Bridges & Measurement Circuits: Wheatstone, Kelvin, LVDT, Potentiometers

Key Concepts & Working Principles

1. Bridge Circuits: The Core Idea

Bridge circuits compare two resistances (or impedances) to measure an unknown quantity. The balance condition (zero output voltage) is used to determine the unknown value. The most common types are:

Wheatstone Bridge

• Definition: A DC bridge circuit used to measure unknown resistance by balancing two legs of a bridge.
• Working Principle:
  • Four resistors form a diamond shape with a voltage source across one diagonal and a galvanometer (or voltmeter) across the other.
  • At balance: $\frac{R_1}{R_2}=\frac{R_x}{R_3}$, where $R_x$ is the unknown resistance.
  • No current flows through the galvanometer when balanced.

graph LR
    A["Voltage Source"] --> B["R1"]
    A --> C["R2"]
    B --> D["Galvanometer"]
    C --> D
    B --> E["Rx"]
    C --> F["R3"]
    E --> D
    F --> D

• Applications:
  • Strain gauges (load cells in Daraz’s warehouse scales).
  • Temperature measurement (RTD sensors in NTC’s power plants).
  • Humidity sensors (industrial hygrometers).

Kelvin (Thomson) Bridge

• Purpose: Measures low resistances (e.g., copper wire, shunt resistors) with high accuracy.
• Key Feature: Uses four terminals (two current and two potential leads) to eliminate lead resistance errors.
• Balance Condition: $$\frac{R_1}{R_2}=\frac{R_x}{R_3}\text{and}\frac{R_4}{R_5}=\frac{R_x}{R_3}$$ (Simplified for ideal case.)

graph LR
    A["Voltage Source"] --> B["R1"]
    A --> C["R2"]
    B --> D["Galvanometer"]
    C --> D
    B --> E["Rx (4-terminal)"]
    C --> F["R3 (4-terminal)"]
    E --> G["Current Leads"]
    F --> G
    D --> H["Potential Leads"]

• Real-World Use:
  • Nepal Electricity Authority (NEA/NTC) uses Kelvin bridges to measure shunt resistors in high-current transformers.
  • Battery internal resistance testing in electric vehicles (e.g., e-rickshaws).

Anderson Bridge

• Purpose: Measures inductance (L) and capacitance (C).

• Balance Condition: $$\frac{R_1}{R_2}=\frac{R_3}{R_x}\text{and}{L}_x=R_3{R}_{4}C$$ (For inductance measurement.)

• Applications:

  • Motor winding inductance tests in Ncell’s telecom equipment.
  • Capacitor quality checks in smartphone charging circuits (e.g., Xiaomi, Samsung).

2. Linear Variable Differential Transformer (LVDT)

• Definition: A displacement sensor that converts linear motion into an AC voltage signal via electromagnetic induction.
• Working Principle:
  • A primary coil is energized with AC, and two secondary coils are connected in series opposition.
  • The core (ferromagnetic) moves inside the coils, inducing voltages $V_1$ and $V_2$ in the secondaries.
  • Output voltage $V_o=V_1-V_2$ changes linearly with core position.

graph TD
    A["AC Supply"] --> B["Primary Coil"]
    B --> C["Ferromagnetic Core"]
    C --> D["Secondary Coil 1"]
    C --> E["Secondary Coil 2"]
    D --> F["Output Voltage V1"]
    E --> G["Output Voltage V2"]
    F --> H["Differential Amplifier"]
    G --> H
    H --> I["V_o = V1 - V2"]

• Key Features:

  • High resolution (sub-micron measurements).
  • No contact wear (ideal for harsh environments).
  • Bidirectional measurement (core can move left or right).

• Real-World Example:

  • Pathao’s ride-hailing system uses LVDTs in automated parking sensors to detect vehicle positions in garages.
  • NTC’s power plant turbines use LVDTs to measure valve displacement for precise control.

Worked Example: LVDT for Hydraulic Cylinder Position

Problem: An LVDT has a sensitivity of 10 V/mm and a core displacement of 5 mm. Calculate the output voltage if the core is 2 mm to the right of center.

Solution:

1. Sensitivity = 10 V/mm means 1 mm displacement → 10 V output.
2. Core at +2 mm (right of center) → $V_o=10\times 2=20$ V.
3. If core were at -3 mm (left of center) → $V_o=10\times (-3)=-30$ V.

Application Tie-In:

• Nepal’s irrigation pumps use LVDTs to monitor water valve positions in canals, ensuring precise flow control.

3. Potentiometers in Measurement

• Definition: A three-terminal variable resistor used as a voltage divider or precision measurement tool.

• Types:

  • Linear: Resistance changes uniformly with rotation.
  • Rotary: Manual adjustment (e.g., volume knobs).
  • Sliding: Used in precision measurement (e.g., lab potentiometers).

