Use of Wheatstone bridge in AC circuits
Use of Wheatstone Bridge in AC Circuits
The Wheatstone bridge is a fundamental circuit used for measuring unknown electrical resistances. It was originally developed by Samuel Hunter Christie in 1833 and improved and popularized by Sir Charles Wheatstone in 1843. While it is commonly used in DC circuits, its principles can also be applied to AC circuits with some modifications. In AC circuits, the Wheatstone bridge can measure not only resistances but also the impedance of circuit elements like capacitors and inductors.
Principle of the Wheatstone Bridge
The Wheatstone bridge consists of four resistive arms forming a quadrilateral, a voltage source, and a detector (like a galvanometer) for measuring current. The bridge is considered "balanced" when the voltage across the detector is zero, indicating that the ratio of resistances in one pair of opposite arms is equal to the ratio in the other pair.
In an AC circuit, the Wheatstone bridge operates on the same principle, but the resistances are replaced with impedances, which take into account the resistance, inductance, and capacitance of the circuit elements.
AC Wheatstone Bridge Circuit
The AC Wheatstone bridge circuit includes:
- Four impedances (Z1, Z2, Z3, Z4) instead of resistances
- An AC voltage source instead of a DC source
- A detector capable of measuring AC current, such as an AC voltmeter or an oscilloscope
Formulas for Wheatstone Bridge
In a balanced Wheatstone bridge, the following condition holds for DC circuits:
$$ \frac{R1}{R2} = \frac{R3}{R4} $$
For AC circuits, the condition for a balanced bridge becomes:
$$ \frac{Z1}{Z2} = \frac{Z3}{Z4} $$
where ( Z ) represents the impedance of each arm.
The impedance ( Z ) for different circuit elements is given by:
- For a resistor: ( Z_R = R ) (where ( R ) is the resistance)
- For an inductor: ( Z_L = j\omega L ) (where ( L ) is the inductance and ( \omega ) is the angular frequency)
- For a capacitor: ( Z_C = \frac{1}{j\omega C} ) (where ( C ) is the capacitance)
Differences Between DC and AC Wheatstone Bridge
| Feature | DC Wheatstone Bridge | AC Wheatstone Bridge |
|---|---|---|
| Elements | Resistors | Impedances (Resistors, Inductors, Capacitors) |
| Voltage Source | DC | AC |
| Detector | Galvanometer | AC Voltmeter, Oscilloscope |
| Balance Condition | ( \frac{R1}{R2} = \frac{R3}{R4} ) | ( \frac{Z1}{Z2} = \frac{Z3}{Z4} ) |
| Frequency Sensitivity | No | Yes (due to reactive components) |
Examples
Example 1: Measuring an Unknown Resistance in an AC Circuit
Suppose we have an AC Wheatstone bridge with the following impedances:
- ( Z1 = R1 )
- ( Z2 = R2 )
- ( Z3 = R3 )
- ( Z4 = R_x ) (unknown resistance)
If the bridge is balanced, we can find the unknown resistance ( R_x ) using the formula:
$$ R_x = R3 \left( \frac{R2}{R1} \right) $$
Example 2: Measuring an Unknown Capacitance in an AC Circuit
Let's say we have an AC Wheatstone bridge with:
- ( Z1 = R )
- ( Z2 = \frac{1}{j\omega C_x} ) (unknown capacitance)
- ( Z3 = R )
- ( Z4 = \frac{1}{j\omega C} ) (known capacitance)
For a balanced bridge:
$$ \frac{R}{\frac{1}{j\omega C_x}} = \frac{R}{\frac{1}{j\omega C}} $$
Solving for ( C_x ), we get:
$$ C_x = C $$
This means that the unknown capacitance ( C_x ) is equal to the known capacitance ( C ) when the bridge is balanced.
Conclusion
The Wheatstone bridge is a versatile tool in electrical measurements. When adapted for AC circuits, it allows for the precise measurement of not only resistances but also the reactive components of impedance, such as inductance and capacitance. Understanding the principles and differences between DC and AC Wheatstone bridges is essential for students and professionals working with electrical circuits.