Using Nodal Analysis to Determine Equivalent Resistance in an Unbalanced Wheatstone Bridge

Understanding how to calculate the equivalent resistance in an unbalanced Wheatstone bridge is essential for various fields such as electrical engineering and physics. This article explains the process step-by-step, including the configuration of the bridge, nodal analysis, and how to find the resulting equivalent resistance.

Introduction to the Wheatstone Bridge

A Wheatstone bridge consists of four resistors arranged in a diamond shape, with two resistors on one branch and the other two on the opposite branch. An unbalanced Wheatstone bridge, unlike its balanced counterpart, does not have equal ratios of resistances across the diagonals, leading to a non-zero voltage at the center node.

Components and Configuration

The resistors in an unbalanced Wheatstone bridge are labeled as follows:

R1, R2: These form one branch of the bridge. R3, R4: These form the other branch of the bridge. Voltage Source: This is connected between the nodes formed by R1 and R2 on one side and R3 and R4 on the other.

Nodal Analysis Method

Nodal analysis is a method used to analyze electrical circuits by defining a set of nodes and applying Kirchhoff's Current Law (KCL).

Step-by-Step Procedure

Label Nodes: Identify and label the nodes where the resistors meet. There are three nodes: Node A: Between R1 and R2. Node B: Between R3 and R4. Node C: The common ground or reference node. 2. Assign Voltages: Assign voltages to each node. Let VA be the voltage at Node A and VB be the voltage at Node B.

3. Apply KCL at Nodes: Node A: VA/R1 (VA - VB)/R2 0 This equation can be rearranged to: VA(R1 R2) - VB(R2) 0 Node B: VB/R3 (VB - VA)/R4 0 This equation can be rearranged to: VB(R3 R4) - VA(R4) 0 4. Rearrange Equations: From the KCL equations, derive the system of equations.

5. Form the System of Equations:

[begin{bmatrix} frac{1}{R_1} frac{1}{R_2} -frac{1}{R_2} -frac{1}{R_4} frac{1}{R_3} frac{1}{R_4} end{bmatrix} begin{bmatrix} V_A V_B end{bmatrix} begin{bmatrix} 0 0 end{bmatrix}]

6. Solve the Equations: Solve the system of equations for VA and VB using methods such as substitution or matrix methods.

Calculating Equivalent Resistance

To find the equivalent resistance Req seen from the input terminals:

1. Test Voltage: Apply a test voltage Vtest across the input terminals.

2. Calculate Currents: Use the voltages VA and VB derived from nodal analysis to calculate the currents I1, I2, I3, and I4 through each resistor.

3. Average Current: Use Ohm's law to find the total current and the equivalent resistance:

Req Vtest / Itotal, where Itotal is the total current flowing through the input terminals.

Conclusion

By solving the equations derived from nodal analysis, you can find the voltages at nodes A and B and, subsequently, the equivalent resistance Req seen from the input terminals of the Wheatstone bridge. The exact values will depend on the specific resistor values you have.