Download Wheatstone bridge circuit

CHAPTER 5
 Bridge circuit (DC or AC) is an instrument to measure
resistance, inductance, capacitance and impedance.
 Operate on a null-indication principle. This means the
indication is independent of the calibration of the
indicating device or any characteristics of it.
# Very high degrees of accuracy can be achieved using
the bridges.
 Used in control circuits.
# One arm of the bridge contains a resistive element
that is sensitive to the physical parameter
(temperature, pressure, etc.) being controlled.
Types of bridge circuits are used in measurement:
1) DC bridge:
a) Wheatstone Bridge
b) Kelvin Bridge
2) AC bridge:
a) Similar Angle Bridge
b) Opposite Angle Bridge/Hay Bridge
c) Maxwell Bridge
d) Wein Bridge
e) Radio Frequency Bridge
f) Schering Bridge
The Wheatstone bridge is an
electrical bridge circuit used
to measure resistance.
This bridge consists of a
galvanometer and TWO (2)
parallel branches containing
FOUR (4) resistors.
One parallel branch contains
one known resistance and one
unknown; the other parallel
branch contains resistors of
known resistances.
Figure 5.1:
Wheatstone Bridge Circuit
To operate the bridge, a voltage source is connected to two terminals
of the bridge.
In the circuit at right, if R4 is the
unknown resistance; R1, R2 and R3
are resistors of known resistance
where the resistance of R3 is
adjustable.
How to determine the
resistance of the unknown
resistor, R4?
“The resistances of two resistors
are fixed and the resistance of
other one is adjusted until the
current passing through the
galvanometer decreases to zero”.
Figure 5.1:
Wheatstone Bridge Circuit
When no current flows through the galvanometer, the bridge is called
in a balanced condition.
A
B
D
C
Figure 5.1:
Wheatstone Bridge Circuit
Figure 5.2:
A variable resistor; the amount of
resistance between the connection
terminals could be varied.
When the bridge is in balanced
condition, we obtain,
A
 voltage drops across R1 and R2
are equal,
B
D
I1R1 = I2R2
(2.1)
 voltage drops across R3 and R4
are also equal,
I3R3 = I4R4
(2.2)
C
Figure 5.1:
Wheatstone Bridge Circuit
A
 In this point of balance, we also
obtain;
I1 = I 3
(2.3)
I2 = I 4
(2.4)
Therefore, the ratio of two resistances
in the known leg is equal to the ratio
of the two in the unknown leg;
R3 R4

R1 R2
or
B
D
R2
R4  R3
R1
C
Figure 5.1:
Wheatstone Bridge Circuit
(2.5)
Example 1
Figure 5.3
Find Rx at the balance condition?
Sensitivity of the Wheatstone Bridge
When the pointer of a
galvanometer deflects
towards right or left hand
side, this means that current
is flowing through the
galvanometer and the bridge
is called in an unbalanced
condition.
The amount of deflection is
a function of the sensitivity
of the galvanometer. For the
same current, greater
deflection of pointer indicates
more sensitive a
galvanometer.
Figure 5.4.
(Cont…..)
Sensitivity S can be expressed in linear or angular units as follows:
S
S
S
S
Deflection D


Current
I
mil lim eters

or ;
A
deg rees

or ;
A
radians

A
(2.6)
How to find the current
value?
Figure 5.4.
Thevenin’s Theorem
Thevenin’s theorem is an approach
used to determine the current flowing
through the galvanometer.
Thevenin’s equivalent voltage is
found by removing the galvanometer
from the bridge circuit and computing
the open-circuit voltage between
terminals a and b.
Fig. 5.5:
Thevenin’s equivalent voltage
Applying the voltage divider equation, we express the voltage at point a
and b, respectively, as
R3
Va  E
R1  R3
(2.7)
R4
Vb  E
R2  R4
(2.8)
A
(Cont…..)
The difference in Va and Vb represents
Thevenin’s equivalent voltage. That is,
D
 R3
R4 
 (2.9)
VTh  Va  Vb  E

 R1  R3 R2  R4 
B
C
Fig. 5.5: Wheatstone bridge
with the galvanometer removed
Thevenin’s equivalent resistance is
found by replacing the voltage source
with its internal resistance, Rb as
shown in Figure 5.6.
Fig. 5.6: Thevenin’s resistance circuit
(Cont…..)
Since Rb is assumed to be very low (Rb
≈ 0 Ω), we can redraw the bridge as
shown in Fig. 5.6 to facilitate
computation of the Thevenin’s
resistance as follows:
RTh  R1 // R3  R2 // R4
R1 R3
R2 R4
RTh 

