AC bridges: Maxwell, Hay, Schering and Wien

General AC bridge balance (magnitude and angle), Maxwell and Hay bridges for inductance and Q, Schering bridge for capacitance and loss angle, and the frequency-dependent Wien bridge, with worked balances.

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Why it matters

Inductors, capacitors and insulation are characterised by their behaviour on AC: inductance and Q of a coil, capacitance and loss angle of a cable or bushing. AC bridges measure these by null comparison with stable resistors and capacitors. The Schering bridge, for example, is the standard test for the dielectric loss of high-voltage insulation.

Key ideas

General balance condition. Number the arms 1–4 so that 1 is opposite 4 and 2 opposite 3. At balance the detector voltage is zero in both magnitude and phase, so Z1·Z4 = Z2·Z3 with complex impedances. This single complex equation gives two conditions:

  • magnitudes: |Z1|·|Z4| = |Z2|·|Z3|;
  • angles: θ1 + θ4 = θ2 + θ3. Two independent adjustable elements are therefore needed, and balance is reached by adjusting them alternately. The angle condition gives a quick check of whether a configuration can balance at all: for example, if arms 2 and 3 are pure resistors, an inductive arm 1 needs a capacitive arm 4 opposite it.

Sources and detectors. Supply: an oscillator (typically 1 kHz for L and C, 50 Hz for insulation tests). Detectors: headphones (audio range), vibration galvanometer (power frequency), tuned amplifier or electronic null detector (most common today).

Maxwell inductance–capacitance bridge. The unknown coil (L1 with series resistance R1) is balanced against a capacitor C4 in parallel with R4 in the opposite arm; arms 2 and 3 are resistors. Both balance equations are independent of frequency. Q = ωC4R4, so it suits medium-Q coils (about 1 < Q < 10): for high Q, R4 would have to be impractically large. (Maxwell's inductance bridge compares the unknown with a standard inductor instead.)

Hay bridge. Same as Maxwell but with C4 in series with R4. The balance equations contain ω, but for high Q the term ω²C4²R4² is small and L1 ≈ R2R3C4. Suits high-Q coils (Q > 10).

Schering bridge. Measures capacitance C1 and dissipation factor of a capacitor or insulation, modelled as C1 in series with a loss resistance r1. Arm 2 is a loss-free standard capacitor C2 (air or gas-filled), arm 3 a variable resistor R3, arm 4 a fixed R4 in parallel with variable C4. In high-voltage use the high-voltage corner is at the junction of arms 1 and 2, so the operator adjusts only arms 3 and 4, which are near earth potential; arms 1 and 2 have very high impedance and carry most of the voltage. Stray capacitances are controlled by screening and a Wagner earth device, which brings the detector to earth potential at balance.

Wien bridge. One arm is R1 in series with C1, the adjacent arm R2 in parallel with C2, the other two arms pure resistors R3 and R4. Balance occurs at only one frequency, so the bridge measures frequency (or C if f is known). With equal R and C the frequency is 1/(2πRC) and R3 = 2R4. It is also the feedback network of the Wien-bridge oscillator and a notch filter for harmonic-distortion meters.

Q and D. For a coil, Q = ωL/R; for a capacitor, dissipation factor D = tan δ = ωC·r (series model) — δ is the loss angle, 90° minus the phase angle. Power factor ≈ D when D is small.

Formulas

Z1·Z4 = Z2·Z3 ; |Z1||Z4| = |Z2||Z3| and θ1 + θ4 = θ2 + θ3 General AC balance (arms 1, 4 opposite).

L1 = R2·R3·C4 ; R1 = R2·R3 / R4 ; Q = ω·C4·R4 (Maxwell L–C) L1 in H; R in Ω; C4 in F; ω = 2πf (rad/s).

L1 = R2·R3·C4 / (1 + ω²C4²R4²) ; R1 = ω²C4²R2R3R4 / (1 + ω²C4²R4²) ; Q = 1 / (ω·C4·R4) (Hay)

C1 = C2·R4 / R3 ; r1 = R3·C4 / C2 ; D = tan δ = ω·C1·r1 = ω·C4·R4 (Schering) C1, C2, C4 in F; r1, R3, R4 in Ω.

f = 1 / (2π·√(R1R2C1C2)) ; R3/R4 = R1/R2 + C2/C1 (Wien; R1, C1 series arm; R2, C2 parallel arm; R3 adjacent to the series arm) With R1 = R2 = R and C1 = C2 = C: f = 1/(2πRC), R3 = 2R4.

