86

LECTURE 7 AC Bridges and Their Applications

Contents :

1 Maxwell Bridge..................................................................................................... 87 2 Hay Bridge............................................................................................................ 88 3 Schering Bridge..................................................................................................... 90 4 Unbalance Conditions........................................................................................... 92 5 Wien Bridge.......................................................................................................... 92 6 Wagner Ground Connection................................................................................. 94 7 Universal Impedance Bridge................................................................................. 95

7.1 Capacitance Comparison Bridge........................................................................... 82 7.2 Inductance Comparison Bridge............................................................................. 84

87

7.1 Maxwell Bridge [1]

The Maxwell bridge, whose schematic diagram is shown in Fig. 7-1, measures an

unknown inductance in terms of a known capacitance.

FIGURE 7-1 Maxwell bridge for inductance measurements [1]

Rearranging the general equation for bridge balance, we obtain

132 YZZZ x = .........................................................................................................(7-1)

where Y1 is the admittance of arm 1. Reference to Fig. 7-1 shows that

22 RZ = ; 33 RZ = ; and 11

11 CjR

Y ω+=

Substitution of these values in Eq. (7-1) gives

+=+= 132

1 CjR

RRLjRZ xxx ωω ......................................................................(7-2)

Separation of the real and imaginary terms yields

1

32

RRR

Rx = ..............................................................................................................(7-3)

and 132 CRRLx = ...........................................................................................................(7-4)

where the resistances are expressed in ohms, inductance in henrys, and capacitance in farads.

The Maxwell bridge is limited to the measurement of medium-Q coils (1<Q<10) High-

Q coils are measured on the Hay bridge

The usual procedure for balancing the Maxwell bridge is by first adjusting R3 for

inductive balance and then adjusting R1 for resistive balance. Returning to the R3 adjustment,

we find that the resistive balance is being disturbed and moves to a new value. This process is

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repeated and gives slow convergence to final balance. For medium-Q coils, the resistance

effect is not pronounced, and balance is reached after a few adjustments.

7.2 Hay Bridge [1]

The Hay bridge of Fig. [7-2] differs from the Maxwell bridge by having the resistor R1

in series with standard capacitor C1 instead of in parallel. It is immediately apparent that for

large phase angles, R1 should have a very low value. The Hay circuit is therefore more

convenient for measuring high-Q coils.

FIGURE 7-2 Hay bridge for inductance measurements [1]

The balance equations are derived by substituting the values of the impedances of the

bridge arms into the general equation for bridge balance. For circuits of Fig. 7-2 we find that

( ) 321

1 RRLjRCjR xx =+

− ωω

............................................................................(7-5)

which expands to

32111

1 RRRLjC

jRCL

RR xxx

x =+−+ ωω

Separating the real and imaginary terms, we obtain

321

1 RRCL

RR xx =+ ................................................................................................(7-6)

and 11

RLC

Rx

x ωω

= ........................................................................................................(7-7)

Both Eq. (7-6) and Eq. (7-7) contains Lx and Rx , and we must solve these equations

simultaneously. This yields

89

21

21

2321

21

2

1 RCRRRC

Rx ωω+

= ..............................................................................................(7-8)

21

21

2132

1 RCCRR

Lx ω+= .................................................................................................(7-9)

These expressions for the unknown inductance and resistance both contain the angular

velocity ω and it therefore appears that the frequency of the voltage source must be known

accurately. That this is not true when a high-Q coil is being measured follows from the

following considerations : Remembering that the sum of the opposite sets of sets of phase

angles must be equal, we find that the inductive phase angle must be equal to the capacitive

phase angle, since the resistive angles are zero. Figure 7-3 shows that the tangent of the

inductive phase angle equals

QRL

RX

x

xLL ===

ωθtan ...................................................................................(7-10)

and that of the capacitive phase angle is

11

1tanRCR

X CC ω

θ == .....................................................................................(7-11)

When the two phase angles are equal, their tangents are also equal and we can write

CL θθ tantan = or 11

1RC

Qω

= .......................................................................(7-12)

After substituting Eq. (7-12) in the expression for Lx, Eq. (7-9) reduces to

2132

11

+

=

Q

CRRLx ..............................................................................................(7-13)

For value of Q greater than ten, the term (1/Q)2 will be smaller than 100

1 and can be

neglected. Equation (7-9) therefore reduces the expression derived for the Maxwell bridge,

CRRLx 32= ...................................................................................................(7-14)

The Hay bridge is suited for the measurement of high-Q inductors, especially for those

inductors having a Q greater than ten.

