   │
    │       /    │
    │      /     │
    │     /      │
    │    /       │
    │   /        │
    │  /         │
    │ /          │
    │/           │
  0 ┼────────────┼────→ Current I
    │            │
    └────────────┘
         I_max = ε/r

At $I = 0$: $V = \varepsilon$ (open circuit)
At $I = I_{\text{max}} = \varepsilon/r$: $V = 0$ (short circuit)

Practical Implications

Battery Behavior

Fresh battery: Low internal resistance → terminal voltage stays close to EMF even under load
Used battery: High internal resistance → terminal voltage drops significantly under load
Old batteries: May show near-normal open-circuit voltage but fail under load due to high internal resistance

Measuring Battery Health

Measuring terminal voltage under load is a better indicator of battery condition than open-circuit voltage because it reveals the internal resistance.

Circuit Design

For maximum voltage delivery: Use source with low internal resistance
For maximum power transfer: Match load resistance to internal resistance ($R = r$)
For stable voltage: Use source with $r \ll R$ (internal resistance much smaller than load)

Worked Examples

Example 1: Basic Terminal Voltage

Problem: A 12 V battery has internal resistance 0.1 Ω. What is its terminal voltage when supplying 5 A?

Solution:

$$V = \varepsilon - Ir = 12 - (5)(0.1) = 12 - 0.5 = 11.5 \text{ V}$$

Example 2: Finding EMF from Terminal Voltage

Problem: A battery has terminal voltage 8.5 V when delivering 3 A. Its internal resistance is 0.5 Ω. What is its EMF?

Solution:

$$\varepsilon = V + Ir = 8.5 + (3)(0.5) = 8.5 + 1.5 = 10.0 \text{ V}$$

Example 3: Charging Terminal Voltage

Problem: A 6 V lead-acid battery with internal resistance 0.02 Ω is being charged at 10 A. What is the terminal voltage?

Solution:

$$V = \varepsilon + Ir = 6 + (10)(0.02) = 6 + 0.2 = 6.2 \text{ V}$$

Example 4: Variable Load

Problem: A 9 V battery has internal resistance 0.3 Ω. Find the terminal voltage when connected to: (a) $R = 10$ Ω (b) $R = 1$ Ω (c) $R = 0.1$ Ω

Solution:

(a) $I = \frac{9}{10 + 0.3} = 0.874$ A $$V = IR = (0.874)(10) = 8.74 \text{ V}$$

(b) $I = \frac{9}{1 + 0.3} = 6.92$ A $$V = IR = (6.92)(1) = 6.92 \text{ V}$$

Observation: As load resistance decreases, terminal voltage drops significantly.

Related Concepts

Electromotive Force (EMF) — The ideal voltage of the source
Internal Resistance — The resistance within the source that causes voltage drop
Ohm's Law — Fundamental relationship V = IR
Maximum Power Transfer — Condition for maximum power delivery
FAD1022 L10 — EMF and Internal Resistance — Complete lecture on these concepts

Summary Table

Condition	Formula	Terminal Voltage vs EMF
Open circuit ($I = 0$)	$V = \varepsilon$	Equal
Discharging ($I > 0$)	$V = \varepsilon - Ir$	Less than
Charging ($I < 0$)	$V = \varepsilon + Ir$	Greater than

Key Equations

Equation	Description
$V = \varepsilon - Ir$	Terminal voltage when delivering current
$V = \varepsilon + Ir$	Terminal voltage when charging
$V = \varepsilon$	Terminal voltage at open circuit
$V = IR$	Terminal voltage across external load
$\varepsilon = I(R + r)$	Complete circuit EMF equation