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16.6 Bridge Circuits

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A full bridge is a circuit having four arms which are connected in a lattice form. Four nodes are formed in

this manner. Two opposite nodes are used for the excitation (voltage or current supply) of the bridge and

the remaining two opposite nodes provide the bridge output.

A bridge circuit is used to make some form of measurement. Typical measurements include change in

resistance, change in inductance, change in capacitance, oscillating frequency, or some variable

(stimulus) that causes these. There are two basic methods of making the measurement:

1. Bridge balance method

2. Imbalance output method

A bridge is said to be balanced when the output voltage is zero. In the bridge-balance method, we start

with a balanced bridge. Then, when one is preparing to make a measurement, the balance of the bridge

will be upset due to the associated variation, resulting in a nonzero output voltage. The bridge can be

balanced again by varying one of the arms of the bridge (assuming, of course, that some means is

provided for the fine adjustments that may be required). The change that is required to restore the

balance provides the measurement. In this method, the bridge can be balanced precisely using a

servo device.

In the imbalance output method, we usually start with a balanced bridge. However, the bridge is not

balanced again after undergoing the change due to the variable that is being measured. Instead, the

output voltage of the bridge due to the resulting imbalance is measured and used as an indication of the

measurement.

There are many types of bridge circuits. If the supply to the bridge is DC, then we have a DC bridge.

Similarly, an AC bridge has an AC excitation. A resistance bridge has only resistance elements in its

four arms. An impedance bridge has impedance elements consisting of resistors, capacitors, and

inductors in one or more of its arms. If the bridge excitation is a constant-voltage supply, we

have a constant-voltage bridge. If the bridge supply is a constant current source, we have a constantcurrent

bridge.

16.6.1 Wheatstone Bridge

The Wheatstone bridge is a resistance bridge with a constant DC voltage supply (i.e., a constant-voltage

resistance bridge). A Wheatstone bridge is used in strain-gage measurements, and also in force, torque,

and tactile sensors that employ strain-gage techniques. Since a Wheatstone bridge is used primarily in the

measurement of small changes in resistance, it could be used in other types of sensing applications as well

(for example, in resistance temperature detectors or RTDs).

Consider the Wheatstone bridge circuit shown in Figure 16.17(a). The bridge output, vo; may be

expressed as

vo ¼ ðR1R4 2 R2R3Þ

ðR1 þ R2ÞðR3 þ R4Þ

vref ð16:68Þ

Signal Conditioning and Modification 16-43

Note that the bridge-balance requirement is

R2 ¼

R4 ð16:69Þ

Suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the

beginning. The bridge is balanced according to

Equation 16.69 and then R1 is increased by dR: For

example, R1 may represent the only active strain

gage and the remaining three elements in the

bridge are identical, dummy elements. Then, in

view of Equation 16.68, the change in output due

to the change dR is given by

dvo ¼ ½ðR þ dRÞR 2 R2􀀉

ðR þ dR þ RÞðR þ RÞ

vref 2 0

dvo

vref ¼

dR=R

ð4 þ 2dR=RÞ ð16:70Þ

Note that the output is nonlinear in dR=R:

If, however, dR=R is assumed to be small

in comparison to 2, we have the linearized

relationship.

dvo

vref ¼

4R ð16:71Þ

The error due to linearization, which is a

measure of nonlinearity, may be given as the

percentage

NP ¼ 100 1 2

Linearized output

Actual output

􀀏 􀀐

ð16:72Þ

Hence, from Equation 16.70 and Equation 16.71,

we have

NP ¼ 50

% ð16:73Þ

16.6.2 Constant-Current Bridge

When large resistance variations dR are required for a measurement, the Wheatstone bridge may not be

satisfactory due to its nonlinearity, as indicated by Equation 16.70. The constant-current bridge has less

nonlinearity and is preferred in such applications. However, it requires a current-regulated power supply,

which is typically more costly than a voltage-regulated power supply.

As shown in Figure 16.17(b), the constant-current bridge uses a constant-current excitation, iref ;

instead of a constant-voltage supply. Note that the output equation for the constant-current bridge can

be determined from Equation 16.68 simply by knowing the voltage at the current source. Suppose that

this is the voltage, vref ; with the polarity as shown in Figure 16.17(a). Now, since the load current is

assumed small (high-impedance load), the current through R2 is equal to the current through R1 and is

(a)

(b)

vref

(Constant)

−

R1 +

R3 R4

− +

Load

Small i

iref

(Constant)

−

R1 + R2

R3 R4

vo Load

Small i

FIGURE 16.17 (a) Wheatstone bridge (the constantvoltage

resistance bridge); (b) the constant-current

bridge.

