12 ac bridges rev 3 080423

Chapter 12: AC Bridges

0908341 Measurements & Instrumentation

Chapter 12
Bridges II: AC Bridges
and the measurement of L&C
(Revision 3.0, 4/5/2008)
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.
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

V
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

∠Z X + ∠Z 4 = ∠Z 2 + ∠Z 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 2008: 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.
2.1 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).
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:

QC =

XC
1
=
R
ω ⋅C ⋅ R

4. Deflection AC bridges
As with DC bridges, AC bridge can be used in two configurations: null and
deflection modes. In the null mode, the bridge is used to find the value of an
L or C component accurately. In a deflection type bridge, the physical variable
to be measured is converted to an output voltage. The deflection bridge is
especially necessary in cases where the variable to be measured is changing
rapidly (i.e., faster than a human operator can achieve a null condition).
The next section will discuss the various types of AC null bridges. This
section will provide an overview of the construction of deflection bridges.
Deflection AC bridge are usually of the ‘symmetrical’ type. This is
because they consist of two resistors and two identical reactive components
(i.e., two capacitors or two inductors). An example AC inductive deflection
bridge is shown in Figure 2 where the varying inductor is Lx. In this case the
inductor would change its value with the change in the variable that it is
measuring.
An AC capacitive deflection bridge is shown in Figure 3 where the
varying capacitor is Cx. In this case, the capacitor would change its value with
the change in the variable that it is measuring (e.g., displacement).


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RX

R2

LX

L2

Vi

V
Vo
R3

R4

Figure 2: AC deflection bridge (inductive).

C2

Cx

Vi

V
Vo
R3

R4

Figure 3: AC deflection bridge (capacitive).

5. AC Null Type Bridges
This section examines the various types of AC null type bridges.


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5.1 Maxwell Bridge
The Maxwell Bridge is used to measure inductors with low to medium values
of Q1. A typical arrangement is shown in Figure 4 below [1].

RX
R2
LX
Vi

V
Vo
R3

C4

R4

Figure 4: Maxwell Bridge.

At Balance2,

Z X = Y4 ⋅ Z 2 ⋅ Z 3
This gives:

⎛1
⎞
RX + j ⋅ ω ⋅ LX = R2 ⋅ R3 ⋅ ⎜ + j ⋅ ω ⋅ C4 ⎟
⎜R
⎟
⎝ 4
⎠
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:
1

A low value of Q is considered to be below 10 or below 5.
Depending on the setup of the bridge, it might be simpler in some cases to multiply by an
admittance rather than divide by an impedance.

2


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LX = R2 ⋅ R3 ⋅ C4
5.2 Hay Bridge
The Hay Bridge (shown in Figure 5 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 (or more than 5). At balance conditions, we have the following:

RX
R2
LX
Vi

V
Vo
R3

R4

C4

Figure 5: Hay Bridge.

⎞
⎛
1
⎟ = R2 ⋅ R3
+ j ⋅ ω ⋅ LX ) ⋅ ⎜ R4 +
X
⎜
j ⋅ ω ⋅ C4 ⎟
⎠
⎝
RX
L
RX ⋅ R4 + j ⋅ ω ⋅ LX ⋅ R4 +
+ X = R2 ⋅ R3
j ⋅ ω ⋅ C4 C4

(R

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:


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ω ⋅ LX ⋅ R4 =
LX =

RX
ω ⋅ C4

RX
ω 2 ⋅ R4 ⋅ C4

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 =

R ⋅ R ⋅C − L
RX
= 2 23 24 2 X
ω 2 ⋅ R4 ⋅ C4
ω ⋅ R4 ⋅ C4

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)
Note that in this case, the results depend on the value of the frequency (as
opposed to the Maxwell Bridge, where the results were independent of the
value of the frequency).
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:


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

Rx

R2

Cx
Vi

V
Vo

C3
R4
C4

Figure 6: 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 ⋅ω ⋅ C ⎟
⎟
⎠ ⎝ 4
⎠
⎝
3 ⎠

⎛
1
⎜ RX +
⎜
j ⋅ω ⋅ CX
⎝

⎞ ⎛ R ⋅C ⎞
⎞ ⎛
R2
⎟+⎜ 2 4 ⎟
⎟ = ⋅⎜
⎟ ⎜ j ⋅ω ⋅ C ⋅ R ⎟ ⎜ C ⎟
⎠ ⎝
⎝
⎠
3
4 ⎠
3

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


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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.
5.4 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 Wien3. The source of unknown
frequency is used to excite the a.c. bridge as shown in Figure 7. 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

R2

R4

C4

Figure 7: The Wien Bridge.

At balance conditions:

R1 Z 3
=
= Z 3 × Y4
R2 Z 4
3

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.


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Developing this, gives:
⎞ R C
R1 ⎛
1 ⎞ ⎛ 1
1
⎟ × ⎜ + jωC4 ⎟ = 3 + 4 + jωR3C4 +
= ⎜ R3 +
⎟ R C
R2 ⎜
jωC3 ⎟ ⎜ R4
jωR4C3
4
3
⎠
⎠ ⎝
⎝

Equating real parts from both sides gives:

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

jωR3C4 +

1
=0
jωR4C3

jωR3C4 = −

ω=

1
jωR4C3

1
1
⇒ f =
R3 R4C3C4
2π 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 linked 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
8. The dashed line on electrical diagrams indicates a mechanical connection
between electrical components.


