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Electricity and Magnetism

AC Bridges

AC bridges are network circuits used to measure unknown impedances, inductances, capacitances, and frequency using null detection methods with alternating current sources. They operate on the principle of balancing complex potential drops across bridge arms.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic conditions for bridge balance, application to simple Maxwell and de Sauty bridges, and frequency-independent balance equations.

PGT extension

Complex impedance balancing, derivation of bridge balance conditions for Maxwell's inductance-capacitance bridge, Schering bridge, Wien bridge, and Wagner ground connection.

Prerequisites

Complex numbers and phasor algebra
Series and parallel L-C-R circuits in alternating current
Conditions for bridge balance in Wheatstone network

Principle of AC Bridge Balance

An AC bridge consists of four arms containing complex impedances Z₁, Z₂, Z₃, and Z₄, an AC source, and a null detector (such as headphones or an electronic voltmeter).
At balance, no current flows through the detector, meaning the potential difference across the detector terminals is zero.
The fundamental balance equation is given by the product of opposite arm impedances being equal: Z₁Z₄ = Z₂Z₃.

Complex Balance Condition

Since impedances are complex quantities expressed as $Z = R + jX$, the bridge balance equation Z₁Z₄ = Z₂Z₃ must hold in both magnitude and phase.
Equating real parts and imaginary parts yields two independent balance conditions that must be satisfied simultaneously.
This implies that balancing an AC bridge generally requires two variable components (such as a variable resistor and a variable capacitor or inductor).

Maxwell's Inductance-Capacitance Bridge

Used for measuring an unknown inductance in terms of a known standard capacitance.
One arm contains an unknown inductor with resistance r₁ and inductance L₁ in series with a resistor R₁, while the opposite arm contains a standard capacitor C₃ in parallel with a resistor R₃.
Yields independent balance equations for both resistance and inductance, meaning the balance is independent of supply frequency.

Schering Bridge

Primarily used for the measurement of unknown capacitance, relative permittivity (dielectric constant), and dissipation factor (dielectric loss) of capacitors.
Features a standard mica capacitor in one arm and a parallel combination of a variable resistor and variable capacitor in the adjacent arm.
Extremely precise for high-voltage applications because the loss angle can be directly determined from the resistance and capacitance values at balance.

Wien Bridge

Used for measuring audio frequencies and as a frequency-determining network in oscillators.
Contains series and parallel RC combinations in adjacent arms.
The balance conditions depend explicitly on the angular frequency ω of the AC supply.

Derivations

General Balance Condition for an AC Bridge

Conditions: Four arms with complex impedances Z₁, Z₂, Z₃, and Z₄ forming a closed loop with an AC detector between opposite junctions.

Let an alternating voltage E be applied across diagonally opposite corners of the bridge network.
At balance, the current through the detector connected across the other two diagonal corners is zero.
Therefore, the current flowing through Z₁ must equal the current flowing through Z₃ (let it be I_A), and the current through Z₂ must equal that through Z₄ (let it be I_B).
Equating the potential drops across the two arms meeting at a detector terminal: I_AZ₁ = I_BZ₂.
Using the other pair of arms: I_AZ₃ = I_BZ₄.
Dividing the two equations eliminates the currents: (Z₁ / Z₃) = (Z₂ / Z₄).

Result: Z_1 $Z_4 = Z_2 Z_3$

Maxwell's Inductance Bridge Derivation

Conditions: Arm 1: L₁ and r₁ in series with R₁; Arm 2: Pure resistance R₂; Arm 3: Pure resistance R₃; Arm 4: Parallel combination of R₄ and C₄.

Write complex impedances for each arm: Z₁ = (R₁ + r₁) + jωL₁, Z₂ = R₂, Z₃ = R₃, and 1/Z₄ = (1/R₄) + jωC₄.
Substitute into the balance condition Z₁Z₄ = Z₂Z₃, which can be rearranged as Z₁ = Z₂Z₃(1/Z₄).
Substitute the explicit terms: (R₁ + r₁ + jωL₁) = R₂R₃ [ (1/R₄) + jωC₄ ].
Equate real parts: R₁ + r₁ = R₂R₃ / R₄.
Equate imaginary parts: ωL₁ = R₂R₃ωC₄, leading to L₁ = R₂R₃C₄.

