Chapter 27: AC Bridge

27.1 What is an AC Bridge?

AC bridges are used for measuring the values of inductors and capacitors, or for converting the signals measured from inductive or capacitive components into a suitable form such as a voltage.[cite: 466, 472, 473, 478] Inductors and capacitors can also be measured using voltage division, though this is an approximate method.[cite: 479, 480, 481] AC bridges work the same way as a Wheatstone bridge, which therefore has an equation capable of measuring complex quantities.[cite: 482, 485, 486, 487, 496]

An oscilloscope is often used for determination of an unknown impedance by comparison with known impedances and to determinations of frequency.[cite: 488, 500, 501, 502] Very small electromechanical movements or headphones can be used if the source is in the audio range.[cite: 497, 498, 499] It is more difficult to obtain balance in an AC bridge than a DC bridge because both the magnitude and phase angle of impedances are related to the balance condition.[cite: 503, 504, 505, 506, 507, 509, 510] AC bridges provide precise ways of measuring inductance, capacitance, and resistance.[cite: 508, 511, 512]

27.2 Balance Conditions in an AC Bridge

Most well known AC bridges are of a standard four-arm form.[cite: 515, 516, 519, 520, 521] At balance, the current flowing through Z₁ is the same as the current flowing through Z₂.[cite: 531, 532, 533, 534] Also, the current that flows through Z₄ must be the same as the current that flows through Z₃.[cite: 536, 537]

The voltage drop between nodes A and B at balance is the same as the voltage drop across A and D.[cite: 538, 539, 540] The voltage drop across B and C is also equal to the voltage drop across D and C.[cite: 541, 543, 544] From the above we can derive these equations:

Dividing the first by the second gives: Z₁ / Z₂ = Z₄ / Z₃.[cite: 553, 581, 582] From which:

If expressed in polar form, Z = |Z|∠α, then: [cite: 584, 585, 586]

There are two conditions to be simultaneously satisfied for balance in an AC bridge: [cite: 589, 590, 591, 592]

Usually one arm of an AC bridge contains the unknown impedance while the other arms contain known fixed or variable components.[cite: 595, 596, 599, 600, 605, 606] When the current in the detector is reduced by adjustments of the variable components at balance, the unknown impedance can be expressed in terms of the fixed and variable components.[cite: 607, 608]

Procedure for Determining Balance Equations

1. Determine for the bridge circuit the impedance in each arm in complex form and write down the balance equation. [cite: 557, 558, 559] Equations are usually easier to manipulate if initially expressed as R and X_C rather than R and 1/jωC.[cite: 560]
2. Isolate the unknown terms on the left hand side of the equation in the form a + jb.[cite: 561, 562, 563]
3. Manipulate the terms on the right hand side of the equation into the form c + jd.[cite: 564, 568, 569]
4. Equate the real parts of the equation, and equate the imaginary parts.[cite: 570] Substitute ωL for X_L and 1/ωC for X_C where appropriate to express the final equations in their simplest form.[cite: 571]

Types of Detector

The types of zero current detector used with an AC bridge depend on the type of bridge and frequency at which it operates.[cite: 572, 573]

• Oscilloscope: The most versatile and is used for a varying range of frequencies.[cite: 574, 575]
• Earphones: Used up to 10kHz but often 1kHz; this region is where the ear is most sensitive.[cite: 576, 578, 579]
• Electronic Detectors: Various electronic detectors which use tuned circuits to detect current at the correct frequency.[cite: 256, 257, 611]
• Vibration Galvanometers: Used for mains operated frequencies from 10Hz to 300Hz.[cite: 258, 612, 614, 615] When a current of the correct frequency is passed through it, a mirror connected to the vibrator reflects a beam of light onto a scale. It appears as a band when vibrating, and as a spot when not vibrating, at which point zero current flows and the bridge is balanced.[cite: 263, 264, 265, 267, 268, 269, 270, 616, 617, 618, 619]

27.3 Types of AC Bridge Circuit

The different types of AC bridge measure different aspects of impedance.[cite: 222, 226] They are Maxwell, Hay, Owen and Maxwell-Wien.[cite: 227, 228] The Maxwell-Wien is for measuring inductance, the De Sauty and Schering bridges are for measuring capacitance and the Wien bridge for measuring frequency.[cite: 229] There are a large number of combinations of components in different bridges.[cite: 230, 231]

