Signal Conditioning for

Electronic Instrumentation AC Bridges

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MCT 3332 : Instrumentation and Measurements Dr. Hazlina Md Yusof Department of Mechatronics Engineering International Islamic University Malaysia

Analog Signal Conditioning AC Bridges

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•  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

Analog Signal Conditioning AC Bridges AC Bridge: Balance Condition

D

Z1Z2

Z4Z3

A C

D

B

I1 I2

all four arms are considered as impedance (frequency dependent components)The detector is an ac responding device: headphone, ac meterSource: an ac voltage at desired frequency

General Form of the ac Bridge

Z1, Z2, Z3 and Z4 are the impedance of bridge arms

At balance point: orBA BC 1 1 2 2E = E I Z = I Z

and1 21 3 2 4

V V I = I =Z + Z Z + Z

V

1 4 2 3Z Z = Z Z

� � � �1 4 1 4 2 3 2 3Z Z =Z ZT T T T� �� � ��

Complex Form:

Polar Form: Magnitude balance:

Phase balance:

1 4 2 3Z Z =Z Z

1 4 2 3=T T T T� �� � ��3

Analog Signal Conditioning AC Bridges

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Example The impedance of the basic ac bridge are given as follows:o

1

2

100 80 (inductive impedance)250 (pure resistance)

: � :

ZZ

Determine the constants of the unknown arm.

SOLUTION The first condition for bridge balance requires that

o3

4

400 30 (inductive impedance)unknown

� :

ZZ

2 34

1

250 400 1,000 100

Z ZZZ

u :

The second condition for bridge balance requires that the sum of the phase angles of opposite arms be equal, therefore

o4 2 3 1= 0 30 80 50T T T T� � �� �� � � �

Hence the unknown impedance Z4 can be written in polar form as o

4 1,000 50 : ��Z

Indicating that we are dealing with a capacitive element, possibly consisting of a series combination of at resistor and a capacitor.

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.

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Example an ac bridge is in balance with the following constants: arm AB, R = 200 :in series with L = 15.9 mH R; arm BC, R = 300 : in series with C = 0.265 PF; arm CD, unknown; arm DA, = 450 :. The oscillator frequency is 1 kHz. Find the constants of arm CD.

SOLUTION

The general equation for bridge balance states that

This result indicates that Z4 is a pure inductance with an inductive reactance of 150 :at at frequency of 1kHz. Since the inductive reactance XL = 2SfL, we solve for L and obtain L = 23.9 mH

D

Z1 Z2

Z4Z3

A C

D

B

I1 I2V 1

2

3

4

200 100 1/ 300 600 450

unknown

R j L jR j C jR

ZZ

� � : � � : :

ZZZZ

1 4 2 3Z Z = Z Z

1

450 (200 100) 150 (300 600)

j jj

u � :

�2 3

4Z ZZ =Z

Analog Signal Conditioning AC Bridges Capacitance Comparison Bridge

Comparison Bridge: Capacitance

Measure an unknown inductance or capacitance by comparing with it with a known inductance or capacitance.

Diagram of Capacitance Comparison Bridge

Unknown capacitance

At balance point: 3x1 2Z Z = Z Z

where1 1 2 2 3 3

3

1= ; ; and R R Rj CZ

�Z Z = Z

1 2 33

1 1x

x

R R R Rj C j CZ Z

§ · § ·� �¨ ¸ ¨ ¸

© ¹ © ¹

Separation of the real and imaginary terms yields: 2 3

1xR RRR

13

2x

RC CR

and

Frequency independentTo satisfy both balance conditions, the bridge must contain two variable elements in its configuration.

D

R2R1

RxC3

R3 Cx

Vs

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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

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Comparison Bridge: Inductance

Measure an unknown inductance or capacitance by comparing with it with a known inductance or capacitance.

Diagram of Inductance Comparison Bridge

Unknown inductance

At balance point: 3x1 2Z Z = Z Z

where1 1 2 2 3 3 3= ; ; and R R R j LZ �Z Z = Z

� � � �1 2x x S SR R j L R R j LZ Z� �

Separation of the real and imaginary terms yields: 2 3

1xR RRR

23

1x

RL LR

and

Frequency independentTo satisfy both balance conditions, the bridge must contain two variable elements in its configuration.

