Transcript Wheatstone bridge circuit

CHAPTER 5
 Bridge circuits (DC & AC) are 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.
TWO (2) 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.
It consists of a voltage source
and a galvanometer that
connects two parallel branches,
containing four resistors.
Figure 5.1: Wheatstone Bridge Circuit
One parallel branch contains one known resistance and one
unknown; the other parallel branch contains resistors of known
resistances.
In the circuit at right, 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 the other three
are adjusted and balanced until
the current passing through the
galvanometer decreases to zero”.
Figure 5.1: Wheatstone Bridge Circuit
R3 is varied until voltage between the two midpoints (B and D) will be
zero and no current will flow through the galvanometer.
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.
A
When the bridge is in balance
condition (no current flows through
galvanometer G), we obtain;
 voltage drop across R1 and R2 is
equal,
I1R1 = I2R2
 voltage drop across R3 and R4 is
equal,
I3R3 = I4R4
D
B
C
Figure 5.1: Wheatstone Bridge Circuit
A
 In this point of balance, we also
obtain;
I1 = I3
and
I2 = I4
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
R2
R4  R3
R1
D
B
C
Figure 5.1: Wheatstone Bridge Circuit
Example 1
Figure 5.3
Find Rx?
Sensitivity of the Wheatstone Bridge
When the pointer of a bridge
galvanometer deflects to right
or to left direction, 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.
Sensitivity of the Wheatstone Bridge (Cont…)
Sensitivity S can be expressed in units of:
S
S
S
S
Deflection D


Current
I
mil lim eters

or;
A
deg rees

or;
A
radians

A
How to find the current
value?
Figure 5.4.
Thevenin’s Theorem
Thevenin’s theorem is a 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
R4
Vb  E
R2  R4
Thevenin’s Theorem (Cont…)
The difference in Va and Vb represents
Thevenin’s equivalent voltage. That is,
 R3
R4 

VTh  Va  Vb  E

 R1  R3 R2  R4 
Thevenin’s equivalent resistance is found
by replacing the voltage source with its
internal resistance, Rb. 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
equivalent resistance as follows:
Fig. 5.5: Wheatstone bridge
with the galvanometer removed
Fig. 5.6: Thevenin’s resistance
Thevenin’s Theorem (Cont…)
RTh  R1 // R3  R2 // R4
R1 R3
R2 R4
RTh 

R1  R3 R2  R4
Fig. 5.6: Thevenin’s resistance
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
Thevenin’s Theorem (Cont…)
If a galvanometer is connected to
terminal a and b, the deflection current
in the galvanometer is
VTh
Ig 
RTh  Rg
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 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 Fig- 5.9, the voltage at point a is given as
R
 R  E
Va  E
 E

RR
 2R  2
The voltage at point b is expressed as
R  r
Vb  E
R  R  r
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 
Slightly Unbalanced Wheatstone Bridge (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 )
RTh  
2 R  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
R + Δr
Figure 5.10: Resistance of a Wheatstone.
Slightly Unbalanced Wheatstone Bridge (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 resistance when
measuring unknown low
resistances.
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.
Fig. 5.12: Basic Kelvin Bridge showing a second set of ratio arms
The resistor Rlc represents the lead and contact resistance
present in the Wheatstone bridge.
The second set of ratio arms (Ra and Rb in figure) compensates
for this relatively low lead-contact resistance.
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
At balance the ratio of Rb to Ra must be equal to the ratio of R3 to R1.
Therefore,
Rx R3 Rb


R2 R1 Ra