ELECTRICAL MEASUREMENT AND
INSTRUMENTATION
1.MAXWELL'S BRIDGE FOR MEASURING UNKNOWN INDUCTANCE OF AN INDUCTOR
2.HAY’S BRIDGE
AADITYA SHARMA 140020204011
MAXWELL'S
BRIDGE
•
A Maxwell's bridge is a type of Wheatstone bridge used to measure
an unknown inductance. Maxwell's bridge can be used to measure
inductance by comparison either with a variable standard self
inductance or with a standard variable capacitance. These two
measurement can be done by using the Maxwell's bridge in two
different forms.
•
Maxwell's Inductance Bridge
MAXWELL’S INDUCTANCE BRIDGE
• Different types of AC Bridges can be used for
measurement of either inductance or capacitance. We can measure unknown inductance by using MAXWELL’S INDUCTANCE BRIDGE.
• CONSTRUCTION:
• There are four different arms in which:
• First arm of the bridge contains series combination of resistance and inductance.
• Third arm consists of series combination of variable resistance and variable inductance.
• Second and fourth arm consists of resistances respectively.
WORKING OF THE BRIDGE
• This coil is connected to AC supply when alternating current flows through the coil according to magnetic effect , it produces magnetic flux lines.
• When these magnetic flux lines are cut by coil1 then according to FARADAY’s LAW of ELECTROMAGNETIC INDUCTION emf is induced in coil1.This emf is called as SELF INDUCED emf ,denoted by E1.
• Mathematically E1=-L1(di/dt),where L is called as ‘SELF INDUCTANCE OF COIL’.
WORKING OF BRIDGE (CONTINUED)
Similarly in this figure there are 2 different coils in which coil1 is connected to AC supply and coil2 gets open.
When AC supply is given then according to the magnetic effect, magnetic flux lines are produced but some of the magnetic lines cut by coil2and all the magnetic lines cut by coil1.
Hence according to FARADAY’S LAW emf will be induced in both the coils. The emf induced in coil1 is called as SELF INDUCED EMF and the emf induced in coil2 is called as MUTUALLY INDUCED EMF.
Using MAXWELL’S INDUCTANCE BRIDGE we can measure only SELF INDUCTANCE of coil1
DERIVATION
•
In adjacent figure ,
parameters L1 and R1 are
unknown parameters.
•
By using MAXWELL’S
INDUCTANCE BRIDGE we
have to find out these
DERIVATION
The mathematical balancing condition of AC bridge is:
---(1) Where,
Z1=(R1+jωL1) Z2=R2
Z3=(R3+jωL3+r3) Z4=R4
Substitute the values in eq.(1)
R1R4+jωL1R4= R2(R3+r3)+jωL3R2
FINAL EQUATION
•
Real part--- R1R4=R2(R3+r3)•
Imaginary part---L1R4=L3R2
From these equations :
The above 2 equations are FREE FROM FREQUENCY i.e the values of unknown resistance and unknown inductance does not depend upon the frequency of supply. So that ANY TYPE OF DETECTOR can be used
R1=(R3+r3)
L1=
EXAMPLE OF MAXWELL’S INDUCTANCE BRIDGE
The arms of an a.c Maxwell bridge are arranged as follows: AB and AC are non-reactive resistor of 100ohm each. DA is a standard variable reactor L, of 32.7ohm and CD comprises a standard variable resistor R in series with a coil of unknown impedance. Balance was obtained with L1 =47.8mH and R=1.36ohm. Find the resistance and inductance of the coil.
MAXWELL'S INDUCTANCE
CAPACITANCE BRIDGE
•
Using this bridge , we can measure inductance by comparing with a variable standard capacitor.The bridge circuit diagram is as shown in figure. One of the ratio arms consist of resistance and capacitance in parallel. Hence it is simple to write the bridge equation in the impedance form.Impedance Z1 =R1 + jL1. Impedance Z2 = R2.
