A.C Bridges & Potentiometer - GATE Notes & Videos for Electrical Engineering - Electrical

AC Bridges

AC bridges are network arrangements used to measure self-inductance, mutual inductance, capacitance and frequency by balancing impedances in a four-arm bridge. They are widely used in laboratory and calibration work where accuracy and phase information are required.

Types of Sources

• At low frequency, the mains (power line) may be used as the source.
• At higher frequencies, electronic oscillators are used to provide a stable AC source.

Types of Detectors

• Vibration galvanometer - used at power frequency and audio frequencies up to about 1 kHz for null detection.
• Headphones - used in the audio band (≈250 Hz to 4 kHz) for listening-type null detection.
• Tunable amplifier or electronic detector - used over a wide range (≈10 Hz to 100 kHz) when greater sensitivity or remote readout is required.

General Bridge Circuit

The general four-arm bridge has impedances Z₁, Z₂, Z₃, Z₄ in the four arms and a detector between the two diagonal nodes. At balance (no current through the detector) the following two conditions must be satisfied:

• Magnitude condition: |Z₁| · |Z₄| = |Z₂| · |Z₃|
• Phase (angle) condition: ∠Z₁ + ∠Z₄ = ∠Z₂ + ∠Z₃

General bridge circuit

Measurement of Self Inductance

Several bridge arrangements exist to measure self inductance. Choice depends on coil Q, required accuracy and convenience.

• Maxwell's inductance bridge
• Maxwell's inductance-capacitance bridge
• Hay's bridge
• Anderson's bridge
• Owen's bridge

Maxwell's Inductance Bridge

Maxwell's bridge measures an unknown inductance by comparing it with a standard capacitance and resistances. It is suitable for coils of moderate Q and gives good accuracy without requiring a standard inductor.

Notation used in the diagram:

• L₁ - Unknown inductance (with internal resistance R₁).
• L₂ - Variable inductance (when present in alternative forms).
• R₂ - Variable (standard) resistance.
• R₃, R₄ - Fixed standard non-inductive resistances.

Maxwell's Inductance-Capacitance Bridge

This form of Maxwell bridge uses a standard capacitor in one arm and is commonly used for medium Q coils (1 < Q < 10). The bridge can be balanced by adjusting resistances and/or capacitance.

The Q-factor expression for the coil (as shown in the figure) is used to determine suitability of this bridge for a given coil.

Typical practical notes:

• Use R₄ as a variable non-inductive resistance and C₄ as a standard variable capacitance.
• If C₄ is fixed, balance may be obtained either by varying R₂ and R₄, or by placing a resistance in series with L₁ and varying R₄.

Hay's Bridge

Hay's bridge is a modification for measuring inductances of high Q coils (Q > 10). It improves accuracy where the resistance of the coil is small compared with its reactance.

A standard capacitor C₄ is used in one arm. The Q-factor expression for Hay's bridge is shown in the diagram.

From the design shown, the inductance relation often used in this arrangement is

• L₁ = R₂ R₃ C₄ (as indicated in the figure).

Anderson's Bridge

Anderson's bridge converts the inductance arm into a combination involving a capacitor so that the bridge becomes effectively resistive in balance. It is especially suitable for low Q coils (Q < 1) and can also be used to measure capacitance in terms of inductance.

Notation:

• L₁ - Unknown self-inductance with internal resistance R₁.
• R₂, R₃, R₄ - Fixed standard non-inductive resistances.
• r - Variable resistance.
• C - Fixed capacitance.

• Anderson's bridge is suitable for measurement when the coil has a small Q-factor (Q < 1).
• It provides a direct relation between L and standard R and C values so that inductance may be measured accurately without a standard inductor.

Key Points: Inductance Bridges

• Maxwell's bridge is suitable for measuring inductance of low-to-moderate Q inductors.
• Hay's bridge is preferred for high Q inductors.
• Anderson's bridge measures inductance (and can measure capacitance in terms of inductance) and is useful when Q is small.

Owen's Bridge

Owen's bridge is another arrangement for measuring inductance where a standard capacitor and resistances are used in the bridge network to obtain balance.

Notation in the Owen's bridge diagrams:

• L₁ - Unknown self-inductance with internal resistance R₁.
• R₂ - Variable non-inductive resistance.
• R₃ - Fixed standard non-inductive resistance.
• C₂ - Standard variable capacitor.
• C₄ - Fixed standard capacitor.

Measurement of Capacitance

Capacitance is measured using AC bridges designed to relate an unknown capacitor to standard capacitors and resistances. Important bridges include:

• De-Sauty Bridge
• Schering Bridge

De-Sauty Bridge

De-Sauty bridge is a simple bridge for measuring lossless capacitances (for example air-cored or gas-filled capacitors). It compares an unknown capacitor with a standard capacitor and uses fixed resistances in the other arms.

Notation:

• C₁ - Unknown capacitor.
• C₂ - Standard capacitor.
• R₃, R₄ - Fixed non-inductive resistances.

Schering Bridge

Schering bridge is widely used to measure capacitance and dielectric loss of insulating materials. It is particularly suited to measure low values of capacitance and to determine dissipation factor and relative permittivity of dielectrics.

The dissipation factor (loss tangent) at the test frequency is given by:

• D₁ = tan δ = ω C₁ r₁ = ω C₄ R₄

Notation:

• C₁ - Unknown capacitor with loss component r₁.
• C₂ - Fixed standard capacitor.
• R₃ - Fixed standard non-inductive resistance.
• C₄ - Variable capacitor.
• R₄ - Variable non-inductive resistance.

