AC bridge and DC Circuit.

1. AC bridge and DC Circuit
⦿ Unit-1
⦿ B.Sc. Semester-2
⦿ Prepared by: Jay Kothari

2. Outline
⦿ AC Bridge and DC Circuit
⦿ A.C.Bridge introduction and general Bridge balance equation
⦿ Maxwell Bridge
⦿ Owen's Bridge
⦿ De-sautty Bridge
⦿ Anderson Bridge
⦿ R.L. Circuit in series growth and decay
⦿ R.C. circuit in series growth and decay Series LCR circuit and
its analysis and condition of oscillation
⦿ Quality factor
⦿ Examples

3. AC bridge introduction and general
bridge balance equation.
AC બ્રીજની સામાન્ય માહિતી તથા બ્રીજ
બેલેન્સ માટેનું વ્યાપક સમીકરણ મેળવો

4. ⦿ પ્રેરકનું આત્મપ્રેરણ, સુંગ્રાહકની ક્ષમતા, વગેર ે
જાણવા માટે A.C. બ્રીજનો ઉપયોગ થાય છે.
⦿ Definition: AC bridges are the circuits that
are used for the measurement of electrical
quantities such as inductance, capacitance,
resistance. Along with these an ac bridge
allows us to measure storage factor, loss
factor, dissipation factor etc. AC bridges
operate with only AC signal.
⦿ વાસ્તવમાું A.C. બ્રીજ એ વ્હહસ્ટન બ્રીજના
મૂળભૂત ખ્યાલમાુંથી ઉદ્ભવ પામ્યો છે.
વ્હહસ્ટન બ્રીજની જેમ જ A.C. બ્રીજમાું ચાર
બાજઓ હોય છે પરુંત આ ચાર ેબાજઓમાું ફકત
અવરોધના બદલે અમક અમક બાજઓમાું
અવબાધ આવેલા છે. ઉપરાુંત હહીસ્ટન
વ્બ્રીજમાું D.C પ્રવાહનો ઉપયોગ થાય છે. જયાર ે
A.C. બ્રીજમાું A.C. પ્રવાહનો ઉપયોગ થાય છે.

5. ⦿ બીજા ફેરફારોમાું વ્હહસ્ટન બ્રીજમાું
ગેલ્વેનોમીટર વાપરવામાું આવે છે. જયાર ે
A.C. બ્રીજમાું યોગ્ય ડીટેકટર વાપરવામાું
આવે છે. સામાન્ય રીતે A.C. પ્રવાહ (સોસસ)
તરીકે ઈલેકટરોનીક ઓસ્સીલેટરનો ઉપયોગ
થાય છે. આવા ઓસ્સીલેટર (આુંદોલકો)
10Hz થી 100 kHz જેટલી આવૃવ્િ ઉત્પન
કરી શકે છે. A.C. બ્રીજમાું વપરાતા
ડીટેકોમાું મખ્યત્વે હેડ ફોન, કુંપન
ગેલ્વેનોમીટર હોય છે. જેમાુંથી કોઈ એકને
પસુંદ કરવામાું આવે છે. જો શ્રાહય આવૃવ્િ
ઉત્પન્ન કરતા A.C. સોસસનો ઉપયોગ થઈ
શકે તેમ હોય તો ડીટેકટર તરીકે હેડફોન
મજબ ઉપયોગી બને છે.

6. AC Bridge network construction
⦿ An AC bridge consists of 4 nodes
with 4 arms, a source excitation and a
balanced detector. Each of the 4 arms
of the bridge consists of impedance.
⦿ AC Bridges source and detector in an
AC bridge network are connected in
opposite nodes. This is so because if
source and detector are connected to
the same node, all the voltage or
current of the source will be displayed
at the detector. So, in this condition, the
bridge will never come into balance
conditions.

