8b Ac Bridges Rev 1 070517

8b AC Bridges

Measurements & Instrumentation Chapter 8b AC Bridges and the measurement of L&C (Revision 1.0, 17/5/2007)

8b.1 Introduction AC bridges are used for measuring the values of inductors and capacitors or for converting the signals measured from inductive or capacitive into a suitable form such as a voltage. Inductors and capacitors can also be measured using an approximate method of voltage division. These methods are discussed in this Chapter. 8b.2 General condition for balance in AC bridges In an AC bridge in general, at balance conditions, the following is true:

Z X ⋅ Z4 = Z2 ⋅ Z3 Where the value of Zx is unknown and the values of the other three impedances are known. The arrangement of this null AC bridge is shown Figure 1.

ZX Vi

Z2

G Null detector

Z3

Z4

Figure 1: General diagram of an AC Bridge.

The null condition equation to be true, it has to satisfy both the magnitude criterion and the phase angle criterion, as follows [2]:

Z X ⋅ Z4 = Z2 ⋅ Z3

∠θ X + ∠θ 4 = ∠θ 2 + ∠θ 3 Note that the capacitive reactance has a negative phase angle, and the inductive reactance has a position phase angle. Resistors have a zero phase angle. © Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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Using the criterion above can help decide whether a bridge can achieve balance conditions or not just by examining the components in the bridge. General Rule for a.c. bridges As a general rule in AC bridges in order to achieve balance conditions, similar reactive components should be placed on adjacent limbs of the bridge, and different reactive components should be placed on opposite limbs of the bridge. For example, if only capacitors are to be used in an a.c. bridge, then they should be placed on adjacent limbs (e.g., the Wien Bridge). If a capacitor and an inductor are to be used in a bridge, then they should be placed on opposite limbs of the bridge (e.g., the Maxwell Bridge). aide-mémoire Similar[C & C; L&L]↔Adjacent Different [C & L]↔Opposite Acronym: SADO 8b.3 Quality Factor for Inductors and Capacitors The Q factor (Quality factor) for an inductor or capacitor is the ratio of the value of its reactance to its resistance. For an inductor:

QL =

XL ω ⋅L = R R

For a capacitor:

QL =

XC 1 = R ω ⋅C ⋅ R

8b.4 Maxwell Bridge The Maxwell Bridge is used to measure inductors with low to medium values of Q. A typical arrangement is shown in Figure 2 below [1].

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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RX R2 LX Vi

V Vo R3

C4

R4

Figure 2: Maxwell Bridge.

At Balance,

Z X = Y4 ⋅ Z 2 ⋅ Z 3 This gives:

  1 RX + j ⋅ ω ⋅ LX = R2 ⋅ R3 ⋅  + j ⋅ ω ⋅ C4    R4 R ⋅R RX + j ⋅ ω ⋅ LX = 2 3 + j ⋅ ω ⋅ R2 ⋅ R3 ⋅ C4 R4 Equating real parts of both sides gives:

RX =

R2 ⋅ R3 R4

And equating imaginary parts of both sides gives:

LX = R2 ⋅ R3 ⋅ C4 8b.5 Hay Bridge The Hay Bridge (shown in Figure 3 below) is used to measure the value of inductors that have a high Q factor [2]. A Q factor is considered high for value of 10 or more. At balance conditions, we have the following: © Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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RX R2 LX Vi

V Vo R4

R3

C4

Figure 3: Hay Bridge.

  1  = R2 ⋅ R3 + j ⋅ ω ⋅ LX ) ⋅  R4 + ω j C ⋅ ⋅  4  RX L RX ⋅ R4 + j ⋅ ω ⋅ LX ⋅ R4 + + X = R2 ⋅ R3 j ⋅ ω ⋅ C4 C4

(R

X

Equating the real parts on both sides gives:

RX ⋅ R4 + RX =

LX = R2 ⋅ R3 C4

R2 ⋅ R3 ⋅ C4 R ⋅ R ⋅ C − LX LX − = 2 3 4 R4 ⋅ C4 R4 ⋅ C4 R4 ⋅ C4

Equating the imaginary parts on both sides gives:

ω ⋅ LX ⋅ R4 = LX =

RX ω ⋅ C4

RX ω 2 ⋅ R4 ⋅ C4

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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The problem in this case is that we have ended up with two equations with two unknowns (as opposed to the previous cases which gave the value of Rx and Lx directly from the real and imaginary part equations). So we have to solve the two simultaneous equations, giving:

LX =

R2 ⋅ R3 ⋅ C4 − LX RX = ω 2 ⋅ R4 ⋅ C4 ω 2 ⋅ R42 ⋅ C42

ω 2 ⋅ R42 ⋅ C42 ⋅L X + LX = R2 ⋅ R3 ⋅ C4 LX =

R2 ⋅ R3 ⋅ C4 (ω 2 ⋅ R42 ⋅ C42 + 1)

Using this to find the formula for Rx gives:

ω 2 ⋅ R2 ⋅ R3 ⋅ R4 ⋅ C42 RX = (ω 2 ⋅ R42 ⋅ C42 + 1) If we remember that the inductor has a high Q value, then if follows from the phase angle balance equation that capacitor will have a high Q value as well (C4, R4). This is because at balance conditions the phase angle for Z4 should equal the angle for Zx (as Z2 and Z3 are resistors). As the capacitor has a high Q, then it follows that:

QC =

1 >> 1 ω ⋅ R4 ⋅ C4

⇒ ω 2 ⋅ R42 ⋅ C42 =

1 ≈0 Q2 C

Using this approximation we can now find the values of LX and Rx for high Q as follows:

RX = ω 2 ⋅ R2 ⋅ R3 ⋅ R4 ⋅ C42 LX = R2 ⋅ R3 ⋅ C4 8b.6 Schering Bridge The Schering Bridge (Figure 4) is used to measure the value of capacitors (especially their insulating properties) [2]. The values of Cx and Rx are unknown.