R1  R3 R2  R4
Fig. 5.6:
Thevenin’s resistance circuit
(2.10)
(Cont…..)
If the values of Thevenin’s equivalent voltage and resistance have been
known, the Wheatstone bridge circuit in Fig. 5.5 can be changed with
Thevenin’s equivalent circuit as shown in Fig. 5.7,
Fig. 5.5: Wheatstone bridge circuit
Fig. 5.7: Thevenin’s equivalent circuit
(Cont…)
If a galvanometer is connected to
terminal a and b, the deflection current
in the galvanometer is
VTh
Ig 
RTh  Rg
(2.11)
Fig. 5.7:
Thevenin’s equivalent circuit
where Rg = the internal resistance in the galvanometer.
Example 2
R2 = 1.5 kΩ
R1 = 1.5 kΩ
Rg = 150 Ω
E= 6 V
G
R3 = 3 kΩ
R4 = 7.8 kΩ
Figure 5.8 : Unbalance Wheatstone Bridge
Calculate the current through the galvanometer ?
Slightly Unbalanced Wheatstone Bridge
If three of the four resistors in a Wheatstone bridge are equal to R and
the fourth differs by 5% or less, we can develop an approximate but
accurate expression for Thevenin’s equivalent voltage and resistance.
Consider the circuit in Figure 5.9, the voltage at point a is given as
R
 R  E
Va  E
 E

RR
 2R  2
(2.12)
The voltage at point b is expressed as:
R  r
Vb  E
R  R  r
(2.13)
Figure 5.9: Wheatstone Bridge with
three equal arms
Slightly Unbalanced Wheatstone Bridge (Cont…)
Thevenin’s equivalent voltage is the difference in this voltage
1
 R  r
 r 
VTh  Vb  Va  E 
   E

 R  R  r 2 
 4 R  2r 
If ∆r is 5% of R or less, Thevenin equivalent voltage can be simplified to
be
 r 
VTh  E 

 4R 
(2.14)
(Cont…..)
Thevenin’s equivalent resistance can be calculated by replacing the
voltage source with its internal resistance and redrawing the circuit as
shown in Figure 5.10. Thevenin’s equivalent resistance is now given as
R
R ( R)( R  r )
RTh  
2 R  R  r
or
o
o
If ∆r is small compared to R,
the equation simplifies to
R R
RTh  
2 2
R
RTh  R
R + Δr
R
Figure 5.10:
Resistance of a Wheatstone.
(2.15)
(Cont…..)
We can draw the Thevenin equivalent circuit as shown in Figure 5.11
Figure 5.11: Approximate Thevenin’s equivalent circuit
for a Wheatstone bridge containing three equal
resistors and a fourth resistor differing by 5% or less
Kelvin bridge is a modified
version of the Wheatstone bridge.
The purpose of the modification is
to eliminate the effects of contact
and lead resistances in low
resistance measurement.
The measurement with a high
degree of accuracy can be done
using the Kelvin bridge for
resistors in the range of 1 Ω to
approximately 1 µΩ.
Fig. 5.12: Basic Kelvin Bridge showing
a second set of ratio arms
Since the Kelvin bridge uses a second set of ratio arms (Ra and Rb),
it is sometimes referred to as the Kelvin double bridge.
The resistor Rlc represents the
resistance of the connecting leads
from R2 to Rx (unknown resistance).
The second set of ratio arms
(Ra and Rb in figure) compensates
for this relatively low lead-contact
resistance.
Fig. 5.12:
Basic Kelvin Bridge showing
a second set of ratio arms
When a null exists, the value
for Rx is the same as that for
the Wheatstone bridge, which
is
R2 R3
Rx 
R1
or
Rx R3

R2 R1
If the galvanometer is connected to
point B, the ratio of Rb to Ra must be
equal to the ratio of R3 to R1.
Therefore,
Rx R3 Rb


R2 R1 Ra
C
D
Fig. 5.12:
Basic Kelvin Bridge showing
a second set of ratio arms
(3.1)
In general, AC bridge has a similar circuit design as DC bridge, except
that the bridge arms are impedances as shown in Figure 5.13.
The impedances can be either
pure resistances or complex
impedances (resistance +
inductance or resistance +
capacitance). Therefore, AC
bridges are used to measure
inductance and capacitance.
Some impedance bridge circuits
are frequency-sensitive while
others are not. The frequencysensitive types may be used as
frequency measurement devices
if all component values are
accurately known.
Fig 5.13: General AC bridge circuit
The usefulness of AC bridge
circuit is not restricted to the
measurement of an unknown
impedance. These circuits find
other application in many
communication systems and
complex electronic circuits,
such as for:
shifting phase, providing
feedback paths for oscillators
or amplifiers;
filtering out undesired signals;
measuring the frequency of
audio signals.
Fig 5.13: General AC bridge circuit
AC bridge is excited by an AC source and its galvanometer is replaced
by a detector. The detector can be a sensitive electromechanical meter
movements, oscilloscopes, headphones, or any other device capable of
registering very small AC voltage levels.
AC bridge circuits work on the
same basic principle as DC
bridge circuits: that a balanced
ratio of impedances (rather than
resistances) will result in a
“balanced” condition as
indicated by the null-detector.
Fig 5.13: General AC bridge circuit
When an AC bridge is in null or balanced condition, the detector
current becomes zero. This means that there is no voltage
difference across the detector and the bridge circuit in Figure 5.13
can be redrawn as in Figure 5.14.
Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit
The dash line in the figure indicates
that there is no potential difference
and no current between points b
and c. The voltages from point a to
point b and from point a to point c
must be equal, which allows us to
obtain:
I1Z1  I 2 Z 2
(4.1)
Similarly, the voltages from point d
to point b and point d to point c
must also be equal, leading to:
I1Z 3  I 2 Z 4
(4.2)
Fig. 5-14: Equivalent of balanced
(nulled) AC bridge circuit
Dividing Eq. 4.1 by Eq. 4.2, we obtain:
Z1 Z2

