LITAR ELEKTRIK LITAR ELEKTRIK II II EET 102/4 EET 102/4
LITAR ELEKTRIK LITAR ELEKTRIK IIII
EET 102/4EET 102/4
SILIBUS LITAR SILIBUS LITAR ELEKTRIK IIELEKTRIK II
Mutual InductanceMutual Inductance Two port NetworkTwo port Network Pengenalan Jelmaan LaplacePengenalan Jelmaan Laplace Kaedah Jelmaan Laplace Dlm Analisis Kaedah Jelmaan Laplace Dlm Analisis
LitarLitar Sambutan Frekuensi Litar ACSambutan Frekuensi Litar AC Siri FourierSiri Fourier Jelmaan FourierJelmaan Fourier
MUTUAL INDUCTANCEMUTUAL INDUCTANCE
Self inductanceSelf inductance Concept of mutual inductanceConcept of mutual inductance Dot conventionDot convention Energy in a coupled circuitEnergy in a coupled circuit Linear transformer Linear transformer Ideal transformerIdeal transformer
MUTUAL INDUCTANCEMUTUAL INDUCTANCE
INTRODUCTIONINTRODUCTION magnetically coupledmagnetically coupled When two loops with or without When two loops with or without
contacts between them, affect each contacts between them, affect each other through magnetic field other through magnetic field generated by one of them – they are generated by one of them – they are said to be magnetically coupled.said to be magnetically coupled.
Example of device using this Example of device using this concept-transformer.concept-transformer.
TransformerTransformer
Use magnetically coupled coils to Use magnetically coupled coils to transfer energy from one circuit to transfer energy from one circuit to another.another.
Key circuit element where it is used Key circuit element where it is used for stepping down or up ac voltages for stepping down or up ac voltages or currents.or currents.
Also used in electronic circuits such Also used in electronic circuits such as radio and tv receiver.as radio and tv receiver.
Consider a single inductor with N turns.When current i, flow through coil, magnetic flux is produced around it.
Faraday’s LawFaraday’s Law
Induced voltage, v in the coil Induced voltage, v in the coil is proportional to number of is proportional to number of turns N and the time rate of turns N and the time rate of change of magnetic flux, change of magnetic flux, . .
dt
dNv
But we know that the flux But we know that the flux is is produce by current produce by current i,i, thus thus any change in the current any change in the current will change in flux will change in flux as well. as well.
dt
di
di
dNv
dt
diLv @
The inductance L of the The inductance L of the inductor is thus given byinductor is thus given by
di
dNL
Self-Inductance
Self InductanceSelf Inductance
Inductance that relates the Inductance that relates the induced voltage in a coil with induced voltage in a coil with a time-varying current in the a time-varying current in the same coil.same coil.
Mutual InductanceMutual Inductance When two inductors or coils When two inductors or coils
are in close proximity to are in close proximity to each other, magnetic flux each other, magnetic flux caused by current in one caused by current in one coil links with the other coil links with the other coil, therefore producing coil, therefore producing the induced voltage.the induced voltage.
Mutual Inductance Mutual Inductance
Magnetic flux Magnetic flux 11 originating from originating from coil 1 has 2 components:coil 1 has 2 components:
Since entire flux Since entire flux 11 links coil 1, links coil 1, the voltage induced in coil 1 is:the voltage induced in coil 1 is:
12111
dt
dNv 1
11
Only flux Only flux 1212 links coil 2, so the links coil 2, so the voltage induced in coil 2 is:voltage induced in coil 2 is:
As the fluxes are caused by As the fluxes are caused by current icurrent i11 flowing in coil 1, flowing in coil 1, equation vequation v11 can be written as: can be written as:
dt
dNv 12
22
dt
diL
dt
di
di
dNv 1
11
1
111
Self inductance of coil 1
Similarly for equation vSimilarly for equation v22::
dt
diM
dt
di
di
dNv 1
211
1
1222
Mutual inductance of coil 2With respect to coil 1
Coil 2
Magnetic flux Magnetic flux 22 comprises of 2 comprises of 2 components:components:
The entire flux The entire flux 22 links coil 2, so links coil 2, so the voltage induced in coil 2 is:the voltage induced in coil 2 is:
22212
dt
diL
dt
di
di
dN
dt
dNv 2
22
2
22
222
Self-inductance of coil 2
Since only flux Since only flux 21 21 links with links with coil 1, the voltage induced in coil 1, the voltage induced in coil 1 is:coil 1 is:
dt
diM
dt
di
di
dN
dt
dNv 2
122
2
211
2111
Mutual inductance of coil 1 with respect to coil 2
For simplicity, M12 and For simplicity, M12 and M21 are equal:M21 are equal:
MMM 2112
Mutual inductance between two coils
ReminderReminder
Mutual coupling exists when Mutual coupling exists when inductors or coils are in close inductors or coils are in close proximity and circuit are driven by proximity and circuit are driven by time-varying sources.time-varying sources.