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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Rx

R2

Cx Vi

V Vo

C3 R4 C4

Figure 4: Schering Bridge.

At balance conditions (it is easier in this case to use the admittance Y4 and multiply it by the other side, as it is a parallel combination of a capacitor and resistor, as follows):

Z X = Y4 ⋅ Z 2 ⋅ Z 3 Which gives:

 1  RX + j ⋅ω ⋅ CX 

    1  1   =  + j ⋅ ω ⋅ C4  ⋅ R2 ⋅  R j C ⋅ ⋅ ω   4   3 

 1  RX + j ⋅ω ⋅ CX 

  R ⋅C    R2  +  2 4   = ⋅   j ⋅ ω ⋅ C3 ⋅ R4   C3 

Equating real and imaginary parts in both sides of the equation, gives:

RX =

R2 ⋅ C4 C3

CX =

C3 ⋅ R4 R2

Note that the each of the resulting equations solves for one of the unknowns without the need to solve two simultaneous equations. Also note that the result in this case does not depend on the value of the frequency. © Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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8b.7 Wien Bridge One of the four methods for measuring the frequency of a signal is the Wien Bridge. The Wien Bridge is named after Max Wien1. The source of unknown frequency is used to excite the a.c. bridge as shown in Figure 5. The variable resistors R3 and R4 are varied until balance conditions are achieved (as indicated by the lack of signal in the null detector). If the frequency is known to be in the audio range, the null detector used could be a pair of headphones.

C3 R1

R3

Unknown frequency source Null Detector

R4

R2

C4

Figure 5: The Wien Bridge.

At balance conditions:

R1 Z 3 = = Z 3 × Y4 R2 Z 4 Developing this, gives:  R C R1  1   1 1  ×  + jωC4  = 3 + 4 + jωR3C4 + =  R3 + R2  jωC3   R4 jωR4C3  R4 C3

Equating real parts from both sides gives:

1

Max Wien (1866 – 1938) a German physicist and the director of the Institute of Physics at the University of Jena. In 1891. Wien invented the Wien Bridge oscillator but did not have a means of developing electronic gain so a workable oscillator could not be achieved.

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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R1 R3 C4 = + R2 R4 C3 Equating imaginary parts from both sides gives:

jωR3C4 +

1 =0 jωR4C3

jωR3C4 = −

ω=

1 jωR4C3

1 1 ⇒ f = 2π R3 R4C3C4 R3 R4C3C4

In practice, R3 is set as equal to R4, and C3 is set as equal to C4. Thus the unknown frequency is found as:

f =

1 2πR3C3

This also results in the following: R1 = 2 R2 In order to make balancing the bridge easier, the variable resistors R3 and R4 are by a common shaft (i.e., ganged) such that they are always equal as the arm is rotated to achieve balance conditions. This is shown in Figure 6. The dashed line on electrical diagrams indicates a mechanical connection between electrical components.

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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C3 R1

R3

Unknown frequency source Null Detector

R2

R4

C4

Figure 6: The mechanical link between the two resistors (ganged).

The Problem of Harmonics in the Wien Bridge Due to the sensitivity of this bridge, it might be difficult to balance it unless the source waveform is a pure sinusoid. A distorted sinusoid will contain harmonics and these will not be balanced by the bridge at the true balance point. 8b.8 Approximate Methods for Measuring L and C Simpler methods of measuring L and C are also available, although they do not yield the same accuracy as the AC bridge methods. It is possible to measure the value of Lx as shown in Figure 7 below [1]. Rx represents the resistance of the inductor. The value of Rx is first measured by one of the resistance measurement methods discussed earlier. Then using the circuit shown below, the value of R1 is changed until the voltage across it is equal to the voltage across Zx (Rx and Lx).

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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8b AC Bridges

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R1

VR

Vi RX VX LX

Figure 7: Approximate Method of measuring Inductance.

Once the two voltages are equal, the following equation can be used to find the value of Lx:

Z X = R1 RX2 + ω 2 ⋅ L2X = R1 LX =

R12 − RX2

ω

R12 − RX2 = 2 ⋅π ⋅ f

So the finding the value of Lx depends on the values of R1, Rx as well as the frequency, f. Achieving the balance depends on reading two voltages. For these reasons this method is less accurate than the null type bridge methods, as all the tolerances/errors in these quantities will accumulate in the final reading. A similar method can be used to measure the value of an unknown capacitor, as shown in Figure 8 [1]. The voltages across the known resistor and the capacitor are measured. Their values are then used to find the unknown capacitor as follows:

© Copyright held by the author 2007: Dr. Lutfi R. Al-Sharif

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8b AC Bridges

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R1

VR

Vi

CX VC

Figure 8: Approximate Method of measuring capacitance.

VR VC = R1 X C VC VR = 1 R1 ω ⋅ CX CX =

VR 2 ⋅ π ⋅ f ⋅ VC ⋅ R1 ⋅

Note that the capacitor value depends on the frequency, the value of the two voltages and the value of the resistor. This leads to low accuracy with this method, due to the accumulation of the error in these quantities. Another approximate method of measuring the capacitor is measure the time constant of the capacitor with a known resistor. By knowing the time constant and the value of the resistor, the value of the capacitor can be calculated. References & Bibliography [1] “Measurement & Instrumentation Principles”, Alan S. Morris, Elsevier, 2001. [2] “Modern Electronic Instrumentation and Measurement Techniques”, Albert D. Helfrick and William D. Cooper, Prentice Hall International Editions, 1990.

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