DC & AC BRIDGESPart 2 (AC Bridge)

Objectives

• Ability to explain operation of ac bridge circuit.

• Ability to identify bridge by name

• Ability to compute the values of unknown impedance following ac bridges.

AC BridgesAC bridges are used to measure inductance and capacitances and all ac bridge circuits are based on the Wheatstone bridge. The general ac bridge circuit consists of 4 impedances, an ac voltage source, and detector as shown in Figure below. In ac bridge circuit, the impedances can be either pure resistance or complex impedances.

4

2

3

1

Z

Z

Z

Z

Fig. 5-7: General ac bridge circuit

Inductance Capacitance

A simple bridge circuits are shown below;

Cont.Applications - in many communication system and complex electronic circuits. AC bridge circuits - are commonly used for shifting phase, providing feedback paths for oscillators and amplifiers, filtering out undesired signals, and measuring the frequency of audio signals.

The operation of the bridge depends on the fact that when certain specific circuit conditions apply, the detector current becomes zero. This is known as the null or balanced condition. Since zero current means that there is no voltage difference across detector, the bridge circuit may be redrawn as in Fig. 5-8. The voltages at point a and b and from point a to c must be equal.

Definition of electrical impedance• The impedance of a circuit element is defined as the ratio of the

phasor voltage across the element to the phasor current through the element:

• It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.

• For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor:

• where the VR and IR above are DC (constant real) values.

r

rR I

VZ

R

R

I

VR

Definition of Reactance, X

• A resistor's impedance is R (its resistance) and its reactance, XR is 0.

• A capacitance impedance: XC = -1/C

= -1/(2fC)

• An inductive impedance: XL = L = 2fL

Reactance is the imaginary part of impedance, and is caused by the presence of inductors or capacitors in the circuit. Reactance is denoted by the symbol X and is measured in ohms.

Z and Y passive elements

Element Impedance Admittance

R Z= R Y= 1/R

L Z= jωL Y=1/j ωL

C Z=-j(1/ωc) Y=j ωc

Cont.

Fig. 5-8: Equivalent of balanced ac bridge circuit

Fig. 5-7: General ac bridge circuit

Cont. I1Z1 = I2Z2 (1)

Similarly, the voltages from point d to point b and point d to point c must also be equal, therefore

I1Z3 = I2Z4 (2)

equation (1) divided by equation (2)

4

2

3

1

Z

Z

Z

Z

• If impedance is written in the form

where Z represents magnitude and the phase angle of complex impedance, its can be written as,

ZZ

)()(

))(())((

32324141

11221111

ZZZZ

where

ZZZZ

Example 5-5

The impedances of the AC bridge in Fig. 5-7 are given as follows:

Determine the constants of the unknown arm.

01 30200Z

02 0150Z

03 40250Z

unknownZZ x 4

Solution

The first condition for bridge balance requires that

Z1Zx =Z2Z3

Zx = (Z2Z3/Z1)=[(150x250)/200]

= 187.5

Cont.

The second condition for balance requires that the sums of the phase angles of opposite arms be equal

1+ x = 2 + 3

x = 2 + 3 - 1

= 0 + (-40) – 30 = -70o

Cont.

Hence, the unknown impedance Zx, can bewritten as

Zx = 187.5 -700 = (64.13 – j176.19)

Where Zx = Zx cos + j Zx sin

Indicating that we are dealing with acapacitive element, possibly consisting of aseries resistor and a capacitor

Example 5-6

Given the AC bridge of Fig. 5-8 in balance, find the components of the unknown arms Zx.

Fig. 5-9: AC bridge in balance

Similar Angle BridgeThe similar angle bridge (refer figure below) is used to measure the impedance of a capacitive circuit. This bridge is sometimes called the capacitance comparison bridge of the series resistance capacitance bridge.

Z1 = R1

Z2 = R2

Z3 = R3 –jXc3

Zx = Rx –jXcx

31

2 RR

RRx

32

1 CR

RC x

Maxwell Bridge• to determine an unknown inductance with capacitance standard

11

1 11

CjR

Z

22 RZ

33 RZ

Lxxx jXRZ

1

32

R

RRRx 132 CRRLx X - reactance

Z = R + jX

Opposite Angle BridgeThe Opposite Angle Bridge or Hay Bridge (see Figure below) is used to measure the resistance and inductance of coils in which the resistance is small fraction of the reactance XL, that is a coil having a high Q, meaning a Q greater than 10.

21

21

2

21321

2

1 CR

CRRRRx

21

21

2132

1 CR

CRRLx

Wein Bridge

The Wein Bridge shown in Figure below has a series RC combination in one arm and a parallel combination in the adjoining arm. It is designed to measure frequency (extensively as a feedback arrangement for a circuit). It can also be used for the measurement of an unknown capacitor with great accuracy.

444 cjXRZ

11 RZ

33

3 111

cjXR

Z

22 RZ

Cont..

2

44

24

2

13

1

CRR

R

RR

42

4

2

4

2

1

23 )

1

1( C

CRR

RC

2

3

2

3

23

2

14

1

CRC

R

RC

2

3

2

3

2

3

1

24 1 CR

R

R

RR

Equivalent parallel

component

Equivalent series component

Radio Frequency Bridge

The radio frequency bridge shown in figure below is often used in laboratories to measure the impedance of both capacitance and inductive circuits at higher frequencies.

)( 1'1

2

3 CCC

RRx

)11

(1

4'4 CC

X x

C’1 & C’4 : new values of C1 & C4 after rebalancing

Schering Bridge

• used for measuring capacitors and their insulating properties for phase angle of nearly 90o.

3

12

C

CRRx

2

31

R

CRC x

Zx =Rx –j/Cx

Z2 = R2

Z3 = -j/C3

Z1 = 1/(R1 + jC1)

Summary• The Wheatstone Bridge – most basic bridge

circuit. Widely used to measure instruments and control circuits. Have high degree of accuracy.

• Kelvin Bridge – modification of Wheatstone Bridge and widely used to measure very low resistance.

• Thevenin’s theorem – analytical tool to analyzing an unbalance Wheatstone bridge.

• AC bridge – more general form of Wheatstone bridge.

• Different types of AC bridges differ in the types of impedances in the arms