1. CHAPTER 5
DC AND AC BRIDGES

2. 1. Introduction
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.

3. 1. Introduction 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

4. DIRECT-CURRENT (DC)BRIDGE

5. 2. Wheatstone 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.

6. 2. Wheatstone 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

7. 2. Wheatstone Bridge
When no current flows through the galvanometer, the bridge is called in a balanced condition. 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

8. 2. Wheatstone Bridge I1R1 = I2R2 I3R3 = I4R4
When the bridge is in balanced condition, we obtain,
voltage drops across R1 and R2
are equal,
I1R1 = I2R2
voltage drops across R3 and R4
are also equal,
I3R3 = I4R4 A B C D
(2.1)
(2.2)
Figure 5.1:
Wheatstone Bridge Circuit

9. 2. Wheatstone Bridge I1 = I3 I2 = I4 In this point of balance, we also
obtain;
I1 = I3
I2 = I4
Therefore, the ratio of two resistances in the known leg is equal to the ratio of the two in the unknown leg;
(2.3)
(2.4)
Figure 5.1:
Wheatstone Bridge Circuit or
(2.5)

10. 2. Wheatstone Bridge Example 1 Find Rx at the balance condition?
Figure 5.3
Find Rx at the balance condition?

11. 2. Wheatstone Bridge 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.

12. 2. Wheatstone Bridge (Cont…..) How to find the current value?
Sensitivity S can be expressed in linear or angular units as follows:
(2.6)
How to find the current value?
Figure 5.4.

13. 2. Wheatstone Bridge 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
(2.7)
(2.8)

14. Fig. 5.6: Thevenin’s resistance circuit
2. Wheatstone Bridge A B C D
(Cont…..)
The difference in Va and Vb represents Thevenin’s equivalent voltage. That is,
(2.9)
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

15. 2. Wheatstone Bridge (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:
Fig. 5.6:
Thevenin’s resistance circuit
(2.10)

16. 2. Wheatstone Bridge (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

17. 2. Wheatstone Bridge (Cont…)
If a galvanometer is connected to terminal a and b, the deflection current in the galvanometer is
(2.11)
Fig. 5.7:
Thevenin’s equivalent circuit
where Rg = the internal resistance in the galvanometer.

18. 2. Wheatstone Bridge Example 2
E= 6 V
Figure 5.8 : Unbalance Wheatstone Bridge
R2 = 1.5 kΩ
R1 = 1.5 kΩ
Rg = 150 Ω
R3 = 3 kΩ
R4 = 7.8 kΩ
Calculate the current through the galvanometer ?

19. 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
(2.12)
The voltage at point b is expressed as:
(2.13)
Figure 5.9: Wheatstone Bridge with three equal arms

20. 2. Wheatstone Bridge Slightly Unbalanced Wheatstone Bridge (Cont…)
Thevenin’s equivalent voltage is the difference in this voltage
If ∆r is 5% of R or less, Thevenin equivalent voltage can be simplified to be
(2.14)

21. 2. 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 Thevenin’s equivalent resistance is now given as o R
R + Δr
If ∆r is small compared to R, the equation simplifies to
Figure 5.10:
Resistance of a Wheatstone. or
(2.15)

22. 2. 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

23. 3. Kelvin Bridge 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.

24. 3. Kelvin 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

25. 3. Kelvin Bridge
When a null exists, the value for Rx is the same as that for the Wheatstone bridge, which is C or D
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,
Fig. 5.12:
Basic Kelvin Bridge showing a second set of ratio arms
(3.1)

26. ALTERNATING-CURRENT (AC)BRIDGES

27. 4. AC Bridge Introduction
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 frequency-sensitive types may be used as frequency measurement devices if all component values are accurately known.
Fig 5.13: General AC bridge circuit

28. 4. AC Bridge Introduction
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

29. 4. AC Bridge Introduction
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

30. 4. AC Bridge Introduction
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

31. 4. AC Bridge Introduction
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:
(4.1)
Similarly, the voltages from point d to point b and point d to point c must also be equal, leading to:
Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit
(4.2)

32. 4. AC Bridge Introduction
Dividing Eq. 4.1 by Eq. 4.2, we obtain:
which can also be written as
(4.3)
Fig. 5-14: Equivalent of balanced (nulled) AC bridge circuit

33. 4. AC Bridge Introduction
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 or
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. 20 can be written in the form
(4.4)
Eq. 4.4 shows two conditions when ac bridge is balanced;
First condition shows that the products of the magnitudes of the
opposite arms must be equal: Z1Z4 = Z2Z3
Second condition shows that the sum of the phase angles of the
opposite arms is equal: ∟θ1+ ∟θ4 = ∟θ2+ ∟θ3

34. Fig. 5-15: Similar angle bridge
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. R2
C3 = a known standard capacitor
in series with R3.
Cx = an unknown capacitor.
Fig. 5-15: Similar angle bridge

35. 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
Fig. 5-15: Similar-angle bridge

36. 5. Similar-Angle Bridge The condition for balance of the bridge is
Two complex quantities are equal when both real and imaginary terms are equal. Therefore, or

37. 5. Similar-Angle Bridge
(5.1)
and, or
(5.2)

38. 6. Maxwell Bridge
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

39. 6. Maxwell Bridge
The impedance of the arms of the bridge can be written as
Fig. 5-15: Maxwell Bridge

40. 6. Maxwell Bridge The general equation for bridge balance is
Fig. 5-15: Maxwell Bridge

41. 6. Maxwell Bridge Equating real terms and imaginary terms we have
(6.1)
(6.2)
Fig. 5-15: Maxwell Bridge

42. 7. Opposite-Angle 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
Fig. 5-16: Opposite-angle bridge

43. 7. Opposite-Angle Bridge
At balance: Z1Zx = Z2Z3, and substituting the values in the balance equation we obtain
Equating the real and imaginary terms we have
(7.1)
and
(7.2)

44. 7. Opposite-Angle Bridge
Solving for Rx we have, Rx = ω2LxC1R1.
Substituting for Rx in Eq.7.2,
Multiplying both sides by C1 we get

45. 7. Opposite-Angle Bridge
Therefore,
(7.3)
Substituting for Lx in Eq.7.3 into Eq.7.2, we obtain
(7.4)
The term ω in the expression for both Lx and Rx indicates that the bridge is frequency sensitive.

46. 8. Wien Bridge
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 = R Z2 = R2
Fig 5-17: Wien Bridge

47. 8. Wien Bridge Fig 5-17: Wien Bridge
The impedance of the parallel arm is
The impedance of the series arm is
Fig 5-17: Wien Bridge

48. 8. Wien Bridge
Using the bridge balance equation, Z1Z4 = Z2Z3 we obtain:
Equivalent parallel components
(8.1)
(8.2)
Equivalent series components
(8.3)
(8.4)

49. 8. Wien Bridge
Knowing the equivalent series and parallel components, Wien’s bridge can also be used for the measurement of a frequency.
(8.5)

50. 9. Radio-Frequency Bridge
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:
(9.1)
(9.2)

51. 9. Radio-Frequency Bridge
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,
(9.3)
However, if C’4 < C4, and thus 1/C’4 > 1/C4, then Xx is positive and inductive and
(9.4)
Thus, once the magnitude and sign of Xx are known, the value of inductance or capacitance can be found.

52. 10. Schering Bridge
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

53. 10. Schering Bridge
The impedance of the arms of the Schering bridge can be written as
Fig 5-19: Schering bridge

54. 10. Schering Bridge
Substituting these values into general balance equation gives:

55. 10. Schering Bridge
Equating the real and imaginary terms, we find that
(10.1)
(10.2)