In standard bridges it is found that two of the balancing impedances will be of the same nature, and often consist of both pure components (either both capacitors or both resistors) for a bridge to balance quickly.[cite: 232, 233, 234, 235, 236]

• Ratio Arm Bridge: A pair of adjacent arms are both pure components.[cite: 237, 240, 241] It can only be used to measure reactive quantities of the same type.[cite: 244, 246, 247]
• Product-Arm Bridge: A pair of opposite arms are pure components.[cite: 239, 242, 243] The reactive component of the balancing impedance must be of opposite sign to the unknown reactive component.[cite: 252, 253]

A commercial universal bridge can be used to measure resistance, inductance or capacitance.[cite: 254, 255]

(a) The Simple Maxwell Bridge

This bridge measures the resistance and inductance of a coil having a high Q-factor.[cite: 272, 273, 274] At balance, expressions for R_x and L_x may be derived in terms of known components R₂, R₃, R₄ and L₄.[cite: 282, 283]

Isolating the unknown impedance on the left side and manipulating the RHS into terms of (c + jd) gives: [cite: 289, 290, 291]

(b) The Owen Bridge

This is used to measure the resistance and inductance of a coil having a large value of inductance.[cite: 304, 307, 308, 309] From this circuit: Z_x = R_x + jX_Lx, Z₂ = R₂ – jX_C2, Z₃ = -jX_C3, Z₄ = R₄.[cite: 318] At balance: Z_xZ₃ = Z₂Z₄.[cite: 319]

Equating real and imaginary parts: [cite: 322, 323]

(c) The Maxwell-Wien Bridge

The bridge is used to measure the resistance and inductance of a coil having a low Q-factor.[cite: 327, 328, 329, 331, 332] Here, Z_x = R_x + jX_Lx, Z₂ = R₂, and Z₄ = R₄.[cite: 337]

Arm 3 consists of two branches in parallel (Product Arm). Z₃ is given by: [cite: 338, 340, 354]

For the Maxwell-Wien Bridge, at Balance: Z_xZ₃ = Z₂Z₄.[cite: 357, 358, 359, 360, 361] By rationalizing and separating the real and imaginary parts we get: [cite: 364, 365, 366]

(d) The De Sauty Bridge

This bridge provides a very simple method of measuring capacitance by comparison with a known capacitance.[cite: 370, 376, 377, 378] At balance: Z_xZ₃ = Z₂Z₄.[cite: 380]

This simple bridge is usually inadequate in most practical cases because the power factor of the capacitor under test is significant due to internal dielectric losses.[cite: 384, 385]

(e) The Schering Bridge

This bridge is used to measure the capacitance and equivalent series resistance of a capacitor.[cite: 386, 387, 388] From the measured values the power factor of insulating materials may be determined.[cite: 388, 391] C_x is the unknown capacitance and R_x is the equivalent series resistance.[cite: 390, 392]

At balance: Z_xZ₃ = Z₂Z₄.[cite: 399] Equating real and imaginary parts gives: [cite: 403]

The loss in a dielectric may be represented by a resistor in parallel with a capacitor or a lossless capacitor in series with a resistor.[cite: 407, 410, 411, 412] The power factor is given by cosφ, and δ = tan^-1(ωC₃R₃).[cite: 414, 416, 419, 422, 423]

(f) The Wien Bridge

The circuit can be used for three purposes: To measure frequency in terms of known components, to measure capacitance if the frequency is known, or as a frequency stabilising network.[cite: 429, 430, 431, 432] In this circuit: Z₁ = R₁, Z₂ = 1 / ((1 / R₂) + jωC₂), Z₃ = R₃ – jX_C3, and Z₄ = R₄.[cite: 443, 444, 445]

At balance Z₁Z₃ = Z₂Z₄.[cite: 446] Equating real parts gives: [cite: 451]

Equating imaginary parts gives: [cite: 453]

If C₂ = C₃ = C and R₂ = R₃ = R, frequency = 1 / 2πCR.[cite: 463, 464]