Formula Sheets: Magnetic Circuits & Core Losses | Basic Electrical Technology - Electrical Engineering (EE) PDF Download

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Net w ork Theory: Magnetically Coupled Circuits
F orm ula Sheet for Electrical GA TE
Mutual Inductance Basics
• Mutual Inductance (M ) : Inductance due to magnetic coupling b et w een t w o coils.
V
2
=M
dI
1
dt
, V
1
=M
dI
2
dt
where V
1
, V
2
are induced v oltages, I
1
, I
2
are curren ts in coils, M is m utual inductance
(H ).
• Coupling Co e?icien t ( k ) :
k=
M
v
L
1
L
2
, 0=k=1
where L
1
, L
2
are self-inductanc es of the coils.
• Maxim um Mutual Inductance : M
max
=
v
L
1
L
2
(when k=1 , p erfect coupling).
Imp edance in Coupled Circuits
• Phasor Domain : F or sin usoidal steady state, use s=j? .
V
1
=j?L
1
I
1
±j?MI
2
, V
2
=j?L
2
I
2
±j?MI
1
where sign dep ends on dot con v en tion.
• Dot Con v en tion :
– Same direction (dots aligned) : P ositiv e M term.
– Opp osite direction: Negativ e M term.
• Equiv alen t Imp edance (Series-Aiding) :
L
eq
=L
1
+L
2
+2M
• Equiv alen t Imp edance (Series-Opp osing):
L
eq
=L
1
+L
2
-2M
T-Equiv alen t Circuit
• T-Mo del for Coupled Coils :
L
a
=L
1
-M, L
b
=L
2
-M, L
c
=M
where L
a
, L
b
are self-inductances min us m utual inductance, L
c
is m utual inductance.
• Application : Simplifies analysis b y con v erting coupled coils in to uncoupled inductors.
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Page 2


Net w ork Theory: Magnetically Coupled Circuits
F orm ula Sheet for Electrical GA TE
Mutual Inductance Basics
• Mutual Inductance (M ) : Inductance due to magnetic coupling b et w een t w o coils.
V
2
=M
dI
1
dt
, V
1
=M
dI
2
dt
where V
1
, V
2
are induced v oltages, I
1
, I
2
are curren ts in coils, M is m utual inductance
(H ).
• Coupling Co e?icien t ( k ) :
k=
M
v
L
1
L
2
, 0=k=1
where L
1
, L
2
are self-inductanc es of the coils.
• Maxim um Mutual Inductance : M
max
=
v
L
1
L
2
(when k=1 , p erfect coupling).
Imp edance in Coupled Circuits
• Phasor Domain : F or sin usoidal steady state, use s=j? .
V
1
=j?L
1
I
1
±j?MI
2
, V
2
=j?L
2
I
2
±j?MI
1
where sign dep ends on dot con v en tion.
• Dot Con v en tion :
– Same direction (dots aligned) : P ositiv e M term.
– Opp osite direction: Negativ e M term.
• Equiv alen t Imp edance (Series-Aiding) :
L
eq
=L
1
+L
2
+2M
• Equiv alen t Imp edance (Series-Opp osing):
L
eq
=L
1
+L
2
-2M
T-Equiv alen t Circuit
• T-Mo del for Coupled Coils :
L
a
=L
1
-M, L
b
=L
2
-M, L
c
=M
where L
a
, L
b
are self-inductances min us m utual inductance, L
c
is m utual inductance.
• Application : Simplifies analysis b y con v erting coupled coils in to uncoupled inductors.
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Energy in Coupled Circuits
• T otal Energy Stored :
E =
1
2
L
1
I
2
1
+
1
2
L
2
I
2
2
±MI
1
I
2
where p ositiv e for same direction curren ts, negativ e for opp osite (p er dot con v en tion).
T ransformer Analysis
• Ideal T ransformer : V oltage and curren t ratios.
V
2
V
1
=
N
2
N
1
,
I
2
I
1
=
N
1
N
2
where N
1
, N
2
are turns in primary and secondary coils.
• Imp edance T ransformation :
Z
in
=
(
N
1
N
2
)
2
Z
L
where Z
L
is load imp edance on secondary .
• Reflected Imp edance (Coupled Coils) :
Z
reflected
=
?
2
M
2
Z
2
where Z
2
is imp edance in secondary circuit.
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Inductance (H ), Imp edance (? ).
• Dot con v en tion is critical for correct sign of m utual inductance terms.
• F or GA TE, fo cus on phasor domain analysis and equiv alen t circuits (T-mo del or re-
flected imp edance).
• Ensure k=1 ; if M >
v
L
1
L
2
, c hec k for errors.
• Use KVL/K CL with m utual inductance terms for circuit analysis.
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