7. General Equation for AC Bridge
Balance:
⦿ The below figure shows a basic ac bridge.The four
arms of the bridge are impedances Z1,Z2,Z3 & Z4.
⦿ Basically, there are 2 conditions in order to balance the
bridge.
⦿ 1)The detector current Id should be zero.
⦿ 2)The potential difference between the detector node
should be zero.
⦿ This requires that the potential difference between points
b and d should be zero. This will be the case when the
voltage drop from a to b equals to voltage drop from a to
d, both in magnitude and phase.
⦿ In complex notation we can, thus, write :

8. Substituting the value of
I1 and I2
The above equation is the basic equation for a
balanced AC bridge.

9. ⦿ Equation Z1 Z4 = Z2 Z3 states that the product of impedances of one pair
opposite arms must equal the product of impedances of the other pair of
opposite arms expressed in complex notation. This means that both
magnitudes and the phase angles of the impedances must be taken into
account.
⦿ Considering the polar form, the impedance can be written as Z = Z∠θ,
where Z represents the magnitude and θ represent the phase angle of the
complex impedance.Now that equation can be re-written in the form
⦿ (Z1∠θ1)(Z4∠θ4) = (Z2∠θ2)(Z3∠θ3)
⦿ Thus for balance, we must have,
⦿ Z1 Z4 ∠θ1 + θ4 = Z2 Z3 ∠θ2 + θ3
⦿ The above equation shows that two conditions must be satisfied
simultaneously when balancing an ac bridge.The first condition is that the
magnitude of impedances satisfies the relationship :
Z1 Z4 = Z2 Z3 (magnitude criteria)
⦿ The second condition is that the phase angles of impedances satisfy the
relationship :
⦿ ∠θ1 + θ4 = ∠θ2 + θ3 (Phase criteria)
So,in a bridge balance condition, magnitude and phase criteria should be satisfied Simultaneously.

10. ⦿ The phase angles are positive for an
inductive impedance and negative for capacitive
impedance.
⦿ It is customary to use the angle by which the
voltage leads the current. This leads to a positive
phase for inductive circuits since current lags the
voltage in an inductive circuit. The phase is
negative for a capacitive circuit since the current
leads the voltage. The useful mnemonic ELI the
ICE man helps to remember the sign of the
phase. The phase relation is often depicted
graphically in a phasor diagram.

11. ⦿ 1)A.C.બ્રીજને સુંતલનમાું છે કે નહિ તે ચકાસવા માટેની
શરતો ધ્રવીયયામ અને કાતેઝીયનયામમા મેળવો
⦿ Derive the Conditions to Check A.C. Bridge balance, in
Polar and Cartesian coordinates.
⦿ (2) ડીસોટી બ્રીજ સમજાવો અને આપેલો A.C.બ્રીજ માટે
નીચે મજબના અવબાધ અને ખણા માટે સુંતલન
⦿ વ્સ્થવ્ત ચકાસો.
⦿ Z=400 ∟50°, Z2=200 ∟ 40°, Z3=800 ∟ -50, Z4=400
∟ 20°
⦿ Explain the Desauty Bridge and verify the given
A.C.bridge balance position from following
impedance and angle.
⦿ Z=400 ∟50°, Z2=200 ∟ 40°, Z3=800 ∟ -50, Z4=400
∟ 20°

12. ⦿ Z1 Z4 ∠θ1 + θ4 = Z2 Z3 ∠θ2 + θ3
⦿ (400*400 )(∟50+ ∟ 20) = 800*200 (∟ 40- ∟ 50)
⦿ Phasor algebra
⦿ Z1 Z4 = Z2 Z3 (magnitude criteria)
⦿ 1600=1600 so 1st condition is satisfied
⦿ ∠θ1 + θ4 = ∠θ2 + θ3 (Phase criteria)
⦿ ∟ 70 = ∟-10 so 2nd condition is not satisfied .
⦿ From above equation we can say that bridge is not in
balance.

13. Desautty Bridge
⦿ DeSauty's Bridge is the simplest
method of comparing
two capacitances. The connections
of DeSauty's Bridge are shown in the
below figure.
⦿ Let C1 = capacitor whose capacitance
is to be measured,
⦿ C2 = a standard capacitor, and
⦿ R3, R4 = non-inductive resistors.

