Measurement of Self Inductance - GATE Notes & Videos for Electrical Engineering -

Measurement of Self Inductance by Maxwell's Inductance Bridge

Maxwell's inductance bridge measures an unknown inductance by comparison with a variable standard self-inductance. The bridge is arranged so that at balance the phasor sum of voltages around each arm satisfies the usual bridge balance conditions. The circuit connections and phasor diagram for the balance condition are shown in the figure below.

Let the circuit variables be:

• L₁ = unknown inductance (with series resistance R₁)
• L₂ = variable standard inductance (with fixed resistance r₂)
• R₂ = adjustable resistance placed in series with L₂
• R₃, R₄ = known non-inductive resistances used as ratio arms

Resistors R₃ and R₄ are normally selected from standard decade values such as 10, 100, 1 000 and 10 000 Ω. The variable resistance r₂ is usually a decade resistance box. In practice, an additional known resistance may be placed in series with the unknown coil to obtain precise balance. The detailed derivation of the bridge balance equations follows from equating complex impedances in appropriate arms (see phasor diagram) and has been treated in standard AC bridge theory.

Measurement of Self Inductance by Maxwell's Inductance-Capacitance Bridge

Maxwell's inductance-capacitance bridge measures an unknown inductance by comparison with a standard variable capacitance. The circuit connections and the phasor diagram at balance are shown in the figure below.

Define the elements as:

• L₁ = unknown inductance to be measured
• R₁ = effective series resistance of the inductor L₁
• R₂, R₃, R₄ = known non-inductive resistances
• C₄ = variable standard capacitance

Writing the complex balance equation for the bridge yields the algebraic relations shown in the figures below.

Separating real and imaginary parts produces two independent balance equations as shown.

Because the two balance conditions involve R₄ and C₄ in a way that makes the equations independent, the bridge offers straightforward determination of both the unknown inductance and its effective resistance. The quality factor (Q) at the measurement frequency is obtained from the relation

Q = ωL₁ / R₁ = ωC₄ R₄

Advantages of Maxwell's Inductance-Capacitance Bridge

1. The two balance equations are independent when R₄ and C₄ are chosen as the variable elements, allowing simultaneous determination of L₁ and R₁.
2. The measurement expressions do not explicitly contain the angular frequency ω, so the bridge result is frequency-independent provided the capacitor and resistors behave ideally.
3. The bridge gives simple expressions for the unknowns L₁ and R₁ in terms of known bridge elements. For example, if the product R₂R₃ is chosen as 10⁶, then L₁ = C₄ × 10⁶. Thus the numeric value of C₄ (in μF) directly gives L₁ in henry for that choice of R₂R₃.
4. The bridge is useful for measuring a wide range of inductances at power and audio frequencies.

Disadvantages of Maxwell's Inductance-Capacitance Bridge

1. The bridge ideally requires a variable standard capacitor calibrated to high accuracy, which can be expensive. A fixed high-accuracy capacitor is sometimes used instead, but this requires alternative balancing strategies:

• varying resistances R₂ and R₄ (difficult because R₂ appears in both balance equations), or
• adding a small known series resistance with the unknown coil and varying this resistance along with R₄.

2. The Maxwell bridge is best suited for coils with moderate quality factor (1 < Q < 10). For high-Q coils (Q > 10) R₄ must be very large (≈10⁵-10⁶ Ω), and such high-value resistance boxes are expensive; therefore Maxwell's bridge is unsuitable for very high Q measurements.
3. The bridge is also ill-suited for very low Q coils (Q < 1). When Q is small, the resistive component dominates and achieving simultaneous inductive and resistive balance becomes slow because changing one element disturbs the other (a sliding balance). Hence many successive adjustments are needed to converge to balance for low-Q coils.
4. Overall, Maxwell's bridge is most appropriate for measurement of medium-Q coils and when a suitable variable capacitor or accurate fixed standards are available.

