This document discusses various types of AC bridges used to measure inductances and capacitances, including their operating principles and examples. It describes AC bridges such as the basic AC bridge, Maxwell bridge, Hay bridge, Schering bridge, Wien bridge, and the use of Wagner ground connection to reduce stray capacitances in bridge circuits. The key applications of AC bridges are measuring inductances, capacitances, frequencies, and characterizing capacitors.

1. Signal Conditioning for
Electronic Instrumentation
AC Bridges
1
MCT 3332 : Instrumentation and Measurements
Dr. Hazlina Md Yusof
Department of Mechatronics Engineering
International Islamic University Malaysia

2. Analog Signal Conditioning
AC Bridges
2
• Used to measure inductance and capacitances
• Applied in communication systems and complex
electronic circuits
- Used for shifting phase, providing feedback paths for
oscillators or amplifiers, filtering out undesired signals and
measuring the frequency of audio signals
• Operates on balanced condition
- Reactance and resistive
components are balanced

3. Analog Signal Conditioning
AACC BBrrididgge:e Bsa lance Condition
B
I1 I2
D
Z1
Z2
A C
Z4 Z3
D
all four arms are considered as impedance
(frequency dependent components)
The detector is an ac responding device:
headphone, ac meter
Source: an ac voltage at desired frequency
Z1, Z2, Z3 and Z4 are the impedance of bridge arms
At balance point: or BA BC 1 1 2 2 E =E IZ =IZ
General Form of the ac Bridge
I = V I = V
and 1 2
Z +Z Z +Z
1 3 2 4
V
1 4 2 3 Z Z =Z Z
Complex Form:
Polar Form: Magnitude balance:

5. 1 4 1 4 2 3 2 3 Z Z ‘T ‘T =Z Z ‘T ‘T
Phase balance:
1 4 2 3 Z Z =Z Z
1 4 2 3 ‘T ‘T =‘T ‘T 3

6. Analog Signal Conditioning
AC Bridges
4
Exam ple The impedance of the basic ac bridge are given as follows:
o
Z
100 :‘
80 (inductive impedance)
o
1
3
Z
250 :
(pure resistance)
2
4
Determine the constants of the unknown arm.
SOLUTION The first condition for bridge balance requires that
400 30 (inductive impedance)
unknown
‘ :
Z
Z
2 3
4
1
250 400 1,000
100
Z Z Z
Z
u
:
The second condition for bridge balance requires that the sum of the phase angles of
opposite arms be equal, therefore
o
4 2 3 1 ‘T =‘T ‘T ‘T 0 30 80 50
Hence the unknown impedance Z4 can be written in polar form as
o
4 Z 1,000 : ‘ 50

7. Analog Signal Conditioning
AC Bridges
Example 7
An ac bridge is in balance
with the following constants:
arm AB, R = 200 Ω in series
with L = 15.9 mH R; arm BC, R
= 300 Ω in series with C =
0.265 μF; arm CD, unknown;
arm DA, = 450 Ω. The
oscillator frequency is 1 kHz.
Find the constants of arm CD.
Example an ac bridge is in balance with the in series with L = 15.9 mH R; arm BC, R = 300 unknown; arm DA, = 450 :. The oscillator frequency arm CD.
SOLUTION
B
V I1 I2 1
A C
The general equation for bridge balance states 5
This result indicates that Z4 is a pure inductance at at frequency of 1kHz. Since the inductive obtain L = 23.9 mH
D
Z1
Z2
Z4 Z3
D
450 (200 (300 u
Z = Z Z
2 3
4
Z

8. Analog Signal Conditioning
AC Bridges
Comparison Bridge: Capacitance
Capacitance Comparison Bridge
Measure an unknown inductance or
capacitance by comparing with it with a known
inductance or capacitance.
Rx C3
Unknown
capacitance
D
R2
R1
R3 Cx
Diagram of Capacitance
Comparison Bridge
At balance point: 1 x 2 3 Z Z =Z Z
where
1 1 2 2 3 3
3
1 = ; ; and R R Rj
ZC
Z Z = Z
§ · § ·
¨ ¸ ¨ ¸
© ¹ © ¹
R R R R
1 2 3
jZC jZC
3
1 1
x
x
R R R
C C R
1
Separation of the real and imaginary terms yields: 2 3
1
x
R
3
2
x
R
and
Frequency independent
To satisfy both balance conditions, the bridge must contain two variable
elements in its configuration.
Vs
6

9. Capacitance Comparison Bridge
Example 8
A similar angle bridge is used to measure a
capacitive impedance at a frequency of 2kHz. The
bridge constant at balance are
C3 =100μF R1=10k Ω
R2=50k Ω R3=100k Ω
Find the equivalent series circuit of the unknown
impedance
7

10. Comparison Bridge: Inductance
Measure an unknown inductance or
capacitance by comparing with it with a known
inductance or capacitance.
D
R2
R1
L3
Rx
Lx
R3
Diagram of Inductance
Comparison Bridge
At balance point: 1 x 2 3 Z Z =Z Z
where
Unknown
inductance
1 1 2 2 3 3 3 Z =R ;Z = R ; and Z R jZ L

12. 1 x x 2 S S R R jZ L R R jZ L
R R R
L L R
2
Separation of the real and imaginary terms yields: 2 3
1
x
R
3
1
x
R
and
Frequency independent
To satisfy both balance conditions, the bridge must contain two variable
elements in its configuration.
Vs
8
Analog Signal Conditioning
AC Bridges
Inductance Comparison Bridge