Z3 Z 4
which can also be written as
Z1Z 4  Z 2 Z3
(4.3)
Fig. 5-14: Equivalent of balanced
(nulled) AC bridge circuit
If the impedance is written in the form Z = Z∟θ where Z represents
the magnitude and θ the phase angle of the complex impedance,
Eq. 19 can be written in the form
Z11 Z 44   Z 2 2 Z33 
or
I (1   4 )  Z 2 Z 3( 2  3 )
Z1Z 4
(4.4)
f
Eq. 4.4 shows twot conditions when ac bridge is balanced;
First condition shows
that the products of the magnitudes of the
h
e be equal: Z1Z4 = Z2Z3
opposite arms must
Second condition shows that the sum of the phase angles of the
i
opposite arms is equal: ∟θ1+ ∟θ4 = ∟θ2+ ∟θ3
m
p
Similar-angle bridge is an AC bridge used to measure the impedance of a
capacitive circuit. This bridge is sometimes known as the capacitance
comparison bridge or series resistance capacitance bridge.
The following are some components
used to construct a similar-angle
bridge:
 R1 = a variable resistor.
 R2 = a standard resistor.
 R3 = an added variable resistor
needed to balance the
bridge.
 Rx = an unknown resistor used to
indicate the small leakage
resistance of the capacitor.
 C3 = a known standard capacitor
in series with R3.
 Cx = an unknown capacitor.
R2
Fig. 5-15: Similar angle bridge
By referring to Figure 5.15, the
impedance of the arms of this
bridge can be written as
R2
Z1  R1
Z 2  R2
Z 3  R3 in series with C3
j
 R3 
 C3
Z x  Rx in series with C x
j
 Rx 
 Cx
Fig. 5-15: Similar-angle bridge
The condition for balance of the bridge is
Z1 Z x  Z 2 Z 3


j 
j 
  R2  R3 

R1  Rx 
C x 
C3 


jR1
jR2
R1 Rx 
 R2 R3 
C x
C3
Two complex quantities are equal when both real and imaginary
terms are equal. Therefore,
R1Rx  R2 R3
or
R2 R3
Rx 
R1
(5.1)
and,
jR1
jR2

C x C3
or
C3 R1
Cx 
R2
(5.2)
Maxwell bridge is an ac bridge used to measure an unknown
inductance in terms of a known capacitance. This bridge is
sometimes called a Maxwell-Wien Bridge.
Using capacitance as a standard
has several advantages due to:
Capacitance of capacitor is
influenced by less external
fields.
Capacitor has small size.
Capacitor is low cost.
Fig. 5-15: Maxwell Bridge
The impedance of the arms of the bridge can be written as
Z1 
1
1
 j C1
R1
Z 2  R2
Z 3  R3
Z 4  Rx  j X Lx
Fig. 5-15: Maxwell Bridge
The general equation for bridge balance is
Z1 Z x  Z 2 Z 3
1
1
 j C1
R1
Rx  j X Lx   R2 R3
1

Rx  j X Lx   R2 R3
1  j C1
R1
Rx  j X Lx R2 R3

1  j C1
R1
Rx  j X Lx
R2 R3

 j R2 R3C1
R1
Fig. 5-15: Maxwell Bridge
Equating real terms and imaginary terms we have
R2 R3
Rx 
R1
(6.1)
j Lx  jR2 R3C1
Lx  R2 R3C1
(6.2)
Fig. 5-15: Maxwell Bridge
Opposite-angle bridge is an AC bridge for measurement of
inductance. To construct this bridge can be done by replacing the
standard capacitor of the similar-angle with an inductor as shown
in Figure 5-14.
Opposite-angle bridge is sometimes
known as a Hay Bridge. It differs
from Maxwell bridge by having a
resistor R1 in series with a
standard capacitor C1.
The impedance of the arms of the
bridge can be written as
j
Z1  R1 
 C1
Z 2  R2
Z 3  R3
Z x  Rx  j Lx
Fig. 5-16: Opposite-angle bridge
At balance: Z1Zx = Z2Z3, and substituting the values in the balance
equation we obtain