Mutual inductance is the ability of Mutual inductance is the ability of one inductor to induce voltage one inductor to induce voltage across a neighboring inductor, across a neighboring inductor, measured in henrys (H).measured in henrys (H).
Dot ConventionDot Convention A dot is placed in the circuit A dot is placed in the circuit
at one end of each of the two at one end of each of the two magnetically coupled coils to magnetically coupled coils to indicate the direction of indicate the direction of magnetic flux if current enters magnetic flux if current enters that dotted terminal of the that dotted terminal of the coil.coil.
Dot convention is stated as Dot convention is stated as follows:follows:
If a current If a current entersenters the dotted terminal the dotted terminal of one coil, the reference polarity of of one coil, the reference polarity of mutual voltage in second coil is mutual voltage in second coil is positivepositive at dotted terminal of second coil.at dotted terminal of second coil.
If a current If a current leavesleaves the dotted terminal the dotted terminal of one coil, the reference polarity of of one coil, the reference polarity of mutual voltage in second coil is mutual voltage in second coil is negativenegative at dotted terminal of second at dotted terminal of second coil.coil.
Dot convention for coils Dot convention for coils in seriesin series
MLLL 221
MLLL 221
Example 1Example 1
Example 1Example 1
dt
diM
dt
diLRiv 21
1111
dt
diM
dt
diLRiv 12
2222
Coil 1:
Coil 2:
In frequency domain..In frequency domain..
22212
21111
)(
)(
ILjRMIjV
MIjILjRV
Example 2Example 2
Example 2Example 2
221
2111
)(0
)(
ILjZMIj
MIjILjZV
L
Example 3Example 3
Solution..Solution.. For coil 1, we use KVL:For coil 1, we use KVL:
123
03)54(12
21
21
IjjI
IjIjj
For coil 2, For coil 2,
0)612(3 21 IjIj
22
1 )42(3
)612(Ij
j
IjI
Substitute equation 1 into 2:Substitute equation 1 into 2:
12)4()342( 22 IjIjj
Aj
I o04.1491.24
122
Solve for I1:Solve for I1:
A
IjI
o
oo
39.4901.13
)04.1491.2)(43.63372.4(
)42( 21
Energy in a coupled Energy in a coupled circuitcircuit
Energy stored in an inductor is given Energy stored in an inductor is given by:by:
Now, we want to determine energy Now, we want to determine energy stored in magnetically coupled coils.stored in magnetically coupled coils.
2
2
1Liw
Circuit for deriving energy Circuit for deriving energy stored in a coupled circuitstored in a coupled circuit
Power in coil 1:Power in coil 1:
Energy stored in coil 1:Energy stored in coil 1:
dt
diLiivtp 1
11111 )(
2110 11111 2
11
ILdiiLdtpwI
Maintain iMaintain i1 1 and we increase iand we increase i22 to Ito I22. So, the power in coil 2 is:. So, the power in coil 2 is:
dt
diLi
dt
diMI
vidt
diMi
vivitp
222
2121
222
121
22112 )(
Energy stored in coil 2:Energy stored in coil 2:
2222112
0 2220 2112
22
2
1
22
ILIIM
diiLdiIM
dtpw
II
Total energy stored in the coils Total energy stored in the coils when both i1 and i2 have when both i1 and i2 have reached constant values is:reached constant values is:
2112222
211
21
2
1
2
1IIMILIL
www
Since MSince M1212=M=M2121=M, thus=M, thus
21222
211 2
1
2
1IMIILILw
Generally, energy stored in Generally, energy stored in magnetically coupled circuit magnetically coupled circuit is:is:
21222
211 2
1
2
1IMIILILw
Coupling coefficient, kCoupling coefficient, k
A measure of the magnetic A measure of the magnetic coupling between two coils; 0 coupling between two coils; 0 ≤ k ≤ 1≤ k ≤ 1
21LL
Mk
Linear TransformerLinear Transformer
Transformer is generally a four-Transformer is generally a four-terminal device comprising two or terminal device comprising two or more magnetically coupled coils.more magnetically coupled coils.