Result: $L_1 = R_2 R_3 C_4 \text{}$ and } $r_1 = \frac{R_2 R_3}{R_4} - R_1$

Formula sheet

Formula / law	Equation	Variables and units	Conditions
General AC Bridge Balance Equation	$$Z_1 Z_4 = Z_2 Z_3$$	Z₁, Z₂, Z₃, Z₄ are complex impedances of the four bridge arms. Ohm (Ω)	Null detector reads zero current.
Maxwell Bridge Inductance	$$L = R_2 R_3 C$$	L = unknown inductance, R₂, R₃ = resistance arms, C = standard capacitance. Henry (H)	Frequency independent balance.
De Sauty Bridge Capacitance	$$C_1 = C_2 \frac{R_2}{R_1}$$	C₁ = unknown capacitance, C₂ = standard capacitor, R₁, R₂ = non-inductive resistors. Farad (F)	Lossless capacitors.
Schering Bridge Unknown Capacitance	$$C_1 = C_3 \frac{R_4}{R_2}$$	C₁ = test capacitor with loss, C₃ = standard capacitor, R₂, R₄ = bridge arm resistances. Farad (F)	Series equivalent resistance r₁ = R₂(C₄/C₃).
Schering Bridge Dissipation Factor	$$D = \omega C_4 R_4$$	$D = dissipation factor (\tan \delta)$, ω = angular frequency, C₄, R₄ = parallel components. Dimensionless	Low-loss dielectric measurement.
Wien Bridge Frequency	$$f = \frac{1}{2pi \sqrt{R_3 R_4 C_3 C_4}}$$	f = frequency of AC source, R, C = arm resistance and capacitance values. Hertz (Hz)	Bridge arms symmetrical such that R₃ = R₄ and C₃ = C₄.
Complex Impedance Expression	$$Z = R + j\left(\omega L - \frac{1}{\omega C}\right)$$	R = resistance, L = inductance, C = capacitance, ω = angular frequency, j = imaginary unit. Ohm (Ω)	Series L-C-R branch.
Detector Sensitivity Condition	$$S = \frac{\delta I}{\delta Z}$$	S = sensitivity, ΔI = detector current change, ΔZ = impedance imbalance. Ampere per Ohm (A/Ω)	Near bridge balance point.

Graphs and experiments

Wagner Earthing Device Circuit Setup

Draw: A bridge network with an additional auxiliary RC branch connected from the detector junction to ground.

Interpret: Eliminates errors due to stray earth capacitances between bridge components and ground by ensuring the detector terminals are at ground potential at balance.

Exam use: Tested conceptually in PGT exams regarding precision bridge measurement errors.

Null Detector Response Curve

Draw: A plot of detector current or deflection versus variable resistance/capacitance setting showing a sharp minimum at the balance point.

Interpret: Shows how sharpness of null depends on bridge component tuning and detector sensitivity.

Exam use: Used to explain experimental precision in finding balance.

Wien Bridge Frequency Response

Draw: A plot of output-to-input voltage ratio against frequency f, peaking sharply at the resonance frequency f₀.

Interpret: Demonstrates frequency selectivity used in audio oscillators.

Exam use: Frequently asked in PGT application-based questions.

Solved numericals

TGT core

In a de Sauty bridge used to compare two capacitances, the arm resistances at balance are R₁ = 400 Ω and R₂ = 800 Ω. If the standard capacitor C₂ has a value of 0.5 μF, find the unknown capacitance C₁.

Recall the balance condition for de Sauty bridge: C₁R₁ = C₂R₂.
Rearrange the formula to solve for C₁: C₁ = C₂ (R₂ / R₁).
Substitute the given values: C₁ = 0.5 μF × (800 / 400).
Calculate the final value: C₁ = 0.5 × $2 = 1.0 \muF$.

Answer: 1.0 μF

PGT extension

A Maxwell's inductance bridge is balanced with the following component values: R₂ = 1000 Ω, R₃ = 1000 Ω, R₄ = 500 Ω, and C₄ = 0.2 μF. Calculate the unknown inductance L₁ and its effective series resistance r₁ assuming R₁ = 0.

Use the Maxwell bridge balance formula for inductance: L₁ = R₂R₃C₄.
Substitute the values: L₁ = 1000 × 1000 × 0.2 × 10^-6 H = 0.2 H.
Use the resistance balance formula: r₁ = (R₂R₃ / R₄) - R₁.
Substitute the values with R₁ = 0: r₁ = (1000 × 1000) / $500 = 2000 \Omega$.