Key Points: Capacitance Bridges

• Schering bridge is widely used for measurement of capacitance, dissipation factor and dielectric characteristics of insulating materials such as insulating oil, bushings and capacitors.
• High-voltage Schering bridge variants are used for insulation testing at elevated voltages.
• For low-value capacitances, Schering bridge provides better sensitivity compared with simpler bridges.

Measurement of Frequency

Frequency can be measured using bridge methods such as the Wein bridge, which balances at a particular frequency determined by the resistances and capacitances in the bridge.

Wein's Bridge

The bridge balance gives a frequency at which the bridge is balanced; the expression appears in the diagram.

For the common symmetrical case when R₁ = R₂ = R and C₁ = C₂ = C, the balance (and the notch or peak frequency for the Wein network) is:

• Wein's bridge is used as a notch filter in harmonic distortion analysers.
• It is used as a frequency isolator in oscillator and amplifier circuits.
• It is commonly employed to isolate and remove the fundamental frequency component in a harmonics analyser, enabling measurement of harmonic content.

Wagner's Earthing Device

Wagner's earthing device (also called Wagner earthing) is used to eliminate the effect of stray earth capacitances when measuring capacitance with a Schering bridge. It improves measurement accuracy by providing a balance path for leakage and earth currents.

Other Bridge and Comparisons

• Carey-Foster bridge is designed to determine the small difference between two nearly equal resistances accurately.
• The potentiometer wire used for precision DC measurements should have a high specific resistance and a low temperature coefficient.
• Sensitivity of a potentiometer increases with increasing length of the potentiometer wire (for a given working current).

DC & AC Potentiometer

Potentiometers are precision instruments used to measure electromotive force (emf) or to compare voltages without drawing current from the source under test. They are also used to calibrate voltmeters, ammeters and wattmeters.

DC Potentiometer

A DC potentiometer compares two emf sources by balancing them against a precisely known voltage drop along a long uniform wire (the slide-wire). The working current in the wire is adjusted (standardised) so that the drop per unit length is known.

• Because a potentiometer measures voltage directly, it can be used to determine current by measuring the voltage drop across a known standard resistance carrying the unknown current.
• The potentiometer is widely used for calibration of voltmeters, ammeters and wattmeters and is often taken as a laboratory standard for voltage comparisons.
• Standardization: Adjust the working current so that the voltage drop per unit length of the slide wire matches a known reference source; this process is called standardization.

Principle of null comparison: if two cells of equal emf are connected head-to-head through a galvanometer and there is no circulating current, the galvanometer shows null deflection. In the potentiometer the sliding contact is moved until the galvanometer indicates null, giving a length on the wire proportional to the emf.

If a standard cell of emf E gives a balance at length L and an unknown cell of emf E₁ gives a balance at length L₁ along the same wire, then

E = L·v

E₁ = L₁·v

Dividing,

E / E₁ = L / L₁

Hence, knowing the standard cell emf E and the measured lengths, the unknown emf E₁ is found directly.

AC Potentiometer

An AC potentiometer compares an unknown AC voltage with a known AC reference and, unlike the DC type, can determine both magnitude and phase of the unknown voltage. The basic balancing principle is similar to the DC potentiometer but additional arrangements are used to control phase.

There are two common types of AC potentiometers:

• Polar type potentiometer
• Coordinate type potentiometer

Polar Type Potentiometer

Polar instruments use separate controls to set magnitude and phase on a polar scale referenced to the unknown emf. They typically include an electrodynamometer type ammeter and a phase-shifting transformer operated from single-phase supply. The phase-shifting transformer employs two stators arranged at 90° to each other with series components that allow small adjustments to maintain constant supply to the potentiometer rotor. The induced voltages from the two stators combine to give a resultant rotor voltage whose phase relative to the supply can be controlled; the mathematical representation of these induced emfs is shown schematically.

The combined induced emf may be written (as shown in the figure) and results in a resultant emf proportional to sin(ωt - φ), where φ is the phase angle provided by the phase shifter.

Coordinate Type Potentiometer

In the coordinate AC potentiometer two linear potentiometers operate at right angles to provide in-phase and quadrature components of the unknown emf. One potentiometer measures the component in phase with a reference (in-phase potentiometer) and the other measures the component 90° out of phase (quadrature potentiometer). By adjusting both potentiometers and their slide contacts, the resultant vector sum of the two components can be set equal to the unknown emf and the galvanometer shows null.

• The in-phase potentiometer provides the real component and the quadrature potentiometer provides the imaginary component of the unknown voltage.
• Rheostats and sliding contacts adjust currents so that the quadrature and in-phase contributions cancel the unknown voltage, producing a null.
• Sign-changing switches S₁, S₂ are provided to reverse polarity if required for balance.
• Two step-down transformers T₁ and T₂ isolate the potentiometer from the mains and provide earthed shielding between windings; they typically supply low voltages (for example about 6 V) to the potentiometer circuits.

The resultant magnitude and phase of the vector sum of the coordinate components give the magnitude and phase angle of the unknown emf.

Applications of AC Potentiometer

• Measurement of self-inductance and capacitance where phase information is required.
• Calibration of AC voltmeters, ammeters and wattmeters.
• Precise comparison of AC sources and verification of phase relationships in networks.