Measurement of Self Inductance by Anderson's Bridge

Anderson's bridge is a modification of Maxwell's inductance-capacitance bridge that measures self-inductance in terms of a standard capacitor. It is designed to give easier convergence and greater accuracy over a wide range of inductances.

The circuit connections and phasor diagram for the balanced condition are shown below.

Let the bridge elements be:

• L₁ = self-inductance to be measured
• R₁ = resistance of the self-inductor
• r = resistance connected in series with the self-inductor (adjustable)
• R₂, R₃, R₄ = known non-inductive resistances
• C = fixed standard capacitor

The balance equations that follow from the circuit are shown in the figures below.

Because the adjustable elements appear in only one of the two balance equations, convergence to balance is easier. In practice the controls r₁ and r are alternately adjusted to obtain rapid and independent convergence of the two balance conditions.

Advantages of Anderson's Bridge

1. When adjustments are carried out using r₁ and r, the two balance controls become effectively independent, avoiding the sliding balance difficulty encountered with Maxwell's bridge for low-Q coils. Hence Anderson's bridge converges more readily for low-Q measurements.
2. A fixed standard capacitor can be used instead of an expensive variable capacitor.
3. Anderson's bridge can also be used, with appropriate rearrangement, for accurate determination of capacitance in terms of inductance.

Disadvantages of Anderson's Bridge

1. The circuit is more complex than Maxwell's bridge; it has more components and the algebraic balance equations are more involved and tedious to manipulate.
2. An extra junction point in the bridge increases difficulty of shielding and may introduce stray inductance or capacitance that need to be considered for high-accuracy work.

Measurement of Self Inductance by Owen's Bridge

Owen's bridge measures inductance in terms of capacitance and is frequently used where both accuracy and ease of balance are required. The circuit and phasor diagram at balance are shown below.

Let the elements be defined as:

• L₁ = unknown self-inductance (with series resistance)
• R₂ = variable non-inductive resistance
• R₃ = fixed non-inductive resistance
• C₂ = variable standard capacitor
• C₄ = fixed standard capacitor

The complex balance condition for the bridge and the separation into real and imaginary parts are shown here.

Separating real and imaginary parts gives the two independent relations:

Advantages of Owen's Bridge

1. When C₂ and R₂ are made variable, the bridge yields two independent balance equations and both adjustments lie in the same arm, making convergence easier.
2. The balance equations are simple and do not explicitly contain frequency, enabling frequency-independent determination of L₁ and R₁.
3. Owen's bridge can be applied over a wide range of inductance measurements.

Disadvantages of Owen's Bridge

1. It requires a variable standard capacitor, which may be expensive and typically has accuracy of order 1% unless a high-quality device is used.
2. For high-Q coils the required capacitance value C₂ can become large, making practical implementation more difficult.

Heaviside Mutual Inductance Bridge

Heaviside bridge is intended principally to measure mutual inductance in terms of a known self-inductance. With suitable modification it can also be used to measure self-inductance in terms of mutual inductance (Campbell's modification).

For measurement of mutual inductance, denote:

• M = unknown mutual inductance
• L₁ = self-inductance of the secondary coil of the mutual inductor
• L₂ = known self-inductance (reference)
• R₁, R₂, R₃, R₄ = non-inductive resistors forming the remaining bridge arms

At balance the voltage drops between corresponding bridge nodes must be equal; writing these conditions yields the balance equations illustrated below.

The form of these equations shows that L₁ (self-inductance of the secondary) must be known if the mutual inductance M is to be determined directly by this method. In the special case when R₃ = R₄ and R₁ = R₂, the equations reduce to simple relations:

M = (L₂ - L₁) / 2 when R₃ = R₄ and R₁ = R₂.

The same Heaviside arrangement can be used to measure a self-inductance. The balance expression for that measurement is given in the figure below.