D

R2R1

L3

Rx

LxR3

Vs

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Analog Signal Conditioning AC Bridges Inductance Comparison Bridge

Maxwell Bridge

Measure an unknown inductance in terms of a known capacitance

D

R2R1

C1

R3Rx

Lx

Diagram of Maxwell Bridge

V

Unknown inductance

At balance point: 3 1x 2Z = Z Z Y

where2 2 3 3 1 1

1

1; ; and =R R j CR

Z �Z = Z Y

2 3 11

1x x xR j L R R j C

RZ Z

§ ·� �¨ ¸

© ¹Z =

Separation of the real and imaginary terms yields: 2 3

1xR RRR

2 3 1xL R R C and

Frequency independentSuitable for Medium Q coil (1-10), impractical for high Q coil: since R1 will be very large. 9

Analog Signal Conditioning AC Bridges Maxwell Bridge

Hay Bridge

Similar to Maxwell bridge: but R1 series with C1

Diagram of Hay Bridge

V

At balance point: 1 3x 2Z Z = Z Z

where1 1 2 2 3 3

1

; ; and jR R RCZ

� Z = Z Z

� �1 2 31

1x xR R j L R R

j CZ

Z§ ·

� � ¨ ¸© ¹

which expands to

Unknown inductance

D

R2

R1

C1

R3 Rx

Lx

1 1 2 31 1

x xx xL jRR R j L R R RC C

ZZ

� � �

1 2 31

xxLR R R RC

�

11

xx

R L RC

ZZ

Solve the above equations simultaneously

(1)

(2)

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Analog Signal Conditioning AC Bridges Hay Bridge

Hay Bridge: continues

2 3 12 2 2

1 11xR R CLC RZ

�

2 21 1 2 3

2 2 21 11x

C R R RRC R

ZZ

�

ZLx

Rx

Z

TL

R1

Z

TC

ZC1�

and

Phasor diagram of arm 4 and 1

tan xLL

x

LX QR R

ZT

1 1

1tan CC

XR C R

TZ

1 1

1tan tan or L C QC R

T TZ

Thus, Lx can be rewritten as2 3 1

21 (1/ )xR R CL

Q

�

For high Q coil (> 10), the term (1/Q)2 can be neglected 2 3 1xL R R C| 11

Analog Signal Conditioning AC Bridges Hay Bridge

Schering Bridge

Used extensively for the measurement of capacitance and the quality of capacitor in term of D

D

R2R1

C1

C3Rx

Cx

V

Unknown capacitance

Diagram of Schering Bridge

At balance point: 3 1x 2Z = Z Z Y

where2 2 3 1

3 1

1 1; ; and =R j Cj C R

ZZ

�Z = Z Y

2 11

1x

x x

j jR R j CC C R

ZZ Z

§ ·§ ·�� �¨ ¸¨ ¸

© ¹© ¹

which expands to 2 1 2

3 3 1x

x

j R C jRRC C C RZ Z

� �

Separation of the real and imaginary terms yields: 12

3x

CR RC

13

2x

RC CR

and

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Analog Signal Conditioning AC Bridges Schering Bridge

Schering Bridge: continues

Dissipation factor of a series RC circuit: xx x

x

RD R CX

Z

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

1 1x xD R C RCZ Z For Schering Bridge:

For Schering Bridge, R1 is a fixed value, the dial of C1 can be calibrated directly in Dat one particular frequency

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Analog Signal Conditioning AC Bridges Schering Bridge

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

C3R4

R3

Vs

Diagram of Wien Bridge

At balance point:2 1 4 3 Z Z Z Y

2 1 4 31 3

1jR R R j CC R

ZZ

§ ·§ · � �¨ ¸¨ ¸© ¹ © ¹

Unknown Freq.

which expands to 4 31 4 42 3 1 4

3 1 3 1

R CR R jRR j C R RR C R C

ZZ

� � �

32 1

4 3 1

CR RR R C

�

3 11 3

1C RC R

ZZ

(1)

(2)

1 3 1 3

12

fC C R RS

Rearrange Eq. (2) gives In most, Wien Bridge, R1 = R3 and C1 = C3

2 42R R 12

fRCS

(1) (2)

4433

3221

11 and ;1;;1 RCjR

RCj

R � � ZYZZ ZZ

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Analog Signal Conditioning AC Bridges Wien Bridge

Wagner Ground Connection

D

R2R1

C3 Rx

1

2

R3 Cx

Rw

Cw

C1 C2

D

A B

C

Diagram of Wagner ground

Wagner ground connection eliminates some effects of stray capacitances in a bridge circuitSimultaneous balance of both bridge makes the point 1 and 2 at the ground potential. (short C1and C2 to ground, C4 andC5 are eliminated from detector circuit) The capacitance across the bridge arms e.g. C6cannot be eliminated by Wagner ground.

Wagner ground

Stray across armCannot eliminate

One way to control stray capacitances is by Shielding the arms, reduce the effect of stray capacitances but cannot eliminate them completely.

C4

C5C6

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Analog Signal Conditioning AC Bridges Wagner Ground