• Working Principle:

  • Acts as a voltage divider: $V_{out}=V_{in}\times \frac{R_2}{R_1+R_2}$.
  • Used in null-balance measurements (e.g., comparing against a standard cell).

graph LR
    A["V_in"] --> B["R1"]
    B --> C["Wiper"]
    C --> D["V_out"]
    B --> E["R2"]
    E --> C

• Applications:
  • Nepal’s NEPSE stock exchange uses potentiometers in analog control panels for voltage calibration.
  • Khalti’s payment terminals employ potentiometers for adjusting reference voltages in current sensors.

Worked Example: Potentiometer as a Voltmeter

Problem: A potentiometer has $R_{total}=1000\Omega$. If $V_{in}=5\text{V}$ and the wiper is set to $R_1=300\Omega$, find $V_{out}$.

Solution: $$V_{out}=V_{in}\times \frac{R_2}{R_{total}}=5\times \frac{700}{1000}=3.5\text{V}$$

Real-World Tie-In:

• eSewa’s payment kiosks use potentiometers to calibrate display voltages for accurate transaction readings.

4. Measurement Circuit Configurations

Bridge and potentiometer circuits can be connected in series, parallel, or combined configurations. Key considerations:

5. Error Analysis in Bridge Circuits

Common errors and their mitigation:

1. Lead Resistance Errors

   • Cause: Resistance of connecting wires affects measurement.
   • Solution: Use Kelvin bridge (4-terminal) or compensation techniques.

2. Temperature Drift

   • Cause: Resistor values change with temperature.
   • Solution: Use temperature-compensated resistors (e.g., Manganin).

3. Nonlinearity

   • Cause: Strain gauges or sensors deviate from linearity.
   • Solution: Calibration and signal conditioning (e.g., amplifiers).

In the Real World

1. Daraz Logistics & Load Cells

   • Wheatstone Bridge in load cells measures weight of packages.
   • How: Strain gauges (resistors) change resistance when compressed; the bridge detects imbalance → weight is calculated.
   • Example: A 10 kg package causes a 0.01 Ω change in $R_x$, which the bridge converts to a 4-20 mA signal for the weighing system.

2. NTC’s Power Measurement Systems

   • Kelvin Bridge measures shunt resistors in high-current transformers.
   • Why: Ensures accurate current sensing for billing (e.g., residential meters).
   • Example: A 0.01 Ω shunt resistor in a 100 A circuit requires Kelvin bridge precision to avoid 0.1% error.

3. Pathao’s Automated Parking

   • LVDT sensors detect car position in multi-level garages.
   • How: Core movement (e.g., 10 cm) → 50 mV output → microcontroller calculates parking slot.
   • Advantage: Eliminates human error in tight spaces.

Exam Tip

What Examiners Look For

1. Balance Conditions

   • Always derive the balance equation for Wheatstone/Kelvin bridges. Memorize:
     • Wheatstone: $\frac{R_1}{R_2}=\frac{R_x}{R_3}$
     • Kelvin: Four-terminal condition (avoid 2-terminal errors).
   • Common Mistake: Forgetting to include $R_4$ and $R_5$ in Kelvin bridge equations.

2. LVDT Operation

   • Explain phase-sensitive detection (why AC is used).
   • Graph: Sketch $V_o$ vs. displacement (linear, passes through zero at center).
   • Exam Question: "Why is LVDT preferred over a potentiometer for industrial applications?" Answer: No contact wear, higher resolution, bidirectional measurement.

3. Potentiometer Applications

   • Null-balance vs. deflection methods:
     • Null-balance: More accurate (e.g., lab standards).
     • Deflection: Faster but less precise (e.g., volume knobs).
   • Worked Example: Always show voltage divider formula and wiper position.

4. Error Sources

   • Bridge Circuits: Lead resistance, temperature, nonlinearity.
   • LVDT: Core eccentricity, frequency drift.
   • Potentiometer: Contact resistance, mechanical wear.

5. Real-World Tie-Ins

   • Nepal Context: Relate to NTC, NEPSE, Daraz, Pathao, or banks.
     • Example: "How would you measure the load on a bridge using a Wheatstone bridge?" Answer: Strain gauges on bridge pillars → resistance change → bridge imbalance → load calculated.

Model Answer Structure

For a 10-mark question like "Explain the working of an LVDT with a neat diagram and applications":

1. Introduction (1 mark): Define LVDT as an electromagnetic displacement sensor.
2. Diagram (2 marks): Draw the 3-coil structure with core and AC supply.
3. Working Principle (4 marks):
   • Primary coil → AC excitation.
   • Secondary coils in series opposition.
   • Core position → $V_o=V_1-V_2$.
   • Phase-sensitive detection for direction.
4. Applications (2 marks): Pathao parking, NTC turbines.
5. Advantages (1 mark): No wear, high resolution.