14. The balance can be obtained by
varying either R3 or R4. The
advantage of DeSauty's Bridge is its
simplicity. But this advantage is
nullified by the fact that it is
impossible to obtain balance if both
the capacitors are not free from
dielectric loss. Thus with DeSauty's
Bridge, only loss-less capacitors like
air capacitors can be compared.
In order to make measurements
on imperfect capacitors (i.e.,
capacitors having a dielectric
loss), DeSauty's Bridge is modified
as shown in the below figure. This
modification is due to Grover. r1,r2= Representing losses of
their corresponding
capacitors.

15. ⦿ Z1Z4 = Z2Z3
⦿ (R1+1/jωC1)⋅R4=(R2+1jωC2)⋅R3
⦿
R1R4+R4/ jωC1=R2R3+R3/jωC2
⦿ Equating both the real and imaginary parts and
separate them,
Real Parts
R1R4 = R2R3
R1 = R2R3 / R4
Imaginary Parts
R4/ jωC1= R3/jωC2
R4/ C1= R3/C2
C1 = C2(R4/R3)
C1 / C2 = R4/ R3.

16. Maxwell’s Bridge
⦿ Definition: The bridge used for the measurement of self
inductance of the circuit is known as the Maxwell bridge. It
is the advanced form of the Wheatstone bridge. The Maxwell
bridge works on the principle of the comparison, i.e., the
value of unknown inductance is determined by comparing it
with the known value or standard value.
Types of Maxwell’s Bridge
Two methods are used for determining the self-
inductance of the circuit.

17. Maxwell’s
Bridge
Maxwell’s Induction
Bridge
મેકસવેલનો પ્રેરક બ્રીજ
Maxwell’s LC
Bridge
મેકસવેલનો LC બ્રીજ

18. Maxwell’s Induction Bridge
Bridgeમેકસવેલનો પ્રેરક બ્રીજ સમજાવો.
⦿ આ બ્રીજ દ્વારા કોઈ અજ્ઞાત પ્રેરકના
પ્રેરકત્વનું મૂલ્ય જાણી શકાય છે. િકીકતે
આ બ્રીજમાું એક ચલીત પ્રમાણભૂત પ્રેરક
સાથે અજ્ઞાતની સરખામણી કરવામાું
આવે છે. આકૃહિમાું મેકસવેલનો પ્રેરક
બ્રીજ દશાાવેલો છે.
Z1 = R1 + jωL1
Z2 = R2 + jωL2
Z3 = R3
Z4= R4
બ્રીજ બેલેન્સના વ્યાપક સમીકરણમાું
બધી જ હકુંમતો અવેજ કરતા.

19. ⦿ Z1Z4 = Z2Z3
⦿ (R1 + jωL1)⋅R4=(R2 + jωL2)⋅R3
⦿
R1R4+R4 ⋅ jωL1=R2R3+R3 ⋅ jωL2
⦿ Equating both the real and imaginary parts and
separate them,
Real Parts
R1R4 = R2R3
R1 = R2R3 / R4
Imaginary Parts
R4 ⋅ jωL1= R3 ⋅ jωL2
L1 = R3L2/ R4
સામાન્ય રીતે R૩તથા R4 ના મૂલ્યો સમાન રાખવામાું આવે છે તથા
R2 તથા L2 ના ક્રમીક ફેરફારો દ્વારા બ્રીજ બેલેન્સ કરવામાું આવે છે.