Campbell's Modification of Heaviside Bridge

• The modified Heaviside bridge due to Campbell is used to measure a self-inductance in terms of a mutual inductance standard. In this modification an additional balancing coil R is included in arm a-d in series with the inductor under test. An additional adjustable resistance r is put in arm a-b. Balance is obtained by varying M and r.
• A short-circuiting switch is placed across the coil R₂, L₂ under measurement. Two sets of readings are taken: one with the switch open and the other with the switch closed. Denote the corresponding readings of mutual inductance and resistance as M₁, r₁ (switch open) and M₂, r₂ (switch closed).

With the switch open the balance equation is shown here:

From the two readings one can eliminate the effects of lead resistances and obtain the self-inductance L₂ and the correction resistance R₂ in terms of the measured quantities:

L₂ = (M₁ - M₂) (1 + R₄ / R₃)

With the switch open the resistances satisfy R₂ + R = (R₁ + r₁) R₄ / R₃

With the switch closed the relation is R = (R₁ + r₂) R₄ / R₃

Hence

R₂ = (r₁ - r₂) R₄ / R₃

When R₃ = R₄ the expressions simplify to

L₂ = (M₁ - M₂)

R₂ = (r₁ - r₂)

This two-reading technique is a practical method to eliminate the influence of lead resistances and stray effects.

Heaviside-Campbell Equal-Ratio Bridge

To improve sensitivity and remove the need for a balancing coil, the equal-ratio version of the Heaviside-Campbell bridge uses the secondary of the mutual inductor made from two identical coils. One coil is placed in arm a-b and the other in arm a-d. The primary couples equally to both secondaries. The coil whose self-inductance and resistance are to be determined is placed in the appropriate arm (L₂, R₂). Resistors R₃ and R₄ are chosen equal and balance is achieved by varying the mutual inductance setting and an adjustable resistance r.

At balance the currents in the two equal secondary coils are equal: I₁ = I₂ = I/2, since I = I₁ + I₂ and R₃ = R₄ imply I₁R₃ = I₂R₄.

Writing the other balance equation and equating real and imaginary components yields the relations:

Equating the real parts gives R₂ = R₁ + r.

Equating the imaginary parts gives L₂ = 2(Mₓ + Mᵧ) = 2M, so the inductance measured is twice the mutual inductance range available.

To eliminate effects of leads and residuals, two readings are again taken with the switch open and closed. Denote the two readings as M₁, r₁ (open) and M₂, r₂ (closed). Then

R₁ = (r₁ - r₂)

L₁ = 2 (M₁ - M₂)

Thus the equal-ratio arrangement improves sensitivity and reduces systematic errors while allowing the effects of leads to be cancelled by paired measurements.

Practical Notes, Selection and Applications

• Choice of bridge depends on the expected quality factor (Q) and the required accuracy. Use Maxwell/Anderson/Owen depending on whether capacitor availability, required ease of balance, or Q range dominates the requirement.
• For low-Q coils Anderson's bridge often gives the best practical performance because of independent controls and use of fixed standard capacitors.
• For medium-Q and audio/power frequency work Maxwell's inductance-capacitance bridge or Owen's bridge are commonly used.
• Mutual inductance bridges such as Heaviside or Campbell's modification are used when standards are mutual inductances or when converting between self and mutual units; the equal-ratio form improves sensitivity.
• In all bridge measurements attention must be paid to stray capacitances and inductances, lead resistances, shielding, and instrument sensitivity (detector and null indicator). Two-reading methods (open/closed or shorted/unshorted) are standard procedures to cancel lead effects.

Summary

This chapter described the principal bridge methods for measurement of self-inductance: Maxwell's inductance bridge, Maxwell's inductance-capacitance bridge, Anderson's bridge, Owen's bridge, and the Heaviside family (including Campbell's modifications and the equal-ratio form). For each bridge the defining circuit elements, principal balance relations (illustrated by figures), advantages and limitations were summarised, together with practical guidance on which bridge to select by coil Q, available standards and required accuracy. The image placeholders above contain the detailed circuit diagrams and algebraic balance expressions necessary to apply each method in laboratory practice.