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FIGURE 7-3 Impedance triangles illustrate inductive and capacitive phase angles [1]

7.3 Schering Bridge [1]

The Schering bridge, one of the most important ac bridges, is used extensively for the

measurement of capacitors. Although the Schering bridge is used for capacitance

measurements in a general sense, it is particularly useful for measuring insulating properties,

i.e., for phase angles very nearly 900. The basic circuit arrangement is shown in Fig. 7-4.

FIGURE 7-4 Schering bridge for the measurement of capacitance [1]

The balance equations are derived in the usual manner, and by substituting the

corresponding impedance and admittance values in the general equation, we obtain

132 YZZZ x = ...........................................................................................................(7-15a)

or

+

−=− 1

132

1 CjRC

jRCjR

xx ω

ωω....................................................................(7-15b)

and expanding

91

13

2

3

12

RCjR

CCR

CjR

xx ωω

−=− ............................................................................(7-15c)

Equating the real terms and the imaginary terms, we find that

3

12 C

CRRx = ........................................................................................................(7-16)

2

13 R

RCCx = .......................................................................................................(7-17)

As can be seen from the circuit diagram of Fig. 7-4, the two variables chosen for the balance

adjustment are capacitor C1 and resistor R2. There seems to be nothing unusual about the

balance equations or the choice of variable components, but consider for a moment how the

quality of a capacitor is defined.

The Power Factor (PF) of a series RC combination is defined as the cosine of the phase

angle of the circuit. For phase angles very close to 900, the reactance is almost equal to the

impedance and we can approximate the power factor to

xxx

x RCXR

PF ω=≈ ...........................................................................................(7-18)

The dissipation factor of a series RC circuit is defined as the cotangent of the phase angle and

therefore, by definition, the dissipation factor

xxx

x RCXR

D ω== ..............................................................................................(7-19)

The dissipation factor tells us something about the quality of a capacitor; ie., how close the

phase angle of the capacitor is to the ideal value of 900. By substituting the value of Cx in Eq.

(7-17) and of Rx in Eq. (7-16) into the expression for the dissipation factor, we obtain

11CRD ω= .........................................................................................................(7-20)

If the resistor R1 in the Schering bridge of Fig. 7-4 has a fixed value, the dial of capacitor C1

may be calibrated directly in dissipation factor D. This is the usual practice in a Schering

bridge. Notice that the term ω appears in the expression for dissipation factor [Eq. (7-20)].

This means, of course, that the calibration of the C1 dial holds for only one particular

frequency at which the dial is calibrated. A different frequency can be used, provided that a

correction is made by multiplying the C1 dial reading by the ratio of the two frequencies.

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7.4 Unbalance Conditions [1]

It sometimes happens that an ac bridge cannot be balanced at all simply because one of

the stated balance conditions cannot be met. Consider, for example, the circuit of Fig. 7-5,

where Z1 and Z4 are inductive elements (pisitive phase angles), Z2 is a pure capacitance (-900

phase angle), and Z3 is a variable resistance (zero phase angle). The resistance of R3 needed to

obtain bridge balance can be determined by applying the first balance condition (magnitudes)

and we find that

Ω=×

== 300400

600200

2

413 Z

ZZR

Hence adjusting R3 to a value of 300 Ω will satisfy the first condition.

Considering the second balance condition (phase angles) yields the following situation :

000

32

00041

90090

903060

−=+−=+

+=++=+

θθ

θθ

Obviously, 3241 θθθθ +≠+ , and the second condition is not satisfied. In this case, bridge

balance cannot be obtained.