Impedance Z3 = R3. Impedance Z4 =
= = For balanced condition of bridge
ZZ = ZZ OR
= R2R3 Or
R1R4 +L1R4 = R2R3 + C4R4R2R3
Equating real and imaginary quantities separately we have R1R4 = R2R3
and
L1R4 = C4R4R2R3
or
L1 = C4R2R3 and R1=R2*R3/R4
The bridge is perfectly balanced by varying C4 and R4 which gives independent settings. Now the quality factor is given by,
EXAMPLE OF MAXWELL’S INDUCTANCE
CAPACITANCE BRIDGE
•
The arms of an a.c Maxwell bridge are arranged as follows: AB is a
non-inductive resistance of 1,00 ohm in parallel with a capacitor of
capacitance 0.5,BC is a non-inductive resistance of 600ohm CD is
an inductive impedance(unknown) and DA is a non-inductive
resistance of 400ohms.If balance is obtained under these
conditions, find the value of the resistance and the inductance of
the branch CD.
ADVANTAGES OF MAXWELL'S BRIDGE
• The frequency does not appear in the final expression of both equations, hence it is independent of frequency.
• Maxwell's inductor capacitance bridge is very useful for the wide range of measurement of inductor at audio frequencies
DISADVANTAGE OF MAXWELL’S BRIDGE
• The variable standard capacitor is very expensive.
• The bridge is limited to measurement of low quality coils (1 < Q < 10) and it is also unsuitable for low value of Q (i.e. Q < 1) from this we conclude that a Maxwell bridge is used suitable only for medium Q coils.
The above all limitations are overcome by the modified bridge which is known as Hey's bridge which does not use an electrical resistance in parallel with the capacitor.
HAY’S BRIDGE
•
Hay’s bridge is a modification to Maxwell bridge.
•
It is useful for measuring high quality factor i.e Q.
•
The modified Maxwell’s bridge or Hay’s bridge is suitable for
measuring Q factor over a wide range.
The circuit arrangement is shown
in the fig.
In the
fig:-L₁ = self inductance
R₁ = resistance of coil under test
R₂, R₃ & R₄ = known non inductive
resistances
C₄ = standard variable capacitor
R₄ and C₄ are in series unlike in
Maxwell’s bridge
.When the bridge is balanced
I₂ = I₁ ; I₄ = I₃ ; V₁ = V₃ and V₂ = V₄
Since ,
V₁ = I₁ Z₁ = I₁ (R₁ + jω L₁ ) and
V₃ = I₃ R₃ ,
I₁ (R₁ + jω L₁ ) = I₃R₃ ……….(1) and
V₂ = I₂R₂ = I₁R₂
V₄ = I₄Z₄ = I₃ (R₄ - j/ ω C₄ )
I₁R₂ = I₃(R₄ - j/ ω C₄ ) ………...(2)
Dividing expression (1) by (2) we get ,
(R₁ + jω L₁)/ R₂ = R₃ / (R₄ - j/ ω C₄ )
Separate real and imaginary values :
R₁R₄ + L₁/C₄ = R₂R₃ ……….(4) and
ωL₁R₄ - R₁/ωC₄ = 0
R₁ = ω²C₄L₁R₄ ………(5) solving equ. (4) & (5)
L₁ = R₂R₃C₄ /(1+ω²C₄²R₄²) ………..(6)
and
R₁ = ω²C₄R₄ x R₂R₃C₄ /(1+ω²C₄²R₄²)
R₁ = R₂R₃R₄C₄²ω²/(1+ω²C₄²R₄²) ……….(7)
Q- factor of the coil (Tan¯¹ ωL/R) :
Q = ωL₁/R₁ = ωR₂R₃C₄/(1+ω²C₄²R₄²)
R₂R₃R₄C₄²ω²/(1+ω²C₄²R₄²)
1/ωR₄C₄ ………(8)
In equⁿ (6) and (7) the term ω²C₄²R₄² is very small as compared to unity. If we compute equⁿ (8) in equⁿ (6) we get:
L₁ = R₂R₃C₄ /[1 + ( 1/Q)²] ………(9)
If Q = 10 , then ωC₄R₄ will be 0.01
For values of Q > 10, the term ωC₄R₄ will be smaller than 0.01 and so it can be neglected. Therefore L₁ = R₂R₃C₄ which is same for Maxwell’s bridge.