13. Analog Signal Conditioning
AC Bridges
Maxwell Bridge
Maxwell Bridge
Measure an unknown inductance in terms of
a known capacitance
D
R2
R1
C1
R3
Rx
Lx
V
Unknown
inductance
Diagram of Maxwell Bridge
At balance point: x 2 3 1 Z =Z Z Y
where
R ; R ; and = 1 j C
Z = Z Y Z
2 2 3 3 1 1
R
1
§ ·
1
x x x R j L RR j C
Z Z
2 3 ¨ 1
¸
R
© 1
¹
Z =
R R R
x 2 3 1 and L R R C
Separation of the real and imaginary terms yields: 2 3
1
x
R
Frequency independent
Suitable for Medium Q coil (1-10), impractical for high Q coil: since R1 will be very
large. 9

14. Hay Bridge
Similar to Maxwell bridge: but R1 series with C1
V
Diagram of Hay Bridge
At balance point: 1 x 2 3 Z Z = Z Z
where
R j ; R ; and R
Z = Z Z
1 1 2 2 3 3
ZC
1
§ ·
¨ ¸
© ¹

15. 1 23
1
1
x x R R j L RR
j C
Z
Z
which expands to
Unknown
inductance
D
R2
R1
C1
R3 Rx
Lx
R R L x jR x
j L R
R R
x x
1 1 2 3
C C
1 1
Z
Z
R R L x
R R
1 23
1
x
C
R L R
C
1
1
x
x
Z
Z
Solve the above equations simultaneously
(1)
(2)
10
Analog Signal Conditioning
AC Bridges
Hay Bridge

16. Analog Signal Conditioning
AC Hay Bridges
Bridge: continues
Hay Bridge
L R R C
2 3 1
2 2 2
x
1 Z C R
1 1
2 2
R C R R R
1 1 2 3
2 2 2
x
1 C R
1 1 Z
Z
ZLx
Z
Rx
TL
R1
Z
TC
ZC1
and
Phasor diagram of arm 4 and 1
X Z
L Q
R R
tan L x
T
L
x
tan C 1
1 1
C
X
R C R
T
Z
tan tan or 1 L C Q
C R
1 1
T T
Z
Thus, Lx can be rewritten as
L R R C
2 3 12
1 (1/ ) x
Q
For high Q coil ( 10), the term (1/Q)2 can be neglected x 2 3 1 L | R R C 11

17. Schering Bridge
Used extensively for the measurement of capacitance
and the quality of capacitor in term of D
D
R2
R1
C1
C3 Rx
Cx
V
At balance point: x 2 3 1 Z =Z Z Y
where
Unknown
capacitance
Diagram of Schering Bridge
R ; 1 ; and = 1 j C
Z = Z Y
2 2 3 1
j C R
3 1
Z
Z
§ ·§ ·
R j R j 1
j C
C C R Z
¨ ¸¨ 2 1
¸
Z Z
© ¹© 1
¹
x
x x
R j R C
jR
which expands to 2 1 2
ZC C ZC R
3 3 1
x
x
R R C
C C R
1
Separation of the real and imaginary terms yields: 1
2
3
x
C
3
2
x
R
and
12
Analog Signal Conditioning
AC Bridges
Schering Bridge

18. Schering Bridge: continues
D R Z
R C
Dissipation factor of a series RC circuit: x
x x
x
X
Dissipation factor tells us about the quality of a capacitor, how close the
phase angle of the capacitor is to the ideal value of 90o
x x 1 1 For Schering Bridge: D Z R C Z R C
For Schering Bridge, R1 is a fixed value, the dial of C1 can be calibrated directly in D
at one particular frequency
13
Analog Signal Conditioning
AC Bridges
Schering Bridge

19. Wien Bridge
Measure frequency of the voltage source using series
RC in one arm and parallel RC in the adjoining arm
D
R2
R1
C1
C3
R4
R3
Vs
Diagram of Wien Bridge
At balance point: Z2 Z1Z4Y3
1 1 1 ; ; 1 j C ; and R
Z R Z Y Z Z
3 4 4
R
3
R
2 2 3
j C
1
Z
§ · § ·
¨ ¸ ¨ ¸
© ¹ © ¹
R R j R 1 j C
2 1 4 3
C R
1 3
Z
Z
Unknown
Freq.
R R R j C R R jR R C
which expands to 1 4 4 4 3
Z
2 31 4
R Z
C R C
3 13 1
R R
C
R R C
2 1 3
4 3 1
C R 1
3 1
C R
1 3
Z
Z
(1)
(2)
1
Rearrange Eq. (2) gives In most, Wien Bridge, R1 = R3 and C1 = C3
1 3 1 3
2
f
S C C R R
2 4 R 2R 1
2
f
S RC
(1) (2)
14
Analog Signal Conditioning
AC Bridges
Wien Bridge

20. Wagner Ground Connection
C
A D
B
R2
R1
1
2
C3 Rx
R3 Cx
Rw
Cw
C1 C2
D
Diagram of Wagner ground
One way to control stray capacitances is by
Shielding the arms, reduce the effect of stray
capacitances but cannot eliminate them
completely.
Wagner ground connection eliminates some
effects of stray capacitances in a bridge circuit
Simultaneous balance of both bridge makes the
point 1 and 2 at the ground potential. (short C1
and C2 to ground, C4 and C5 are eliminated from
detector circuit)
The capacitance across the bridge arms e.g. C6
cannot be eliminated by Wagner ground.
Wagner ground
Stray across arm
Cannot eliminate
C4
C5
C6
15
Analog Signal Conditioning
AC Bridges
Wagner Ground