Coil that is directly connected to Coil that is directly connected to voltage source is voltage source is primary winding.primary winding.
Coil connected to the load is called Coil connected to the load is called secondary winding.secondary winding.
R1 and R2 included to calculate for R1 and R2 included to calculate for losses in coils.losses in coils.
Linear TransformerLinear Transformer
Primary winding Secondary winding
Obtain input impedance, Zin Obtain input impedance, Zin as seen from source because as seen from source because Zin governs the behaviour of Zin governs the behaviour of primary circuit.primary circuit.
Apply KVL to the two loops:Apply KVL to the two loops:
2221
2111
)(0
)(
IZLjRMIj
MIjILjRV
L
Input impedance ZInput impedance Zinin::
L
in
ZLjR
MLjR
I
VZ
22
22
11
1
RZ
Equivalent circuit of Equivalent circuit of linear transformerlinear transformer
Equivalent T circuit Equivalent T circuit
Equivalent ∏ circuitEquivalent ∏ circuit
Voltage-current Voltage-current relationship for primary relationship for primary and secondary coils give and secondary coils give
the matrix equation:the matrix equation:
2
1
2
1
2
1
I
I
LjMj
MjLj
V
V
By matrix inversion, this By matrix inversion, this can be written as:can be written as:
2
1
221
12
21
221
221
2
2
1
)()(
)()(V
V
MLLj
L
MLLj
MMLLj
M
MLLj
L
I
I
Matrix equation for Matrix equation for equivalent T circuit:equivalent T circuit:
2
1
2
1
)(
(
I
I
LLjLj
LjLLj
V
V
cbc
cca
If T circuit and linear circuit If T circuit and linear circuit are the same, then:are the same, then:
MLLa 1
MLLb 2
MLc
For ∏ network, nodal For ∏ network, nodal analysis gives the terminal analysis gives the terminal
equation as:equation as:
2
1
2
1
111
111
V
V
LjLjLj
LjLjLjI
I
CBC
CCA
Equating terms in admittance Equating terms in admittance matrices of above, we obtain:matrices of above, we obtain:
ML
MLLLA
2
221
ML
MLLLB
1
221
M
MLLLC
221
IDEAL TRANSFORMERIDEAL TRANSFORMER
Properties of ideal Properties of ideal transformer:transformer:
Coils have very large reactances (LCoils have very large reactances (L11, , LL22, M, M→∞)→∞)
Coupling coefficient is equal to unity Coupling coefficient is equal to unity (k=1)(k=1)
Primary and secondary winding are Primary and secondary winding are lossless (Rlossless (R11=0=R=0=R22))
Ideal transformer is a unity-coupled, losslesstransformer where primary and secondary coils have infinite self-inductance.
Transformation ratioTransformation ratio
We know that:We know that:
Divide v2 with v1, we get: Divide v2 with v1, we get:
dt
dNv
11
dt
dNv
22
nN
N
v
v
1
2
1
2
Energy supplied to the primary Energy supplied to the primary must equal to energy absorbed by must equal to energy absorbed by secondary since no losses in ideal secondary since no losses in ideal transformer.transformer.
Transformation ratio is: Transformation ratio is:
2211 iviv
nV
V
I
I
1
2
2
1
Types of transformer:Types of transformer:
Step-down transformerStep-down transformer
One whose secondary voltage is less One whose secondary voltage is less than its primary voltage.than its primary voltage.
Step-up transformerStep-up transformer
One whose secondary voltage is One whose secondary voltage is greater than its primary voltage.greater than its primary voltage.
Typical circuits in ideal Typical circuits in ideal transformertransformer
Complex PowerComplex Power
From:From:
Complex power in primary winding for Complex power in primary winding for ideal txt:ideal txt:
212
1 @ nIIn
VV
2*
22*
22*
111 )( SIVnIn
VIVS
Input impedanceInput impedance
We know that:We know that:
Since VSince V2 2 / I/ I2 2 = Z= ZLL , thus , thus
2
22
1
1 1
I
V
nI
VZ in
2n
ZZ Lin Reflected
impedance
Example Example
Find IFind I11 dan I dan I2 2 for given circuit:for given circuit:
Solution…Solution…
11stst: Find input impedance: Find input impedance
2
210
nZ in
3
1n
RZ
Therefore, solve for Therefore, solve for II1 1
281810inZ
AIo
5.028
0141
An
II 5.1
)(
5.0
31
12
THE ENDTHE END