Wheatstone Bridge

The Wheatstone bridge is a precision instrument used to measure unknown resistances by comparing them against known resistances. It operates on the principle of balanced potential differences.

Definition

A Wheatstone bridge is an electrical circuit used to measure resistances precisely by balancing potential drops. When balanced, no current flows through the detector (galvanometer), allowing accurate resistance determination.

Circuit Configuration

Basic Bridge Layout

flowchart TB
    subgraph "Wheatstone Bridge"
    A((a)) --- B[R1] --- C((b))
    A --- D[Rx<br/>Unknown] --- E((c))
    C --- F[R2] --- G((d))
    E --- H[R3<br/>Variable] --- G
    C --- I[Galvanometer<br/>G] --- E
    G --- J[Source<br/>ε] --- A
    end
    
    style D fill:#ffccbc
    style H fill:#e1f5e1
    style I fill:#fff9c4

Components

Component Symbol Description
Known resistors $R_1$, $R_2$ Precision resistors with known values
Variable resistor $R_3$ Adjustable resistor (rheostat or resistance box)
Unknown resistor $R_x$ The resistance to be measured
Galvanometer G Sensitive current detector (null detector)
Voltage source ε Battery or DC power supply

Operating Principle

Balance Condition

The bridge is balanced when:

  • The galvanometer shows zero deflection (no current through it)
  • Points b and c are at the same potential
  • Potential difference across the galvanometer: $V_{bc} = 0$
  • Current through galvanometer: $I_{bc} = 0$

Procedure

flowchart TD
    A[Place unknown Rx in circuit] --> B[Adjust variable R3]
    B --> C{Galvanometer reads zero?}
    C -->|No| B
    C -->|Yes| D[Bridge is balanced]
    D --> E[Calculate Rx using balance condition]

Derivation of Balance Condition

Current Relationships (at balance)

When balanced, the galvanometer carries no current, so:

Left branch (abc): $I_{R_1} = I_{R_x} = I_1$

Right branch (adc): $I_{R_2} = I_{R_3} = I_2$

Voltage Relationships (at balance)

Since points b and c are at equal potential:

  • $V_{R_1} = V_{R_2}$ (voltage drops from a to b and a to d are equal)
  • $V_{R_x} = V_{R_3}$ (voltage drops from b to c and d to c are equal)

Applying Ohm's Law

From $V = IR$:

$$I_1 R_1 = I_2 R_2 \quad \text{--- (1)}$$

$$I_1 R_x = I_2 R_3 \quad \text{--- (2)}$$

Deriving the Formula

Divide equation (1) by equation (2):

$$\frac{I_1 R_1}{I_1 R_x} = \frac{I_2 R_2}{I_2 R_3}$$

$$\frac{R_1}{R_x} = \frac{R_2}{R_3}$$

Wheatstone Bridge Balance Condition

$$\boxed{\frac{R_1}{R_2} = \frac{R_x}{R_3}}$$

Or equivalently:

$$\boxed{R_x = \frac{R_1 \cdot R_3}{R_2}}$$

Alternative forms:

$$\boxed{R_1 \cdot R_3 = R_2 \cdot R_x}$$

$$\boxed{\frac{R_1}{R_x} = \frac{R_2}{R_3}}$$

Alternative Notation (P, Q, R, S)

Some textbooks use the notation P, Q, R, S:

flowchart TB
    A((a)) --- B[P] --- C((b))
    A --- D[R<br/>Unknown] --- E((c))
    C --- F[Q] --- G((d))
    E --- H[S<br/>Variable] --- G
    C --- I[G] --- E

Balance condition:

$$\boxed{\frac{P}{Q} = \frac{R}{S}}$$

Or:

$$P \cdot S = Q \cdot R$$

Applications

1. Resistance Measurement

The primary application — measuring unknown resistances with high precision.