Answer: $L_1 = 0.2 H$, $r_1 = 2000 Ω$

PGT extension

In a Wien bridge oscillator network, the frequency-determining arms consist of equal resistors $R = 10 k\Omega$ and equal capacitors C = 10 nF. Determine the frequency of oscillation.

The frequency formula for a symmetrical Wien bridge is $f = 1 / (2\piRC)$.
Substitute $R = 10 \times 10^{3} \Omega$ and $C = 10 \times 10^{-9} F$.
Calculate RC product: R$C = 10^{4} \times 10^{-8} = 10^{-4} s$.
Compute f: $f = 1 / (2 \times 3.1416 \times 10^{-4}) = 10^{4} / 6.2832 \asymp 1591.5 Hz$.

Answer: 1591.5 Hz

Assumptions and limitations

AC sources are assumed to be purely sinusoidal with negligible harmonic distortion; presence of harmonics prevents a true sharp null.
Stray capacitances and mutual inductances between bridge leads can introduce measurement errors unless shielded and grounded properly (Wagner earth).
Components are assumed to be linear; nonlinear elements cause bridge balance to depend on signal amplitude.

Common misconceptions

Common error: An AC bridge can be balanced by adjusting a single variable resistor.

Correct: Balancing an AC bridge generally requires independent adjustment of two variables to satisfy both magnitude and phase (real and imaginary) conditions.

Complex impedance matching requires simultaneous real and imaginary part cancellation.

Common error: All AC bridge balance conditions depend on the frequency of the AC source.

Correct: Certain bridges like Maxwell's inductance-capacitance bridge have balance equations independent of frequency.

When the frequency term cancels out algebraically in both real and imaginary balance equations, frequency does not affect the balance point.

Quick revision

General AC bridge balance condition is Z₁Z₄ = Z₂Z₃.
Maxwell bridge measures unknown inductance using a standard capacitor: $L = R_{2}R_{3}C$.
Schering bridge measures high capacitance and dielectric loss factor.
Wien bridge is used for audio frequency measurement and oscillator frequency determination.
Wagner earthing device eliminates stray earth capacitance errors in sensitive bridge circuits.

Must remember

The fundamental balance equation for any AC bridge is $Z_1 Z_4 = Z_2 Z_3$, where $Z_1, Z_2, Z_3, Z_4$ are complex impedances.
Balancing an AC bridge requires two independent conditions to be met simultaneously: equating real parts and equating imaginary parts.
Maxwell's bridge measures unknown inductance in terms of known capacitance and is independent of supply frequency.
Schering bridge is used for high-precision measurement of capacitance, relative permittivity, and dielectric loss (dissipation factor).
Wien bridge is utilized for measuring audio frequencies and as a frequency-determining network in oscillators.

Glossary

Null DetectorA sensitive device such as headphones or an electronic voltmeter used to detect zero current at bridge balance.

Complex ImpedanceExpressed as $Z = R + jX$, combining resistance $R$ and reactance $X$ in an AC circuit.

Dissipation Factor ($D$)A measure of loss in a dielectric material, given by $D = \omega C_4 R_4$ in a Schering bridge.

Wagner Earth ConnectionAn auxiliary circuit used in AC bridges to eliminate errors caused by stray capacitances to ground.

Comparison of AC Bridges

Bridge Name	Primary Application	Key Components
Maxwell Bridge	Measurement of unknown inductance	Standard capacitance, resistances, unknown inductor
Schering Bridge	Capacitance, dissipation factor, dielectric constant	Standard mica capacitor, variable capacitor/resistor
Wien Bridge	Audio frequency measurement & oscillators	Series and parallel RC combinations

Practice Quiz

Choose an option to reveal the explanation.

1. What is the fundamental balance condition for an AC bridge with complex impedances $Z_1, Z_2, Z_3,$ and $Z_4$?

Explanation: At balance, no current flows through the detector, and the product of opposite arm impedances is equal, given by $Z_1 Z_4 = Z_2 Z_3$.

2. Why do most AC bridges require at least two variable components for complete balance?

Explanation: Since impedances are complex quantities ($Z = R + jX$), both magnitude and phase conditions must be met, yielding two independent equations requiring two variable components.