Worked examples

Example 1 — Maxwell bridge. A Maxwell L–C bridge balances at 1 kHz with R2 = 400 Ω, R3 = 600 Ω, C4 = 0.5 μF and R4 = 1 kΩ. Find L1, R1 and Q.

  1. L1 = R2R3C4 = 400 × 600 × 0.5 × 10⁻⁶ = 0.12 H.
  2. R1 = R2R3/R4 = 400 × 600/1000 = 240 Ω.
  3. Q = ωC4R4 = 2π × 1000 × 0.5 × 10⁻⁶ × 1000 = 3.14.

Answer: L1 = 0.12 H, R1 = 240 Ω, Q = 3.14 (a medium-Q coil, suitable for Maxwell).

Example 2 — Schering bridge (GATE level). A Schering bridge at 50 Hz balances with C2 = 100 pF, R3 = 130.5 Ω, R4 = 1000/π Ω and C4 = 0.5 μF. Find the capacitance, series loss resistance and dissipation factor of the specimen.

  1. C1 = C2R4/R3 = 100 × (318.31/130.5) = 243.9 pF.
  2. r1 = R3C4/C2 = 130.5 × 0.5 × 10⁻⁶ / 100 × 10⁻¹² = 652.5 kΩ.
  3. D = ωC4R4 = 2π × 50 × 0.5 × 10⁻⁶ × 1000/π = 0.05.
  4. Check: ωC1r1 = 314.16 × 243.9 × 10⁻¹² × 652 500 = 0.05.

Answer: C1 = 243.9 pF, r1 = 652.5 kΩ, tan δ = 0.05. Choosing R4 = 1000/π Ω at 50 Hz makes D read directly as 0.1 × C4 in μF.

Example 3 — Hay bridge. At 1 kHz, a Hay bridge balances with R2R3 = 10⁶ Ω², C4 = 0.1 μF and R4 = 100 Ω. Find L1, R1 and Q.

  1. ωC4R4 = 6283.2 × 0.1 × 10⁻⁶ × 100 = 0.06283; 1 + (0.06283)² = 1.003948.
  2. L1 = 10⁶ × 0.1 × 10⁻⁶ / 1.003948 = 0.09961 H.
  3. R1 = ω²C4²R2R3R4 / 1.003948 = (6283.2)² × (10⁻⁷)² × 10⁶ × 100 / 1.003948 = 39.48/1.003948 = 39.32 Ω.
  4. Q = 1/(ωC4R4) = 1/0.06283 = 15.9.

Answer: L1 = 99.6 mH, R1 = 39.3 Ω, Q = 15.9. Neglecting the denominator would give 100 mH, only 0.4 % high — the approximation used for high-Q coils.

Common mistakes

  • Balancing magnitudes only and forgetting the phase condition; an AC bridge needs two adjustments.
  • Putting the balancing capacitor in the wrong arm: with resistive arms 2 and 3 the capacitor must be opposite the inductor.
  • Using Maxwell's bridge for Q > 10 or Hay's for Q < 10 — swapping their ranges.
  • Mixing up Schering arms: C1 = C2R4/R3, not C2R3/R4.
  • Forgetting that the Wien bridge balance depends on frequency and that a distorted supply never gives a perfect null.
  • Ignoring stray capacitance at high frequency or high voltage; screening and a Wagner earth are needed.

For GATE IN

Expect NAT questions on L, R, C and loss factor from Maxwell, Hay and Schering balances, and the balance frequency of a Wien bridge. MCQs ask which bridge suits which Q range, whether a given arm configuration can balance (angle condition), and the role of the Wagner earth. Practise deriving balance equations from Z1Z4 = Z2Z3 by equating real and imaginary parts.

Quick check

  1. Which bridge would you use for a coil with Q = 30?
  2. In a Wien bridge with R = 10 kΩ and C = 10 nF in both RC arms, what is the balance frequency?
  3. A Schering bridge gives ωC4R4 = 0.01. What is the loss angle?
  4. Arms 2 and 3 are resistors and arm 1 is capacitive. What must arm 4 be? Answers: 1. Hay bridge; 2. 1591.5 Hz; 3. δ = tan⁻¹ 0.01 ≈ 0.57°; 4. inductive (or an arm whose angle is positive), so that θ1 + θ4 = 0.