2. Slide Wire Bridge (Meter Bridge)

A practical laboratory implementation:

flowchart LR
    subgraph "Slide Wire Bridge"
    A[Batt] --- B[Rp<br/>Known] --- C[Jockey]
    C --- D[Galvanometer] --- E
    A --- F[X<br/>Unknown] --- E
    E --- G[Resistance Wire<br/>Uniform] --- C
    end

The resistance wire has uniform cross-section, so resistance is proportional to length:

$$\frac{R_1}{R_2} = \frac{l_1}{l_2}$$

Where $l_1$ and $l_2$ are the lengths on either side of the balance point.

3. Sensor Applications

Wheatstone bridges are used with:

  • Strain gauges — measure mechanical deformation
  • Thermistors — temperature sensing
  • Load cells — weight measurement
  • Pressure sensors — detect pressure changes

In these applications, the sensor resistance changes with the measured quantity, and the bridge detects small changes in resistance.

Sensitivity and Precision

Factors Affecting Sensitivity

Factor Effect
Galvanometer sensitivity Higher sensitivity detects smaller imbalances
Source voltage Higher voltage increases sensitivity
Resistor matching Bridge arms should have comparable resistances

Precision

  • Typical precision: 0.1% to 1% of measured value
  • With care: up to 0.01% precision possible
  • Accuracy depends on the precision of known resistors $R_1$, $R_2$, and $R_3$

Worked Examples

Example 1: Basic Resistance Measurement

Given:

  • $R_1 = 100$ Ω
  • $R_2 = 50$ Ω
  • At balance: $R_3 = 225$ Ω

Find: $R_x$

Solution:

$$R_x = \frac{R_1 \cdot R_3}{R_2} = \frac{100 \times 225}{50} = \frac{22500}{50}$$

$$R_x = 450 \text{ Ω}$$

Example 2: Slide Wire Bridge

Given:

  • Balance point divides the wire in ratio 3:2
  • Known resistance = 30 Ω (connected to shorter segment)

Find: Unknown resistance

Solution:

$$\frac{R_x}{30} = \frac{3}{2}$$

$$R_x = 30 \times \frac{3}{2} = 45 \text{ Ω}$$

Example 3: Sensor Application

A strain gauge has resistance $R_x$ that changes with strain. In a bridge with:

  • $R_1 = R_2 = 100$ Ω (fixed)
  • $R_3 = 100$ Ω (calibrated)

At no strain, the bridge is balanced ($R_x = 100$ Ω).

Under strain, to rebalance, $R_3$ must be adjusted to 102 Ω.

Find: Strained resistance of gauge

Solution:

$$R_x = \frac{R_1 \cdot R_3}{R_2} = \frac{100 \times 102}{100} = 102 \text{ Ω}$$

The strain increased resistance by 2 Ω.

Unbalanced Bridge

When the bridge is not balanced, current flows through the galvanometer. The current can be calculated using:

  • Kirchhoff's laws
  • Thevenin equivalent circuit

The unbalanced bridge is useful for:

  • Detecting small changes in resistance (sensor applications)
  • Measuring temperature, strain, pressure, etc.

Advantages and Limitations

Advantages

Advantage Explanation
High precision Null detection is very sensitive
No calibration needed Measurement depends only on ratio of known resistances
Independent of source voltage Balance condition does not depend on ε
Versatile Can measure wide range of resistances

Limitations

Limitation Explanation
DC only Traditional bridge works with DC
Range limitation Very high or very low resistances are difficult to measure
Manual balancing Requires adjustment to find balance point
Contact resistance Affects very low resistance measurements

Related Concepts

  • Voltage Divider — Related circuit for voltage division
  • Electrical Measurements — Overview of measurement techniques
  • Ohm's Law — Foundation of bridge analysis
  • Resistor Networks — Series and parallel combinations
  • Galvanometer — Current detection instrument

Sources

Key Takeaways

  1. A Wheatstone bridge balances when $\frac{R_1}{R_2} = \frac{R_x}{R_3}$
  2. At balance, no current flows through the galvanometer
  3. The measurement is independent of the source voltage
  4. High precision is achieved through null detection
  5. Widely used for precision resistance measurement and sensor applications