3. Maxwell's inductance-capacitance bridge measures unknown inductance in terms of which standard component?

Explanation: Maxwell's bridge uses a known standard capacitance to measure an unknown inductance.

4. Which of the following bridges is primarily used for measuring capacitance and dielectric loss angle?

Explanation: The Schering bridge is specifically designed for high-precision measurement of capacitance, relative permittivity, and dissipation factor.

5. What is the frequency expression for a Wien bridge when symmetrical arms are used ($R_3 = R_4$ and $C_3 = C_4$)?

Explanation: For symmetrical arms where $R_3=R_4=R$ and $C_3=C_4=C$, the frequency formula reduces to $f = \frac{1}{2\pi R C}$.

6. In a Maxwell bridge, under what condition is the balance equation independent of frequency?

Explanation: Proper circuit configuration in Maxwell's bridge results in balance equations for resistance and inductance that contain no frequency terms.

7. What is the expression for the unknown capacitance $C_1$ in a basic De Sauty bridge with arm resistances $R_1, R_2$ and standard capacitor $C_2$?

Explanation: The balance condition for a De Sauty bridge yields $C_1 = C_2 \frac{R_2}{R_1}$ for lossless capacitors.

8. What does the dissipation factor $D$ in a Schering bridge represent?

Explanation: The dissipation factor $D = \omega C_4 R_4$ represents the tangent of the dielectric loss angle ($\tan \delta$).

9. Which bridge is commonly employed in audio oscillators as a frequency-determining network?

Explanation: The Wien bridge incorporates series and parallel RC combinations, making it ideal for audio frequency measurement and oscillator frequency control.

10. What is the complex impedance of a series L-C-R branch?

Explanation: The impedance of a series L-C-R circuit is given by $Z = R + jX$, where reactive part $X = \omega L - \frac{1}{\omega C}$.

11. What is the null detector typically used in low-frequency AC bridges?

Explanation: Headphones (for audible frequencies) and electronic voltmeters are standard null detectors in AC bridges.

12. In a Schering bridge, the unknown capacitor test arm is accompanied by a series resistance $r_1$. What is its equivalent value at balance?

Explanation: At balance, the series equivalent resistance of the lossy capacitor arm is given by $r_1 = R_2 \frac{C_4}{C_3}$.

13. If an AC bridge is balanced, what is the current passing through the null detector?

Explanation: By definition, at bridge balance, no current flows through the null detector as the potential difference across its terminals is zero.

14. Which bridge is generally preferred for measuring high inductors (high $Q$ coils)?

Explanation: While Maxwell bridge is for low to medium $Q$ coils, the Hay bridge is better suited for high $Q$ inductors.

15. What is the unit of detector sensitivity $S$ defined as $\frac{\Delta I}{\Delta Z}$?

Explanation: Sensitivity is expressed as detector current change per unit impedance imbalance, having units of $A/\Omega$.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The general balance condition for an AC bridge is given by $Z_1 Z_4 = Z_2 Z_3$. If $Z_1 = R_1$, $Z_2 = R_2$, $Z_3 = \frac{1}{j\omega C_3}$, and $Z_4$ is an unknown impedance $Z_x$, what is $Z_x$?

Explanation: Using $Z_x Z_1 = Z_2 Z_3 \implies Z_x R_1 = R_2 \left(\frac{1}{j\omega C_3}\right)$, giving $Z_x = \frac{R_2}{j\omega C_3 R_1}$.

2. In a Maxwell bridge used for measuring inductance, if $R_2 = 1000\;\Omega$, $R_3 = 1000\;\Omega$, and $C = 1\;\mu F$, what is the unknown inductance $L$?

Explanation: Using formula $L = R_2 R_3 C = 1000 \times 1000 \times 1 \times 10^{-6} = 1\;H$.

3. Which of the following AC bridges does NOT depend on the frequency of the AC source for its balance equation?

Explanation: Maxwell's inductance-capacitance bridge yields independent balance equations for resistance and inductance, making it frequency-independent.

4. In a Schering bridge, the dissipation factor $D$ is given by $\omega C_4 R_4$. If angular frequency $\omega = 3140\;rad/s$, $C_4 = 0.1\;\mu F$, and $R_4 = 1000\;\Omega$, what is the value of $D$?

Explanation: $D = 3140 \times (0.1 \times 10^{-6}) \times 1000 = 3140 \times 10^{-4} = 0.314$.