20. Maxwell LC bridge
મેકસવેલનો LC બ્રીજ સમજાવો.
⦿ આ બ્રીજ દ્વારા અજ્ઞા પ્રેરકનું મૂલ્ય કોઈ પ્રમાણભૂત
ચલીત સુંગ્રાિકની સરખામણી દ્વારા આપી શકાય છે.
આકૃહિમાું L1 = અજ્ઞાત
પ્રેરક
R1 = L1 નો આુંતહરક
અવરોધ
R2, R3 = જ્ઞાત અવરોધો
C = જ્ઞાત સુંગ્રાિક
R4 = ચલીત અવરોધ

23. ⦿ Definition:
⦿ The Anderson’s bridge gives
the accurate
measurement of self-
inductance of the circuit. The
bridge is the advanced
form of Maxwell’s inductance
capacitance bridge. In Anderson
bridge, the unknown
inductance is compared with
the standard fixed
capacitance which is connected
between the two arms of the
bridge.
Anderson’s Bridge

24. Constructions of Anderson’s
Bridge
⦿ The bridge has fours arms ab,
bc, cd, and ad. The
arm ab consists unknown
inductance along with
the resistance. And the other
three arms consist the purely
resistive arms connected in
series with the circuit.
⦿ The static capacitor and the
variable resistor are connected
in series and placed in parallel
with the cd arm. The voltage
source is applied to the
terminal a and c.

25. Let, L1 – unknown inductance having a resistance R1.
R2, R3, R4 – known non-inductive resistance
C4 – standard capacitor
At balance Condition,
Now Pot. At b =
Pot at e. KVL
Theory of Anderson Bridge
Ic/jωC – I1R3 = 0
-------(1)

26. Pot. diff. At branch = Pot. Diff. At branch
dc
ade
ba
dec
The other balance condition equation is expressed as
-------(3)
-------(2)

27. ⦿ By substituting the value of Ic in the equation (2) we get,
By Substituting value of Ic in the Equation (3) we get
-----(4)
-----(5)

28. on equating the equation, we get
Equating the real and the imaginary part, we get
CR3r

29. Equating Real Parts
Eq.(6) & (7) are independent of
each other.
-------(6)
----(7)

30. Disadvantages of Anderson Bridge
The main disadvantages of Anderson’s bridge are as follow.
• The circuit has more arms which make it more complex as compared
to Maxwell’s bridge. The equation of the bridge is also more complex.
• The bridge has an additional junction which arises the difficulty in
shielding the bridge.
Because of the above-mentioned disadvantages, Maxwell’s
inductance capacitance bridge is used in the circuit.
Advantages of Anderson Bridge
The following are the advantages of the Anderson’s Bridge.
• The balance point is easily obtained on the Anderson bridge as
compared to Maxwell’s inductance capacitance bridge.
• The bridge uses fixed capacitor because of which accurate reading
is obtained & because of fixed capacitor circuit becomes inexpensive.
• The bridge measures the accurate capacitances in terms of
inductances.

31. Owen’s Bridge
⦿ Definition: The bridge which measures the ind
uctance in terms of resistance and
capacitance is known as Owen’s bridge.
•It works on
the principle of comparison
i.e., the value of
the unknown
inductor is compared with
the standard capacitor. The
connection diagram of
Owen’s bridge is shown in
the figure.

32. ⦿ The bridge circuit is shown in
Fig. 13.62. L1 – unknown self-
inductance of resistance
⦿ R1,R2 – variable non-
inductive resistance,
⦿ R3 – fixed non-inductive
resistance
C2 – variable standard
capacitor,
⦿ C4 – fixed standard capacitor.
⦿ The coil whose self-
inductance L is to be
determined is connected in the
arm AD in series with R3

33. Theory of Owen’s Bridge
Let theimpedance in the four armsof
the bridge be Z1,Z2,Z3,Z4 at the
balance point

35. Advantages of Owen’s Bridge
The following are the advantages of Owen’s bridge.
• The balance equation is easily obtained.
• The balance equation is simple and does not contain any
frequency component
• The bridge is used for the measurement of the large range
inductance.
Disadvantages of Owen’s Bridge
• The bridge uses an expensive capacitor which increases
the cost of the bridge and also it gives a one percent
accuracy.
• The value of the fixed capacitor C2 is much larger than the
quality factor Q2