Bridges - Electrical and Electronic Measurements - Electrical Engineering (EE)

What is Bridge?

A bridge is an electrical circuit arranged in a loop or ring of four arms (branches) so that the network has two opposite nodes for excitation and the other two opposite nodes for a detector. Each arm may contain one or more passive components (resistors, inductors, capacitors) or combinations of these. Bridges are used for precise measurement of electrical quantities by balancing the network so that the detector shows zero (null) response.

Types of Bridges

Bridge circuits are broadly classified according to the type of excitation that they require:

• DC bridges - operated with a direct-current (DC) source and used mainly for measurement of resistance.
• AC bridges - operated with an alternating-current (AC) source and used for measurement of inductance, capacitance, frequency and complex impedances.

DC Bridges

DC bridges are primarily used to measure an unknown resistance by comparison with known resistances. The simplest and most common DC bridge is the Wheatstone bridge.

The typical bridge has four arms. In a measurement arrangement one diagonal of the bridge is connected to a DC source and the other diagonal has a sensitive null detector (a galvanometer). Commonly, two arms have fixed known resistances, one arm is a variable resistance for balancing and the fourth arm contains the unknown resistance to be measured.

• The bridge is excited by a DC voltage source applied across one diagonal; a galvanometer is connected across the other diagonal.
• The galvanometer shows a deflection whenever the bridge is unbalanced.
• The variable resistor is adjusted until the galvanometer shows null (zero) deflection; this is the balanced condition and allows calculation of the unknown resistance using the balance relation.

Derivation of the balance condition for a Wheatstone bridge with arms R₁, R₂, R₃ and R_x (unknown):

At balance no current flows through the galvanometer; the potentials at the two mid-nodes are equal.

Potential at the node between R₁ and R₂ measured from the negative terminal equals the source voltage multiplied by the lower arm divider fraction.

V₁ = V × R₂ / (R₁ + R₂)

Potential at the node between R₃ and R_x measured from the negative terminal equals the source voltage multiplied by that lower arm divider fraction.

V₂ = V × R_x / (R₃ + R_x)

Equating the two potentials at balance:

R₂ / (R₁ + R₂) = R_x / (R₃ + R_x)

Cross-multiply and rearrange to obtain the standard balance relation:

R₁ × R_x = R₂ × R₃

Or equivalently

R₁ / R₂ = R₃ / R_x

From this relation the unknown resistance R_x can be calculated as:

R_x = (R₂ × R₃) / R₁

Practical notes on DC bridge measurements:

• Sensitivity depends on the galvanometer sensitivity and the ratios of the arms. A small change in the unknown produces a larger null shift when the bridge is more sensitive.
• For very low resistances use four-terminal (Kelvin) connections or Kelvin double bridge to reduce lead and contact resistance errors.
• Temperature stability of the standard resistors and proper shielding reduce measurement errors.

AC Bridges

AC bridges measure unknown impedance values such as inductances and capacitances. Each arm of an AC bridge contains an impedance (Z) that may be resistive, inductive, capacitive or a combination. A suitable AC source is applied to one diagonal of the bridge and a sensitive AC detector (headphone, AC galvanometer, lock-in amplifier or null detector) is connected to the other diagonal.

• Two arms are usually made of known impedances, one arm is made variable, and the fourth arm contains the unknown impedance to be determined.
• The detector indicates null while the bridge is unbalanced and gives zero at exact balance; the variable arm(s) are adjusted until null is obtained.
• At balance the relationship between impedances gives the unknown impedance; for complex impedances both magnitude and phase conditions must be satisfied.

General balance condition for a bridge whose arms are Z₁, Z₂, Z₃ and Z₄ (complex impedances):

At balance the product of opposite arms are equal.

Z₁ × Z₄ = Z₂ × Z₃

Equivalently

Z₁ / Z₂ = Z₃ / Z₄

Because Z are complex quantities, the condition requires equality of both real and imaginary parts (or equality of magnitude and phase). Therefore two independent adjustable elements (e.g., a variable resistor and a variable capacitor) or a combination of adjustable resistor and adjustable frequency are usually required to satisfy both conditions simultaneously.

Common AC bridge types and their typical uses:

• Maxwell bridge - measurement of moderate-Q inductances using a standard capacitor and resistors; commonly used in laboratory inductance measurement.
• Hay bridge - measurement of inductances with higher Q; modification of Maxwell to improve accuracy at higher Q.
• Anderson bridge - an alternative for inductance measurement with different balancing network that can offer advantages in some ranges.
• Schering bridge - measurement of capacitance and dissipation factor (loss) of capacitors.
• De Sauty bridge - measurement of capacitance by comparison with a standard capacitor; simple but less accurate for high loss capacitors.
• Wien bridge - frequency selective bridge; used for precise frequency measurements and oscillator circuits (Wien bridge oscillator).
• Kelvin (double) bridge - precision measurement of low resistances using four-terminal connections to eliminate lead resistance effects.

Detectors used with AC bridges:

• Sensitive headphones or earphones (audio frequency experimental work).
• AC galvanometer (for laboratory bridges at low frequency).
• Lock-in amplifier / synchronous detector (gives very high sensitivity and noise rejection at a chosen frequency).
• Oscilloscope or null indicator for visual detection of balance in some bridges.

Bridge Sensitivity and Sources of Error

Factors affecting sensitivity and accuracy of bridge measurements include:

• Detector sensitivity - more sensitive null detectors permit finer balance and better resolution.
• Source stability - fluctuations in excitation amplitude or frequency change divider ratios and produce errors unless the detector is true null type.
• Temperature variations - temperature coefficients of resistors, inductors and capacitors cause drift in standard values.
• Stray and lead impedances - lead resistance, contact resistance and stray capacitance/inductance affect low and high impedance measurements; four-terminal connections help mitigate these.
• Quality factor (Q) of inductors - bridges that assume ideal inductors may introduce error when inductors have low Q (losses).
• Frequency dependence - reactances depend on frequency; precise AC bridges require stable and known frequency.

Methods to improve accuracy:

• Use precision standard components with low temperature coefficient and known tolerances.
• Adopt four-wire (Kelvin) connections for low-resistance measurements.
• Use shielding and proper earthing to reduce pickup and stray coupling.
• Use synchronous detection (lock-in) to improve signal-to-noise ratio for AC null detection.
• Calibrate frequently against known standards and, where necessary, apply temperature corrections.

Practical Applications

• Precision measurement of resistance, inductance and capacitance in calibration laboratories and instrument workshops.
• Strain gauge and load cell circuits employ Wheatstone bridge arrangements for conversion of small resistance changes into measurable voltages.
• Temperature sensors such as RTDs (resistance temperature detectors) are commonly connected in a bridge to provide a sensitive temperature dependent output.
• Component testing - determining dissipation factor, equivalent series resistance and loss characteristics of capacitors and inductors.
• Wien bridge used in oscillator design (Wien bridge oscillator) where bridge balance determines oscillation frequency.

Summary

Bridge circuits form a fundamental class of measurement networks used to compare an unknown quantity with known standards by balancing the network until a null condition is obtained. DC bridges (Wheatstone and Kelvin bridges) measure resistance precisely. AC bridges extend the principle to measure inductance, capacitance and frequency using complex impedance balance. Proper choice of bridge type, sensitive null detection, good standards and correct connection technique produce accurate and repeatable measurements in electrical and electronic measurement practice.