5. What is the primary purpose of incorporating a Wagner ground connection in AC bridge setups?

Explanation: Wagner earth (ground) connection eliminates stray earth capacitance effects that hinder precise balancing.

6. For a Wien bridge network with $R_3 = R_4 = 10\;k\Omega$ and $C_3 = C_4 = 1\;nF$, operating at frequency $f$, what is the approximate frequency?

Explanation: $f = \frac{1}{2\pi R C} = \frac{1}{2 \times 3.1416 \times 10^4 \times 10^{-9}} = \frac{1}{6.283 \times 10^{-5}} \approx 15915\;Hz \approx 15.9\;kHz$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. An AC bridge has arm impedances $Z_1 = 200\;\Omega$ resistive, $Z_2 = 400\;\Omega$ resistive, and $Z_3 = 150\;\Omega$ resistive. For the bridge to be balanced, what must be the impedance $Z_4$?

Explanation: $Z_1 Z_4 = Z_2 Z_3 \implies 200 \times Z_4 = 400 \times 150 \implies Z_4 = \frac{60000}{200} = 300\;\Omega$.

2. Which of the following conditions is mandatory for balancing any AC bridge network?

Explanation: Complex balance condition requires matching both magnitude and phase (real and imaginary parts).

3. In a De Sauty bridge, standard capacitor $C_2 = 0.5\;\mu F$, $R_1 = 2000\;\Omega$, and $R_2 = 1000\;\Omega$. The unknown capacitance $C_1$ is:

Explanation: $C_1 = C_2 \frac{R_2}{R_1} = 0.5 \times \frac{1000}{2000} = 0.25\;\mu F$.

4. Why is the Schering bridge exceptionally suited for high-voltage insulation testing?

Explanation: Schering bridges safely handle high voltages because the measurement involves low-voltage arms and direct dissipation factor determination.

5. What happens to the balance of a Wien bridge if the supply frequency changes?

Explanation: Wien bridge balance conditions depend explicitly on angular frequency $\omega$; thus, frequency changes disturb the balance.

6. In a Maxwell bridge, if the resistance $R_2$ is increased while keeping other parameters constant, how does the calculated inductance $L = R_2 R_3 C$ change?

Explanation: Since $L$ is directly proportional to $R_2$, increasing $R_2$ increases the calculated inductance $L$.

7. Which bridge is considered the AC equivalent of the Wheatstone bridge for capacitance measurement of lossless capacitors?

Explanation: The De Sauty bridge is the simplest AC bridge for comparing capacitances, analogous to the DC Wheatstone bridge.

8. What is the primary limitation of the simple De Sauty bridge?

Explanation: Real capacitors have losses, making the simple De Sauty bridge (which assumes lossless capacitors) impractical without a modification like Schering's bridge.

9. If an electronic voltmeter used as a null detector reads a non-zero minimum instead of absolute zero during bridge balancing, what is usually the cause?

Explanation: Harmonics and stray pick-up prevent a sharp null, creating a minimum voltage instead of true zero.

10. In a Wien bridge oscillator circuit, what is the role of the bridge network?

Explanation: The Wien bridge provides frequency-selective feedback to sustain oscillations at the specific bridge balance frequency.

AC Bridges

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Principle of AC Bridge Balance

Complex Balance Condition

Maxwell's Inductance-Capacitance Bridge

Schering Bridge

Wien Bridge

Derivations

General Balance Condition for an AC Bridge

Maxwell's Inductance Bridge Derivation

Formula sheet

Graphs and experiments

Wagner Earthing Device Circuit Setup

Null Detector Response Curve

Wien Bridge Frequency Response

Solved numericals

In a de Sauty bridge used to compare two capacitances, the arm resistances at balance are R1 = 400 Ω and R2 = 800 Ω. If the standard capacitor C2 has a value of 0.5 μF, find the unknown capacitance C1.

A Maxwell's inductance bridge is balanced with the following component values: R2 = 1000 Ω, R3 = 1000 Ω, R4 = 500 Ω, and C4 = 0.2 μF. Calculate the unknown inductance L1 and its effective series resistance r1 assuming R1 = 0.