16-44 Vibration and Shock Handbook

given by

vref

ðR1 þ R2Þ

Similarly, current through R4 and R3 is given by

vref

ðR3 þ R4Þ

Accordingly,

iref ¼

vref

ðR1 þ R2Þ þ

vref

ðR3 þ R4Þ

vref ¼ ðR1 þ R2ÞðR3 þ R4Þ

ðR1 þ R2 þ R3 þ R4Þ

iref ð16:74Þ

Substituting Equation 16.74 in Equation 16.68, we have the output equation for the constant-current

bridge; thus,

vo ¼ ðR1R4 2 R2R3Þ

ðR1 þ R2 þ R3 þ R4Þ

iref ð16:75Þ

Note that the bridge-balance requirement is again given by Equation 16.69.

To estimate the nonlinearity of a constant-current bridge, suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the

beginning and R1 is changed by dR while the other resistors remain inactive. Again, R1 will represent the

active element (the sensing element) and may correspond to an active strain gage. The change in output,

dvo; is given by

dvo ¼ ½ðR þ dRÞR 2 R2􀀉

ðR þ dR þ R þ R þ RÞ

iref 2 0

dvo

Riref ¼

dR=R

ð4 þ dR=RÞ ð16:76Þ

By comparing the denominator on the RHS of this equation with Equation 16.70, we observe that the

constant-current bridge is more linear. Specifically, using the definition given by Equation 16.72, the

percentage nonlinearity may be expressed as

Np ¼ 25

% ð16:77Þ

It is noted that the nonlinearity is halved by using a constant-current excitation instead of a constantvoltage

excitation.

16.6.3 Bridge Amplifiers

The output from a resistance bridge is usually very small in comparison to the reference, and it has to be

amplified in order to increase the voltage level to a useful value (for example, in system monitoring or

data logging). A bridge amplifier is used for this purpose. This is typically an instrumentation amplifier or

a differential amplifier. The bridge amplifier is modeled as a simple gain, Ka; that multiplies the bridge

output.

16.6.3.1 Half-Bridge Circuits

A half bridge may be used in some applications that require a bridge circuit. A half bridge has only two

arms and the output is tapped from the mid-point of the two arms. The ends of the two arms are excited

Signal Conditioning and Modification 16-45

by a positive voltage and a negative voltage.

Initially, the two arms have equal resistances so

that, nominally, the bridge output is zero. One of

the arms has the active element. Its change in

resistance results in a nonzero output voltage. It is

noted that the half-bridge circuit is somewhat

similar to a potentiometer circuit.

A half-bridge amplifier consisting of a resistance

half-bridge and an output amplifier is shown in

Figure 16.18. The two bridge arms have resistances

R1 and R2; and the amplifier uses a feedback

resistance Rf : To obtain the output equation, we

use the two basic facts for an unsaturated opamp; the voltages at the two leads are equal (due to high

gain) and current in both leads is zero (due to high input impedance). Hence, voltage at node A is zero

and the current balance equation at node A is

vref

R1 þ ð2vref Þ

R2 þ

Rf ¼ 0

This gives

vo ¼ Rf

􀀏 􀀐

vref ð16:78Þ

Now, suppose that initially R1 ¼ R2 ¼ R and the active element R1 changes by dR: The corresponding

change in output is

dvo ¼ Rf

R þ dR

􀀏 􀀐

vref 2 0

dvo

vref ¼

dR=R

ð1 þ dR=RÞ ð16:79Þ

Note that Rf =R is the amplifier gain. Now, in view of Equation 16.72, the percentage nonlinearity of the

half-bridge circuit is

Np ¼ 100

% ð16:80Þ

It follows that the nonlinearity of a half-bridge circuit is worse than that of the Wheatstone bridge.

16.6.4 Impedance Bridges

An impedance bridge contains general impedance elements, Z1; Z2; Z3; and Z4, in its four arms, as shown

in Figure 16.19(a). The bridge is excited by an AC supply, vref : Note that vref represents a carrier signal

and the output, vo, has to be demodulated if a transient signal representative of the variation in one of the

bridge elements is needed. Impedance bridges can be used, for example, to measure capacitances in

capacitive sensors and changes of inductance in variable-inductance sensors and eddy-current sensors.