j 
 R1 
Rx  j Lx   R2 R3
 C1 

Lx jRx
R1 Rx  
 j Lx R1  R2 R3
C1  C1
Equating the real and imaginary terms we have
Lx
R1 Rx 
 R2 R3
C1
(7.1)
Rx
  Lx R1
 C1
(7.2)
and
Solving for Rx we have, Rx = ω2LxC1R1.
Substituting for Rx in Eq.7.2,
Lx
R1 ( R1C1 Lx ) 
 R2 R3
C1
2
Lx
 R C1 Lx   R2 R3
C1
2
2
1
Multiplying both sides by C1 we get
 R C Lx  Lx  R2 R3C1
2
2
1
2
1
Therefore,
Lx 
R2 R3C1
1   R1 C1
2
2
2
(7.3)
Substituting for Lx in Eq.7.3 into Eq.7.2, we obtain
R1 R2 R3C1
Rx 
2
2
2
1   R1 C1
2
The term ω in the expression for both Lx and Rx indicates that the
bridge is frequency sensitive.
(7.4)
The Wien bridge is an ac bridge having a series RC combination in one
arm and a parallel combination in the adjoining arm.
In its basic form, Wien’s bridge is
designed to measure either the
equivalent-parallel components
or the equivalent-series
components of an impedance.
The impedance of the arms of this
bridge can be written as:
Z1 = R1
Z2 = R2
Fig 5-17: Wien Bridge
The impedance of the parallel arm is
Z3 
1
1
 j  C3
R3
The impedance of the series arm is
j
Z 4  R4 
 C4
Fig 5-17: Wien Bridge
Using the bridge balance equation, Z1Z4 = Z2Z3 we obtain:
Equivalent parallel components
R1
R3 
R2



1
R2 
1
 R4 
 (8.1) C 3 

C 4 (8.2)
2
2
2
2
1  2R C 


R

R
C
1 
4
4 
4 4 

Equivalent series components
R2
R4 
R1


R3


 1   2 R 2C 2 
3
3 

 
R12 
1
1
C 
C4 (8.4)
C43 
(8.3) C
2
2
3
2 2

2 2 
R
1


R
C
R21 
 4R3 4C 3 
Knowing the equivalent series and parallel components, Wien’s
bridge can also be used for the measurement of a frequency.
1
f 
2 R3C3 R4C4
(8.5)
The radio-frequency bridge is an ac bridge used to measure the
impedance of both capacitance and inductance circuits at high
frequency. For determination of impedance:
This bridge is first balanced with the Zx shorted. After the values of C1
and C4 are noted, the unknown impedance is inserted at the Zx
terminals, where Zx = Rx ± jXx.
•Rebalancing the bridge gives new values of C1 and C4, which can be
used to determine the unknown impedance by the following formulas:
R3 '
Rx 
(C1  C1 )
C2
(9.1)
1 1
1 
X x   '  
  C4 C 4 
(9.2)
Notice that Xx can be either capacitive or inductive. If C’4 > C4, and thus
1/C’4 < 1/C4, then Xx is negative, indicating a capacitive reactance.
Therefore,
Lx 
Xx

(9.3)
However, if C’4 < C4, and thus 1/C’4 > 1/C4, then Xx is positive and
inductive and
1
Cx 
Xx
Thus, once the magnitude and sign of Xx are known, the value of
inductance or capacitance can be found.
(9.4)
Schering bridge is a very important AC bridge used for precision
measurement of capacitors and their insulating properties. Its basic
circuit arrangement given in Figure 5-19 shows that arm 1 contains a
parallel combination of a resistor and a capacitor.
The standard capacitor C3 is a
high quality mica capacitor for
general measurements, or an
air capacitor for insulation
measurements.
A high quality mica capacitor has
very low losses (no resistance)
and an air capacitor has a very
stable value and a very small
electric field.
Fig 5-19: Schering bridge
The impedance of the arms of the Schering bridge can be written as
Z1 
1
1
1

R1  jX C1
Z 2  R2
Z 3   jX C 3
Z 4  Rx  jX x
Fig 5-19: Schering bridge
Substituting these values into general balance equation gives:
Z 2Z3
Z4 
Z1
R2 ( jX C 3 )
Rx  jX x 
1
1
1

R1  jX C1
1
j
1 

Rx 
 R2 ( jX C 3 ) 
Cx
 R1  jX C1 
j
R2C1
jR2
Rx 


Cx
C3
 C3 R1
Equating the real and imaginary terms, we find that
C1
Rx  R2
C3
(10.1)
R1
C x  C3
R2
(10.2)