Interview questions

All Electrical and Electronic Measurements interview questions

Try answering each one aloud before you open it.

  1. 1.What is an AC bridge and why is it used in electrical measurements?Concept

    An AC bridge is a circuit used to measure unknown electrical quantities such as inductance, capacitance, and resistance. It operates on the principle of balancing two legs of a bridge circuit, one leg of which includes the unknown component. AC bridges are used because they provide high accuracy and are suitable for measuring components at different frequencies.

  2. 2.Explain the working principle of a Maxwell bridge.Concept

    The Maxwell inductance–capacitance bridge measures a coil (L1 in series with R1) by balancing it against a capacitor C4 in parallel with R4 in the opposite arm, the other two arms being resistors R2 and R3. Equating Z1·Z4 = Z2·Z3 gives L1 = R2R3C4 and R1 = R2R3/R4, both independent of frequency, and Q = ωC4R4. Because a capacitor standard is more accurate and cheaper than a standard inductor, it is widely used, but only for medium-Q coils (roughly 1 to 10).

  3. 3.How does a Hay bridge differ from a Maxwell bridge?Concept

    In the Hay bridge the standard capacitor C4 is in series with R4 instead of in parallel. This gives Q = 1/(ωC4R4), so a high Q needs only a small R4, whereas Maxwell's Q = ωC4R4 would need an impractically large R4. The Hay balance equations contain frequency, L1 = R2R3C4/(1 + ω²C4²R4²), but for Q > 10 the correction is under 1 % and L1 ≈ R2R3C4. So Hay suits high-Q coils and Maxwell medium-Q coils.

  4. 4.Describe the purpose and operation of a Schering bridge.Concept

    The Schering bridge measures the capacitance and dissipation factor (tan δ) of capacitors, cables and insulation. The specimen is modelled as C1 in series with a loss resistance r1; the arm next to it is a loss-free standard capacitor C2, and the other two arms are a variable resistor R3 and a fixed R4 shunted by a variable C4. Balance gives C1 = C2R4/R3 and tan δ = ωC4R4. In high-voltage use the adjustable arms are near earth potential, which makes it safe to operate, and a Wagner earth removes stray-capacitance errors.

  5. 5.What is a Wien bridge and what is its primary application?Concept

    A Wien bridge is an AC bridge used primarily for measuring frequencies. It consists of a series RC network in one arm and a parallel RC network in another. The bridge is balanced at a specific frequency, which is the frequency of the signal being measured. Wien bridges are commonly used in audio frequency oscillators due to their ability to produce low distortion sine waves.

  6. 6.What does it mean when an AC bridge will not balance?Application

    AC balance needs Z1·Z4 = Z2·Z3 in both magnitude and phase, so two elements must be adjusted, usually alternately, until the detector reads minimum. If the phase condition θ1 + θ4 = θ2 + θ3 cannot be met by the chosen components — for example an inductor opposite another inductor when the other arms are resistive — no adjustment of magnitudes will give a null. A poor null can also come from harmonics in the source, stray capacitance or earth loops, which is why shielding and a Wagner earth are used.

  7. 7.How does the presence of parasitic elements affect the accuracy of AC bridge measurements?Application

    Parasitic elements, such as stray capacitance and inductance, can affect the accuracy of AC bridge measurements by introducing additional impedance that is not accounted for in the bridge's design. This can lead to errors in the balance condition and result in inaccurate measurements of the unknown component. Careful design and shielding are often required to minimize the impact of parasitic elements.

  8. 8.A Maxwell L–C bridge balances with C4 = 100 nF and ratio arms R2 = 1 kΩ and R3 = 500 Ω. What is the unknown inductance?Numerical

    For the Maxwell inductance–capacitance bridge, L1 = R2·R3·C4. Substituting, L1 = 1000 × 500 × 100 × 10⁻⁹ = 0.05 H = 50 mH. The coil's series resistance is found from the other balance condition, R1 = R2R3/R4, and needs the value of R4 in parallel with C4.

  9. 9.A Schering bridge balances at 50 Hz with C4 = 0.05 μF and R4 = 2 kΩ in the parallel arm. What is the dissipation factor of the specimen?Numerical

    In a Schering bridge, D = tan δ = ωC4R4. So D = 2π × 50 × 0.05 × 10⁻⁶ × 2000 = 0.0314. The loss angle is about 1.8°. The specimen's capacitance needs the other condition, C1 = C2R4/R3.

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