Also, impedance bridges can be used as oscillator circuits. An oscillator circuit can serve as a constantfrequency

source of a signal generator (in vibration testing) or it can be used to determine an unknown

circuit parameter by measuring the oscillating frequency.

+vref

(Active

Element)

Output

+ vo

A −

−vref

(Dummy)

FIGURE 16.18 A half bridge with an output amplifier.

16-46 Vibration and Shock Handbook

Analyzing using frequency-domain concepts, it

is seen that the frequency spectrum of the

impedance-bridge output is given by

voðvÞ ¼ ðZ1Z4 2 Z2Z3Þ

ðZ1 þ Z2ÞðZ3 þ Z4Þ

vref ðvÞ

ð16:81Þ

This reduces to Equation 16.68 in the DC case of a

Wheatstone bridge. The balanced condition is

given by

Z2 ¼

Z4 ð16:82Þ

This is used to measure an unknown circuit

parameter in the bridge. Let us consider two

examples.

16.6.4.1 Owen Bridge

The Owen bridge shown in Figure 16.19(b) may be

used to measure the inductance L4 or capacitance

C3; by the bridge-balance method. To derive the

necessary equation, note that the voltage – current

relation for an inductor is

v ¼ L

dt ð16:83Þ

and for a capacitor it is

i ¼ C

dt ð16:84Þ

It follows that the voltage/current transfer function

(in the Laplace domain) for an inductor is

vðsÞ

iðsÞ ¼ Ls ð16:85Þ

and that for a capacitor is

vðsÞ

iðsÞ ¼

Cs ð16:86Þ

Accordingly, the impedance of an inductor element

at frequency v is

ZL ¼ jvL ð16:87Þ

and the impedance of a capacitor element at

frequency v is

Zc ¼

jvC ð16:88Þ

(a)

Z1 Z2

Z3 Z4

Output

vref

(AC Supply)

(b)

C1 R2

R3 R4

Output

vref

(c)

R1 R2

vref

FIGURE 16.19 (a) General impedance bridge;

(b) Owen bridge; (c) Wien-bridge oscillator.

Signal Conditioning and Modification 16-47

Applying these results to the Owen bridge, we have

Z1 ¼

jvC1

Z2 ¼ R2

Z3 ¼ R3 þ

jvC3

Z4 ¼ R4 þ jvL4

in which v is the excitation frequency. Now, from Equation 16.82, we have

jvC1 ðR4 þ jvL4Þ ¼ R2 R3 þ

jvC3

􀀏 􀀐

By equating the real parts and the imaginary parts of this equation, we obtain the two equations

C1 ¼ R2R3

and

C1 ¼

Hence, we have

L4 ¼ C1R2R3 ð16:89Þ

and

C3 ¼ C1

R4 ð16:90Þ

It follows that L4 and C3 can be determined with the knowledge of C1; R2; R3; and R4 under balanced

conditions. For example, with fixed C1 and R2; an adjustable R3 could be used to measure the variable L4;

and an adjustable R4 could be used to measure the variable C3:

16.6.4.2 Wien-Bridge Oscillator

Now consider the Wien-bridge oscillator shown in Figure 16.19(c). For this circuit, we have

Z1 ¼ R1

Z2 ¼ R2

Z3 ¼ R3 þ

jvC3

Z4 ¼

R4 þ jvC4

Hence, from Equation 16.82, the bridge-balance requirement is

R2 ¼ R3 þ

jvC4

􀀏 􀀐

R4 þ jvC4

􀀏 􀀐

Equating the real parts, we obtain

R2 ¼

R4 þ

C3 ð16:91Þ

16-48 Vibration and Shock Handbook

and, by equating the imaginary parts, we obtain

0 ¼ vC4R3 2

vC3R4

Hence,

v ¼

1 ffiffiffiffiffiffiffiffiffiffiffiffiffi

C3C4R3R4 p ð16:92Þ

Equation 16.92 tells us that the circuit is an oscillator whose natural frequency is given by this equation

under balanced conditions. If the frequency of the supply is equal to the natural frequency of the circuit,

large-amplitude oscillations will take place. The circuit can be used to measure an unknown resistance

(e.g., in strain-gage devices) by first measuring the frequency of the bridge signals at resonance (natural

frequency). Alternatively, an oscillator that is excited at its natural frequency can be used as an accurate

source of periodic signals (a signal generator).

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