Formula For Unbalanced Wheatstone Bridge

The Wheatstone bridge is one of the most widely used electrical circuits for measuring unknown resistances, analyzing circuit behavior, and understanding the balance of electrical systems. While a balanced Wheatstone bridge often receives more attention due to its simplicity and symmetry, the unbalanced Wheatstone bridge plays an equally significant role in real-world applications where conditions are not ideal or perfectly matched. The formula for an unbalanced Wheatstone bridge is essential for students, engineers, and technicians who wish to calculate current flow, potential difference, or determine resistance in practical scenarios where imbalance naturally occurs.

Understanding the Concept of Wheatstone Bridge

A Wheatstone bridge is a circuit consisting of four resistors connected in a diamond shape with a power source applied across the bridge and a galvanometer connected between two opposite nodes. When the ratio of the resistances in one arm is equal to the ratio of resistances in the other arm, the bridge is said to be balanced, and no current flows through the galvanometer. However, in practical cases, perfect balance is rarely achieved, and this results in an unbalanced Wheatstone bridge.

Key Components of the Bridge

• Four resistors arranged in a diamond-like structure.
• A voltage source applied across the bridge.
• A galvanometer or detection device placed across the middle points.
• Potential differences that change when the circuit is unbalanced.

Formula for Unbalanced Wheatstone Bridge

The general formula is derived by applying Kirchhoff’s laws to the unbalanced network. Let’s assume the resistances are R1, R2, R3, and Rx, where Rx is the unknown resistance. The supply voltage is Vs, and the galvanometer is connected between the midpoints.

The voltage difference (Vg) across the galvanometer in an unbalanced Wheatstone bridge can be expressed as

Vg = Vs à [(R2 / (R1 + R2)) – (Rx / (R3 + Rx))]

This formula shows that the galvanometer deflection depends on the ratio of resistances in each arm of the bridge. When the ratios are not equal, Vg is nonzero, indicating an unbalanced condition.

Current in the Galvanometer

If the galvanometer has a resistance Rg, the current flowing through it can be calculated using Ohm’s law

Ig = Vg / Rg

This provides a practical way to measure the degree of imbalance in the circuit.

Derivation of the Formula

To derive the unbalanced Wheatstone bridge formula, consider two potential dividers

• The first divider consists of R1 and R2, with the potential at their junction being V1 = Vs à (R2 / (R1 + R2)).
• The second divider consists of R3 and Rx, with the potential at their junction being V2 = Vs à (Rx / (R3 + Rx)).

The galvanometer voltage difference is then simply

Vg = V1 – V2

From this, we obtain the earlier mentioned equation that governs the unbalanced Wheatstone bridge.

Applications of the Unbalanced Wheatstone Bridge

Although the balanced form is often preferred for precision measurements, the unbalanced Wheatstone bridge has numerous applications where continuous monitoring of changes is required.

Sensing and Measurement Devices

One of the most common applications is in sensor circuits. For instance, strain gauges, temperature sensors, and pressure sensors use the unbalanced bridge configuration. When physical conditions change, the resistance of one arm of the bridge varies, causing the bridge to become unbalanced. The resulting voltage difference provides a measurable signal that corresponds to the physical change.

Industrial and Engineering Applications

In industrial settings, the unbalanced Wheatstone bridge is widely used in instrumentation and control systems. Monitoring voltage differences across the bridge allows engineers to detect small changes in resistance, making it suitable for precise control of processes.

Educational Importance

In academic environments, unbalanced Wheatstone bridges are used in laboratory experiments to teach students about electrical measurements, error analysis, and the application of Kirchhoff’s laws. Understanding the unbalanced state ensures that learners can interpret real-world conditions rather than relying solely on idealized balanced circuits.

Advantages of Using the Formula

Having a clear formula for an unbalanced Wheatstone bridge provides several benefits

• Enables calculation of potential difference without needing perfect balance.
• Provides insight into circuit sensitivity when resistances vary slightly.
• Facilitates the design of sensor circuits that rely on imbalance for output.
• Allows precise error estimation when working with experimental setups.

Worked Example

Consider a Wheatstone bridge with the following values

• R1 = 100 Ω
• R2 = 200 Ω
• R3 = 150 Ω
• Rx = 160 Ω
• Vs = 10 V

Step 1 Calculate the potential at the junction of R1 and R2

V1 = 10 Ã (200 / (100 + 200)) = 10 Ã (200 / 300) = 6.67 V

Step 2 Calculate the potential at the junction of R3 and Rx

V2 = 10 Ã (160 / (150 + 160)) = 10 Ã (160 / 310) = 5.16 V

Step 3 Calculate the galvanometer voltage

Vg = V1 – V2 = 6.67 – 5.16 = 1.51 V

This means the bridge is unbalanced and the galvanometer would show a deflection corresponding to 1.51 V.

Limitations of the Unbalanced Wheatstone Bridge

While useful, the unbalanced Wheatstone bridge comes with limitations

• Inaccurate when resistance changes are too large or too small to detect effectively.
• Requires a sensitive galvanometer to measure small potential differences.
• Less stable for long-term precision measurements compared to balanced bridges.

The formula for the unbalanced Wheatstone bridge is fundamental for anyone working with electrical measurement systems. By applying the relation between resistances and supply voltage, it is possible to determine the voltage across the galvanometer and understand how far the circuit is from balance. This principle not only enhances theoretical knowledge but also finds direct application in fields such as instrumentation, sensing technology, and engineering research. A strong grasp of the unbalanced Wheatstone bridge ensures better problem-solving skills and prepares learners and professionals to handle real-world circuit conditions effectively.