Module 2. Sinusoidal Steady State Circuit...
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Module 2. Sinusoidal Steady State Circuit analysis Circuit Theory (350004) , 2018-19
Transcript of Module 2. Sinusoidal Steady State Circuit...
Module 2. Sinusoidal Steady State Circuit analysis
Circuit Theory (350004) , 2018-19
Outline
• The sinusoidal signal (SSS)
• Why complex numbers?
• Phasors and impedances
• Impedances of L and C
• The phasor domain
• Example
• Association of Z’s
• Example
• V and I relation
• Power in SSS regimen
• Example
• Effective and mean values
• Magnetically coupled coils• Ampare and Faraday’s law’s• Magnetic flux outside the coil• Two magnetically coupled coils• Three coupled coils in SSS regimen• Dot’s convention• Examples
Magnetically coupled coils
Introduction
Circuit Theory / Magnetically coupled coils
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Ampere’s circuital law for a solenoid
Ampere’s Law
)(t
)(ti
)(tiLingers indcate
i(t) drection
)()( tiN
t
=(t): magnetic flux (webers)
i(t): electric current (amperes)
N: number of turns
: reluctance (henries-1)
Circuit Theory / Magnetically coupled coils
Thumb points to N-pole
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The current i(t) produces a magnetic flux (t):
Right hand rule for the direction of (t):
Faraday’s law
Circuit Theory / Magnetically coupled coils
)(t
)(tv)(ti
)(t
circuit
)(d
d)( t
tNtv =
The time varying flux (t)
produces a voltage as shown in
the figure
According to Len’s law, the coil can
be seen as an electromotive
Force (applying a voltage to the
circuit):
)(d
d)( t
tNt −=
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Ampere + Faraday
Circuit Theory / Magnetically coupled coils
)(t
)(tv
)(ti
)(tv
)(ti
)(tHigh-permeability
material (e.g. soft-
iron or ferrite)
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)(d
d)(
(*)
tit
Ltv =
=
2
:N
L
being L the
self inductance
coefficient:
With units in henry (H)
(*) Note that the
voltage is pointing to
the terminal whose
current is entering
the coil
The time varying current i(t) produces a voltage v(t):
Magnetic flux outside the coil
Circuit Theory / Magnetically coupled coils
)(tC
)(tv
)(ti
)(tS
core
)(11
)(
tNi
t
SC
SC
+
=
+=
C : reluctance of the core,
S : leakage reluctance outside
the core
For high-permeability core
S >> C C>>S
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Two magnetically coupled coils
Circuit Theory / Magnetically coupled coils
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C1
)(1 tv
)(1 ti )(2 ti
)(2 tv
C2
S1
S2
• Voltage difference at the coils terminals.
(*)In this particular example the two magnetic flux inside the core generated by each coil have the same direction and are therefore added !!
( )
( )
++=
++=
CSC
CSC
tNv
tNv
12222
21111
d
dd
d
(*)
Two magnetically coupled coils
Circuit Theory / Magnetically coupled coils
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• Voltage difference at the coils terminals
considering:
+
=
+
+
=
+
=
+
+
=
1
,1
12
2
221
,1
12
,2
22
,2
222
2
,2
21
1
112
,2
21
,1
11
,1
111
d
d
d
d
d
d
d
d
iN
iN
tNi
Ni
Ni
N
tNv
iN
iN
tNi
Ni
Ni
N
tNv
CCSC
CCSC
,111
,111
,2,22,1,11 SCSC +
=
+
=
== CCC ,2,1
thatsupose willWe
Two magnetically coupled coils
Circuit Theory / Magnetically coupled coils
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• Voltage difference at the coils terminals
being:
+=
+=
121222
212111
d
d
d
dd
d
d
d
it
Mit
Lv
it
Mit
Lv
Self induced voltage Mutual induced voltage
1)(0 theis where,
1(H) into 2 coil of the,
and inductance self the,
1212211212
,2
2112
1
2
11
=
=
=
KKLLKM
NNM
NL
C
tcoefficien coupling
inductance mutual
Three magnetically coupled coils
• Voltage diff. at the coils terminals in function of
the currents i1, i2 and i3.
)(3 tv
C2
)(1 tv
)(1 ti )(2 ti
)(2 tv
C1
)(3 ti
C3
1L2L
3L
−−=
−+=
−+=
232131333
323121222
313212111
d
d
d
d
d
dd
d
d
d
d
dd
d
d
d
d
d
it
Mit
Mit
Lv
it
Mit
Mit
Lv
it
Mit
Mit
Lv
.
,
,
311313
322323
211212
LLKM
LLKM
LLKM
=
=
=
(*): 3C generated by coil 3 goes in opposite direction as 1C therefore,
the mutual induced voltage from coil 3 into coil 1 has opposite sign as
the self induced voltage of coil 1
Circuit Theory / Magnetically coupled coils
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Three magnetically coupled coils
• For Sinusoidal Steady State regimen
• Time differentiating
Circuit Theory / Magnetically coupled coils
tjtjj
tjtjj
IeeeIti
VeeeVtv
==
==
)(
)(
−−=
−+=
−+=
323113333
323121222
313212111
IMjIMjILjV
IMjIMjILjV
IMjIMjILjV
( )== units ,2112211212 LLKLLKM 39
Dot’s convention
• Is used for simplifying the representation of coupled coils. It allows to know whether the mutually induced fluxes have the same direction or not, without need of drawing the way the spools are wounded around the core.
• Two coupled coils can be represented in one of the two following ways:
Circuit Theory / Magnetically coupled coils
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Dot’s convention
• If both currents entre or exit their respective coils by their dotted ends, then the generated fluxes inside the core have the same direction, otherwise the fluxes have opposite direction.
Circuit Theory / Magnetically coupled coils
I1I1 I2
I2
2
1
V1 V2
I1
I2I22
1I1
V1 V2
−−=
+=
112222
212111
IMjILjV
IMjILjV
+=
+=
112222
212111
IMjILjV
IMjILjV
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Examples 1
• Equations for V1,V2 and V3
Circuit Theory / Magnetically coupled coils
−=
++=
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131
0 VV
IZVVE gg
( )( )
( )
−−−=
−−+=
−−+=
2231132133
2123112222
211321211 1
IMjIMjIILjV
IIMjIMjILjV
IIMjIMjILjV
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Examples 2
• Equations for V1,V2 and V3
Circuit Theory / Magnetically coupled coils
( )
( )
( )
−
+−
−−
2231132133
2123112222
2113212111
IjωωIMj+I+ILj=V
I+IMjIMj+ILj=V
I+IMjIMj+ILj=V
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