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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 Bridgeb) Kelvin Bridge
2) AC bridge:a) Similar Angle Bridgeb) Opposite Angle Bridge/Hay Bridgec) Maxwell Bridged) Wein Bridgee) Radio Frequency Bridgef) Schering Bridge
Figure 5.1: Wheatstone Bridge Circuit
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.
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
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”.
R3 is varied until voltage between the two midpoints (B and D) will be
zero and no current will flow through the galvanometer.
Figure 5.1: Wheatstone Bridge Circuit
A
B
C
D
Figure 5.2: A variable resistor; the amount of resistance between the connection terminals could be varied.
Figure 5.1: Wheatstone Bridge Circuit
A
B
C
D
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
Figure 5.1: Wheatstone Bridge Circuit
A
B
C
D
2
4
1
3
R
R
R
R
1
234 R
RRR
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;
Example 1
Find Rx?
Figure 5.3
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:
A
radiansS
orA
reesS
orA
etersmilS
I
D
Current
DeflectionS
;deg
;lim
How to find the current value?
Figure 5.4.
Fig. 5.5: Thevenin’s equivalent voltage
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.
31
3
RR
REVa
Applying the voltage divider equation, we express the voltage at point a and b, respectively, as
42
4
RR
REVb
42
4
31
3
RR
R
RR
REVVV baTh
Thevenin’s Theorem (Cont…) The difference in Va and Vb represents Thevenin’s equivalent voltage. That is,
Fig. 5.5: Wheatstone bridge with the galvanometer removed
Fig. 5.6: Thevenin’s resistance
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:
Thevenin’s Theorem (Cont…)
Fig. 5.6: Thevenin’s resistance
4231 //// RRRRRTh
42
42
31
31
RR
RR
RR
RRRTh
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…)
Fig. 5.7: Thevenin’s equivalent circuit
If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is
gTh
Thg RR
VI
where Rg = the internal resistance in the galvanometer
R2 = 1.5 kΩR1 = 1.5 kΩ
R3 = 3 kΩR4 = 7.8 kΩ
Rg = 150 Ω
GE= 6 V
Figure 5.8 : Unbalance Wheatstone Bridge
Calculate the current through the galvanometer ?
Example 2
Slightly Unbalanced Wheatstone Bridge
22
E
R
RE
RR
REVa
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
The voltage at point b is expressed as
rRR
rREVb
Figure 5.9: Wheatstone Bridge with three equal arms
Thevenin’s equivalent voltage is the difference in this voltage
rR
rE
rRR
rREVVV abth 242
1
If ∆r is 5% of R or less, Thevenin equivalent voltage can be simplified to be
R
rEVth 4
Slightly Unbalanced Wheatstone Bridge (Cont…)
rRR
rRRRRTh
))((
2
If ∆r is small compared to R, the equation simplifies to
22
RRRth RRth or
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
o o
R
R
R
R + Δr
Slightly Unbalanced Wheatstone Bridge (Cont…)
Figure 5.10: Resistance of a Wheatstone.
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
Slightly Unbalanced Wheatstone Bridge (Cont…)
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
At balance the ratio of Rb to Ra must be equal to the ratio of R3 to R1. Therefore,
1
32
R
RRRx
1
3
2 R
R
R
Rx or
a
bx
R
R
R
R
R
R
1
3
2