DC and AC Bridges
DC and AC Bridges
DC and AC Bridges
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.
D B
C
Figure 5.2:
Figure 5.1:
A variable resistor; the amount of
Wheatstone Bridge Circuit resistance between the connection
terminals could be varied.
When the bridge is in balanced
A
condition, we obtain,
I1 = I 3 (2.3)
D B
I2 = I 4 (2.4)
R3 R4 R2
or R4 R3 (2.5)
R1 R2 R1
Example 1
Figure 5.3
Deflection D
S (2.6)
Current I
mil lim eters
S or ;
A
deg rees
S or ;
A
radians
S
A
R3 R4
Va E (2.7) Vb E (2.8)
R1 R3 R2 R4
A
(Cont…..)
The difference in Va and Vb represents
Thevenin’s equivalent voltage. That is, D B
R3 R4
VTh Va Vb E (2.9) C
R1 R3 R2 R4 Fig. 5.5: Wheatstone bridge
with the galvanometer removed
R1 R3 R2 R4
RTh (2.10)
R1 R3 R2 R4
(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 (2.11)
RTh Rg Fig. 5.7:
Thevenin’s equivalent circuit
R1 = 1.5 kΩ R2 = 1.5 kΩ
Rg = 150 Ω
E= 6 V G
R3 = 3 kΩ R4 = 7.8 kΩ
R R E
Va E E (2.12)
RR 2R 2
R r
Vb E (2.13)
R R r
Figure 5.9: Wheatstone Bridge with
three equal arms
Slightly Unbalanced Wheatstone Bridge (Cont…)
R r 1 r
VTh Vb Va E E
R R r 2 4 R 2r
r
VTh E (2.14)
4R
(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 r )
RTh o o
2 R R r
If ∆r is small compared to R, R R + Δr
the equation simplifies to Figure 5.10:
Resistance of a Wheatstone.
R R
RTh or RTh R (2.15)
2 2
(Cont…..)
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).
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 C
R2 R3 Rx R3
Rx or D
R1 R2 R1
If the galvanometer is connected to
point B, the ratio of Rb to Ra must be Fig. 5.12:
equal to the ratio of R3 to R1. Basic Kelvin Bridge showing
Therefore, a second set of ratio arms
Rx R3 Rb
(3.1)
R2 R1 Ra
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.
I1Z1 I 2 Z 2 (4.1)
I1Z 3 I 2 Z 4 (4.2)
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)
I (1 4 ) Z 2 Z 3( 2 3 )
Z1Z 4 (4.4)
f
t
Eq. 4.4 shows twohconditions when ac bridge is balanced;
First condition shows
e that the products of the magnitudes of the
opposite arms musti be equal: Z1Z4 = Z2Z3
Second condition m shows that the sum of the phase angles of the
p
opposite arms is equal: ∟θ1+ ∟θ4 = ∟θ2+ ∟θ3
e
d
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.
Z1 R1
Z 2 R2
Z 3 R3 in series with C3
j
R3
C3
Z x Rx in series with C x
Fig. 5-15: Similar-angle bridge
j
Rx
Cx
The condition for balance of the bridge is
Z1 Z x Z 2 Z 3
j j
R1 Rx R2 R3
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 (5.1)
R1
and,
jR1 jR2
C x C3
or
C3 R1
Cx (5.2)
R2
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.
Capacitance of capacitor is
influenced by less external
fields.
Capacitor has small size.
Capacitor is low cost.
1
Z1
1
j C1
R1
Z 2 R2
Z 3 R3
Z 4 Rx j X Lx
Z1 Z x Z 2 Z 3
1
1
Rx j X Lx R2 R3
j C1
R1
1
1 j C1
Rx j X Lx R2 R3
R1
Rx j X Lx R2 R3
1 j C1 R1
Fig. 5-15: Maxwell Bridge
R2 R3
Rx j X Lx j R2 R3C1
R1
Equating real terms and imaginary terms we have
R2 R3
Rx (6.1)
R1
j Lx jR2 R3C1
Lx R2 R3C1 (6.2)
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 (7.1)
C1
and
Rx
Lx R1 (7.2)
C1
Solving for Rx we have, Rx = ω2LxC1R1.
Substituting for Rx in Eq.7.2,
Lx
R1 ( R1C1 Lx )
2
R2 R3
C1
Lx
R C1 Lx R2 R3
2
1
2
C1
Multiplying both sides by C1 we get
R C Lx Lx R2 R3C1
2
1
2
1
2
Therefore,
R2 R3C1
Lx (7.3)
1 R1 C1
2 2 2
R1 R2 R3C1 2
Rx (7.4)
1 R1 C1
2 2 2
The term ω in the expression for both Lx and Rx indicates that the
bridge is frequency sensitive.
The Wien bridge is an ac bridge having a series RC combination in one
arm and a parallel combination in the adjoining arm.
Z1 = R1 Z2 = R2
1
Z3
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:
R1 1 R2 1
R3 R4 (8.1) C 3 C 4 (8.2)
R2 2 2 R 1 2R C
2 2
R 4 4
C 1 4 4
R2 R3 R12 1
R4 C43 C 1 C4 (8.4)
1 2 R 2C 2 (8.3) C 2 2
4R3 4C 3
3 2 2 2 2
R1 3 3
R
R21 1 R C
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 ) (9.1)
C2
1 1 1
X x ' (9.2)
C4 C4
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,
Xx
Lx (9.3)
However, if C’4 < C4, and thus 1/C’4 > 1/C4, then Xx is positive and
inductive and
1
Cx
Xx (9.4)
Thus, once the magnitude and sign of Xx are known, the value of
inductance or capacitance can be found.
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.
1
Z1
1 1
R1 jX C1
Z 2 R2
Z 3 jX C 3
Z 4 Rx jX x
Z 2Z3
Z4
Z1
R2 ( jX C 3 )
Rx jX x
1
1 1
R1 jX C1
j 1 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 (10.1)
C3
R1
C x C3 (10.2)
R2