Determination of Capacitance by de-Sauty's Bridge

To determine the unknown capacitance of a given condenser (capacitor) by using de-Sauty's bridge method.

• Standard capacitor (known capacitance)
• Unknown capacitor (to be measured)
• Non-inductive variable resistances (resistance boxes)
• Galvanometer
• High frequency AC source (audio oscillator or buzzer)
• SPDT switch
• Connecting wires
• Headphones (alternative to galvanometer)

De-Sauty's bridge is an AC bridge circuit used to compare and measure unknown capacitance in terms of a standard capacitance. The bridge operates on the principle similar to Wheatstone bridge, but adapted for impedance measurements with capacitances.

In this bridge circuit, the four arms consist of:

• First arm: Standard capacitor with known capacitance $C_s$
• Second arm: Unknown capacitor with capacitance $C_x$ (to be determined)
• Third arm: Non-inductive resistance $R_1$
• Fourth arm: Non-inductive resistance $R_2$

When an AC source is connected across the bridge and the detector (galvanometer or headphones) shows null deflection or minimum sound, the bridge is said to be balanced.

At balance condition, the impedances of the four arms satisfy the relationship:

The impedance of a capacitor C with an AC voltage of angular frequency $\omega$ is given by:

Substituting the impedances in the balance equation:

Therefore, at balance condition:

This allows us to determine the unknown capacitance $C_x$ when the bridge is balanced.

From the balanced condition of the de-Sauty's bridge, we have:

Where:

• $C_x$ = Unknown capacitance to be determined (in Farad)
• $C_s$ = Standard capacitance (in Farad)
• $R_1$ = Resistance in the arm with the standard capacitor (in Ohm)
• $R_2$ = Resistance in the arm with the unknown capacitor (in Ohm)

1. Set up the circuit as shown in the circuit diagram of the de-Sauty's bridge.
2. Connect the standard capacitor $C_s$ and unknown capacitor $C_x$ in their respective arms of the bridge.
3. Connect the AC source (audio oscillator) across points A and C of the bridge.
4. Connect the detector (galvanometer or headphones) across points B and D.
5. Set initial values for the resistances $R_1$ and $R_2$ (e.g., $R_1 = 1000$ Ω).
6. Switch on the AC source.
7. Adjust the resistance $R_2$ until the galvanometer shows null deflection (or minimum sound in headphones).
8. If null deflection cannot be achieved by adjusting $R_2$ alone, adjust both $R_1$ and $R_2$ alternately.
9. When the bridge is balanced (null deflection), note down the values of $R_1$, $R_2$, and $C_s$.
10. Repeat the experiment for at least 5 different sets of balanced values of $R_1$ and $R_2$.
11. Calculate the unknown capacitance $C_x$ for each set using the formula $C_x = C_s \times \dfrac{R_2}{R_1}$.
12. Find the mean value of the unknown capacitance $C_x$.

Value of standard capacitance, $C_s$ = ............. µF

S.No.	Resistance $R_1$ (Ω)	Resistance $R_2$ (Ω)	Ratio $\frac{R_2}{R_1}$	Unknown Capacitance $C_x = C_s \times \frac{R_2}{R_1}$ (µF)
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Mean value of unknown capacitance $C_x$

For each observation, calculate:

1. The ratio $\frac{R_2}{R_1}$

2. The unknown capacitance using the formula:

Example calculation for the first observation:

Given,

• Standard capacitance, $C_s$ = ............. µF
• Resistance $R_1$ = ............. Ω
• Resistance $R_2$ = ............. Ω

Ratio $\frac{R_2}{R_1}$ = $\frac{R_2 \text{ value}}{R_1 \text{ value}}$ = .............

Unknown capacitance, $C_x = C_s \times \frac{R_2}{R_1}$ = ............. × ............. = ............. µF

3. Calculate the mean value of unknown capacitance:

4. Calculate the percentage error:

Where $C_x (accepted)$ is the value given by the manufacturer or instructor (if available).

The capacitance of the unknown capacitor measured using de-Sauty's bridge is ............. µF.

If a standard value is available for comparison:

The percentage error in measurement is ............. %.

1. All connections should be clean, tight, and properly insulated to avoid contact resistance and leakage currents.
2. Use high-quality, non-inductive resistors for $R_1$ and $R_2$ to get accurate results.
3. The capacitors should be of good quality with minimal leakage currents.
4. Use a stable AC source with appropriate frequency (usually 500-1000 Hz).
5. Ensure that the galvanometer or headphones are sensitive enough to detect the null point accurately.
6. Keep the bridge and components away from external electromagnetic fields to avoid interference.
7. The standard capacitor should be calibrated and its value known precisely.
8. For better accuracy, take multiple readings and calculate the average.
9. Ensure that the capacitors are fully discharged before connecting or disconnecting them from the circuit.
10. Handle the capacitors carefully to avoid damage and ensure accurate measurements.