Superconducting Qubits and Circuits: Artificial Atoms Coupled to ...
Coupled Circuits
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Transcript of Coupled Circuits
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SELF INDUCTANCE:
V = L
where v = induced voltage
= rate of change of current
L = self inductance
dt
di
dt di
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UNIT OF SELF INDUCTANCE IS Henry
INDUCTANCE CAN ALSO BE EXPRESSED AS:
Here L represents the magnitude of selfinductance
dt
d N
dt
di L
di
d N L
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MUTUAL INDUCTANCE
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The induced voltage across coil 2 is given by
It can also be expressed as
Where M = proportionality constant termedas mutual inductance between two coils
dt
d N v L 1222
dt di
M v L1
2
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dt d
N dt di
M 1221
1
122 di
d N M
2
211 di
d NMsimilarly
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COEFFICIENT OF COUPLING:It is defined as the fraction of total flux thatlinks the coil
i.e.K = 12
1
21 2
M 2
N 1 N 2 21
i2
12i1
N 1 N
2
K 2i1
K 1i
2
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K 2
N 1 1
i1
N 2 2
i2
M K 2 L1 L2
Q L N i
K
M
L1 L
2
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ANALYSIS OF COUPLED CIRCUITS:
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How to determine the sign of
mutual emf?
121
111 vdt di M
dt di L Ri
012
222
dt di
M dt di
L Ri
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In the above equation the sign
of the mutual emf is positive
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121
111 ? v
dt di M
dt di L Ri
0? 1222 dt di
M dt
di L Ri
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The sign of mutual emf in aboveequation is
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Equations in time domain form:(instantaneous form):
012222
1
21
111
dt
di M
dt
di L Ri
vdt di
M dt di
L Ri
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Equations in frequency domain form:
I 1 R1 j L1 I 1 Mj I 2 v1 I 1( R1 j L1) j MI 2 v1
I 2 R2 j L2 I 2 Mj I 1 0 I 2 ( R2 j L2 ) Mj I 1 0
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Write the mesh equations in terms
of time and frequency domain
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The equations are
vdt
di
M dt
di
L Ri
vdt
di M
dt
di L Ri
12222
21111
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0)()()(
)()(
112212
1122
222
2211121
Rii
dt di
M dt
iid L
dt iid
M dt di
L Ri
vdt di
M dt iid
L Rii
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Dot rule for coupled circuits :
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Dotted equivalent is:
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Dot Rule: When both assumed currents enter or leavea pair of coupled coils at the dotted
terminals, the signs of the M terms will bethe same as the signs of the L terms.If one current enters at a dotted terminal
and one leaves by a dotted terminal, thesigns of the M terms are opposite to thesigns of the L terms
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M and L terms have the ----------------- sign
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M and L terms have the ----------------- sign
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M and L terms have the ----------------- sign
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M and L terms have the ----------------- sign
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Draw the dotted equivalent and
obtain I1
and I2
0010
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Series aiding of inductor:Leq =L1+L2+2M
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Series opposing of inductor:Leq =L1+L2-2M
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Parallel aiding of inductor:
M L L M L L
Leq 221
221
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Parallel opposing of inductors:
M L L
M L L Leq
221
221
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Single tuned coupled circuit:
11 L j R
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Loop 1:
v MI j L j R I
v MI j I L j R I
2111
21111
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Loop 2:
01
0
2221
1222
22
C L j R I MI j
MI j I L jC
jI R I
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02
1
22
1 v I I
jX R M j M j R
C L X
122
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22221
22221
220
22221
2
22221
1
2
)(
1)(
)(
0
M jX R RC MV
c j M jX R R MV j
C j I
C j
I v
M jX R R
MV j I
M jX R R
M j
v R
I
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Amplification factor
When the secondary is tuned, X 2 =0
A resonance
222210
)( M jX R RC M
V V A
C L 1
2
22
021 M R RC
M
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21Max0
0
21
20
2122
021Resonance0
2
R M
output,maximumFor
V
R RC V V
R
M M
R RC
V
M R RC
MV
o
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DOUBLE TUNED COUPLED CIRCUIT:
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222
1
C L X
111
1C
L X Let
0
V
I
I
jXR M j
M j jXR
2
1
22
11
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222211
21
2 ))((
0
M jX R jX R
M j
V jX R
I
2222112
))(( M jX R jX R
MV j I
]))([( 2222112
2
2220
M jX R jX RC MV
C j I
X I V C