In a Wien bridge oscillator network, the frequency-determining arms consist of equal resistors $R = 10 k\Omega$ and equal capacitors C = 10 nF. Determine the frequency of oscillation.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Comparison of AC Bridges

Practice Quiz

1. What is the fundamental balance condition for an AC bridge with complex impedances $Z_1, Z_2, Z_3,$ and $Z_4$?

2. Why do most AC bridges require at least two variable components for complete balance?

3. Maxwell's inductance-capacitance bridge measures unknown inductance in terms of which standard component?

4. Which of the following bridges is primarily used for measuring capacitance and dielectric loss angle?

5. What is the frequency expression for a Wien bridge when symmetrical arms are used ($R_3 = R_4$ and $C_3 = C_4$)?

6. In a Maxwell bridge, under what condition is the balance equation independent of frequency?

7. What is the expression for the unknown capacitance $C_1$ in a basic De Sauty bridge with arm resistances $R_1, R_2$ and standard capacitor $C_2$?

8. What does the dissipation factor $D$ in a Schering bridge represent?

9. Which bridge is commonly employed in audio oscillators as a frequency-determining network?

10. What is the complex impedance of a series L-C-R branch?

11. What is the null detector typically used in low-frequency AC bridges?

12. In a Schering bridge, the unknown capacitor test arm is accompanied by a series resistance $r_1$. What is its equivalent value at balance?

13. If an AC bridge is balanced, what is the current passing through the null detector?

14. Which bridge is generally preferred for measuring high inductors (high $Q$ coils)?

15. What is the unit of detector sensitivity $S$ defined as $\frac{\Delta I}{\Delta Z}$?

PYQ Patterns

1. The general balance condition for an AC bridge is given by $Z_1 Z_4 = Z_2 Z_3$. If $Z_1 = R_1$, $Z_2 = R_2$, $Z_3 = \frac{1}{j\omega C_3}$, and $Z_4$ is an unknown impedance $Z_x$, what is $Z_x$?

2. In a Maxwell bridge used for measuring inductance, if $R_2 = 1000\;\Omega$, $R_3 = 1000\;\Omega$, and $C = 1\;\mu F$, what is the unknown inductance $L$?

3. Which of the following AC bridges does NOT depend on the frequency of the AC source for its balance equation?

4. In a Schering bridge, the dissipation factor $D$ is given by $\omega C_4 R_4$. If angular frequency $\omega = 3140\;rad/s$, $C_4 = 0.1\;\mu F$, and $R_4 = 1000\;\Omega$, what is the value of $D$?

5. What is the primary purpose of incorporating a Wagner ground connection in AC bridge setups?

6. For a Wien bridge network with $R_3 = R_4 = 10\;k\Omega$ and $C_3 = C_4 = 1\;nF$, operating at frequency $f$, what is the approximate frequency?

Timed Topic Test

1. An AC bridge has arm impedances $Z_1 = 200\;\Omega$ resistive, $Z_2 = 400\;\Omega$ resistive, and $Z_3 = 150\;\Omega$ resistive. For the bridge to be balanced, what must be the impedance $Z_4$?

2. Which of the following conditions is mandatory for balancing any AC bridge network?

3. In a De Sauty bridge, standard capacitor $C_2 = 0.5\;\mu F$, $R_1 = 2000\;\Omega$, and $R_2 = 1000\;\Omega$. The unknown capacitance $C_1$ is:

4. Why is the Schering bridge exceptionally suited for high-voltage insulation testing?

5. What happens to the balance of a Wien bridge if the supply frequency changes?

6. In a Maxwell bridge, if the resistance $R_2$ is increased while keeping other parameters constant, how does the calculated inductance $L = R_2 R_3 C$ change?

7. Which bridge is considered the AC equivalent of the Wheatstone bridge for capacitance measurement of lossless capacitors?

8. What is the primary limitation of the simple De Sauty bridge?

9. If an electronic voltmeter used as a null detector reads a non-zero minimum instead of absolute zero during bridge balancing, what is usually the cause?

10. In a Wien bridge oscillator circuit, what is the role of the bridge network?

In a de Sauty bridge used to compare two capacitances, the arm resistances at balance are R₁ = 400 Ω and R₂ = 800 Ω. If the standard capacitor C₂ has a value of 0.5 μF, find the unknown capacitance C₁.

A Maxwell's inductance bridge is balanced with the following component values: R₂ = 1000 Ω, R₃ = 1000 Ω, R₄ = 500 Ω, and C₄ = 0.2 μF. Calculate the unknown inductance L₁ and its effective series resistance r₁ assuming R₁ = 0.