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C3

R1

R3

Unknown
frequency
source
Null
Detector

R2

R4

C4

Figure 8: The mechanical link between the two resistors (ganged).

5.4.1 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.
6. 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 9 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).


Page 10 of 16



R1

VR

Vi
RX
VX
LX

Figure 9: 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
2
RX + ω 2 ⋅ L2X = R1

LX =

2
R12 − RX

ω

2
R12 − RX
=
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 10 [1]. The voltages across the known resistor and the
capacitor are measured. Their values are then used to find the unknown
capacitor as follows:


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R1

VR

Vi

CX
VC

Figure 10: 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.
7. General guidelines on the use of bridges
Having introduced both types of bridges, it is appropriate at this point to
introduce a general set of guidelines for selection of bridge for different
applications. In order to select a certain type of bridge design, a good
understanding is needed of the nature of the application and any constraints.
The following is a set of guidelines that can be used in such a selection:
a) Type of sensor (R, L or C): The first decision to be made is the
type of sensor to use for the application in order to convert the
non-electrical variable into an electrical variable. If, for example,
a strain gauge is to be used to measure stress or strain, then the

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sensor is resistive.
If however, we wish to measure
displacement by capacitive effects, then the sensor will be a
capacitor. In general, a passive sensor could either be resistive,
capacitive or inductive.
b) Bridge or voltage divider: In certain simple cases, a simple
voltage divider could be used instead of a bridge. This consists
of the sensor that varies with the variable to be measured
connected in series with fixed impedance (usually identical to
the type of component of the sensor). This combination is then
excited by a voltage source and the output is available between
the ground connection and the point of connection of the two
impedances. Obviously there are many disadvantages with this
setup compared to a bridge.
a. It cannot be used in null mode.
b. The output has a large voltage component that is present
even when the input variable is at its neutral position.
This makes it difficult to detect small signal deviations
caused by changes in the measured variable as they are
mixed with a relatively large voltage.
c. The bridge can provide a differential output signal that is
immune to noise, while the voltage divider cannot (its
signal is relative to ground).
c) AC or DC Bridge: Having decided on the type of sensor and the
use of a bridge, the next most important decision is to whether a
dc bridge or an ac bridge is needed. This depends on the type
of components used in the bridge. If any component is reactive
(capacitive or inductive), then the bridge must be an AC bridge.
If all components are resistive, then the natural choice would be
a DC bridge (although an AC bridge could be used in this case if
there is a good reason to do so).
d) Null or deflection: Depending on whether the signal is changing
rapidly or not, the type of bridge mode can be selected. A null
type mode is used for situations where the variable is not
changing (or is changing very slowly) such that there is enough
time to balance the bridge and get a reading. The null type
mode is used where an accurate measurement of impedance is
needed. On the other hand, if the signal is changing rapidly
(e.g., faster than every second), then a deflection type bridge
must be used. The fact that variable is changing for example at
a frequency of 20 Hz, does not have any effect on whether the
bridge is AC or DC. It means that the bridge has to be a
deflection bridge, but it could be AC or DC depending on the
type of components in it (reactive or resistive).
e) When AC deflection bridges are used, it is most appropriate to
use the ‘symmetrical’ arrangements (i.e., 2 adjacent inductors

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and 2 adjacent resistors; or 2 adjacent capacitors and 2
adjacent resistors).
f) In general higher accuracy will be obtained from a null type
bridge, if it is possible to use one. But null type bridges
generally require human intervention, while deflection type
bridges are automatic.
g) 1 arm, 2 arm or 4 arm bridge: If a deflection bridge is to be
used, then it is best to try to have a bridge with the highest
number of elements, in order to maximise the sensitivity. 4 arm
bridges generally require sensors that increase and decrease
their values with changes in the variable that is being measured.
This is not always possible, and in some cases only a 2 arm
bridge is possible. If the cost of the sensor is high, a 1 arm
bridge might be the only option in that case.
h) Feedback control systems: In certain cases, if the output signal
is required to form a feedback signal to a closed loop control
system, then a deflection bridge must be used.
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.
Problems
1. The a.c. bridge shown below is called an ‘Owen’ Bridge and is used to
measure inductance. Assuming balance conditions, find the expressions for
the values of Rx and Lx. Assume that the frequency of the excitation source is
ω rad·s-1.


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RX
R2
LX
Vi

V
Null
detector

R3

C4

C3

Figure 11: Problem 1.

2. The AC bridge that is shown below cannot be balanced.
a) Explain why it cannot be balanced.
b) If R3 is removed from the bridge, suggest how you can make the
bridge balanceable by replacing just ‘one’ component with an
alternative component.

R1

R2

C1
Vi
R3

V
Vo

L3

R4
C4

Figure 12: Problem 2.


Page 15 of 16



3. Describe and draw the setup that gives the best accuracy to measure the
following variables:
a) The level of a non-conducting liquid using two parallel plates.
The liquid level is expected to change every 1 second.

b) The force on a cantilevered beam. The force is a sinusoidal
function of frequency 20 Hz.

c)

The value of an inductor that has an approximate inductance of 1 mH
and an approximate resistance of 10 mΩ and will be used at a frequency
of 50 Hz.


Page 16 of 16

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