FIGURE 7-5 An ac bridge that cannot be balanced [1]

7.5 Wien Bridge [1]

The Wien bridge is presented here not only for its use as an ac bridge to measure

frequency, but also for its application in various other useful circuits. We find, for example, a

Wien bridge in the harmonic distortion analyzer, where it is used as a notch filter,

discriminating against one specific frequency. The Wien bridge also finds application in audio

and HF oscillator as the frequency-determining element. In this chapter, the Wien bridge is

discussed in its basic form, designed to measure frequency.

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The Wien bridge has a series RC combination in one arm and a parallel RC combination

in the adjoining arm (see Figure 7-6). The impedance of arm 1 is 1

11 CjRZ ω−= . The

admittance of arm 3 is 33

31 CjRY ω+= . Using the basic equation for bridge balance and

substituting the appropriate values, we obtain

+

−= 3

34

112

1 CjR

RCjRR ω

ω........................................................................(7-21)

Expanding this expression, we get

( )1

34

31

4413

3

412 C

CRRC

jRRRCj

RRR

R +−+=ω

ω .......................................................(7-22)

Equating the real terms, we obtain

1

34

3

412 C

CRRRR

R += .............................................................................................(7-23)

which reduces to

1

3

3

1

4

2

CC

RR

RR

+= .....................................................................................................(7-24)

Equating the imaginary terms, we obtain

31

4413 RC

RRRC

ωω = .............................................................................................(7-25)

where fπϖ 2= ,

and solving for f, we get

31312

1RRCC

fπ

= ...............................................................................................(7-26)

Notice that the two conditions for bridge balance now result in an expression determining the

required resistance ratio, R2/R4, and another expression determining the frequency of the

applied voltage. In other words, if we satisfy Eq. (7-24) and also excite the bridge with a

frequency described by Eq. (7-26), the bridge will be in balance.

In most Wien bridge circuit, the components are chosen such that R1=R3 and C1=C3.

This reduces Eq. (7-24) to R2/R4=2 and Eq. (7-26) to

RC

fπ21

= .............................................................................................................(7-27)

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which is the general expression for the frequency of the Wien bridge. In a practical bridge,

capacitors C1 and C3 are fixed capacitors, and resistors R1 and R3 are variable resistors

controlled by a common shaft.

Because of its sensitivity, the Wien bridge may be difficult to balance (unless the

waveform of the applied voltage is purely sinusoidal). Since the bridge is not balanced for any

harmonics present in the applied voltage, these harmonics will sometimes produce an output

voltage masking the true balance point.

FIGURE 7-6 Frequency measurement with the Wien bridge [1]

7.6 Wagner Ground Connection [1]

The discussion so far has assumed that the four bridge arms consist of simple lumped

impedances which do not interact in any way. In practice, stray capacitances exist between the

various bridge elements and ground, and also between the bridge arms themselves. These

stray capacitances shunt the bridge arms and cause the measurement errors, particularly at the

hogher frequencies or when small capacitors or large inductors are measured. One way to

control stray capacitances is by shielding the arms and connecting the shields to ground. This

does not eliminate the capacitances but at least makes them constant in value, and they can

therefore be compensated.

One of the most widely used methods for eliminating some of the effects of stray

capacitance in a bridge circuit is the Wagner ground connection. This circuit eliminates the

troublesome capacitance which exists between the detector terminals and ground. Figure 7-7

shows the circuit of capacitance bridge, where C1 and C2 represent these stray capacitances.

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FIGURE 7-7 The Wagner groundconnection eliminates the effect of stray capacitances ecross the detector [1]

7.7 Universal Impedance Bridge [1]

One of the most useful and versatile laboratory bridges is the universal impedance

bridge. Several of the bridge configurations discussed so far are combined in a single

instrument capable of measuring both dc and ac resistance, the inductance and storage factor

Q of an inductor, and the capacitance and dissipation factor Q of a capacitor. A representative

example of a universal impedance bridge is given in Fig. 7-8 which clearly shows the

arrangement of the various front panel controls. Figure 7-9 shows the various bridge

configuration used in this impedance bridge.

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FIGURE 7-8 Universal impedance bridge (courtesy John Fluke Manufacturing Company) [1]

FIGURE 7-9 Bridge configurations of universal impedance bridge of Fig. 7-8. [1]