华中科技大学电路理论课件第10章_颜秋容

φ21
i1
i2
+ - u2 +
= L1i1 + M12i2 = M 21i1 + L2i2
u-i relations of Linear coupled inductors:
2011-9-21
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
= =
dψ 1 dt dψ 2 dt
Linear system
电路理论－颜秋容
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
i1 R1
us L1
R2 •M •
L2
i2
20I1 + j30I1 − j10I2 = 100∠0°
RL
10I2 + j20I2 − j10I1 +10I2 = 0
100∠0°
I1 20Ω j10Ω 10Ω
+
••
j30Ω
j20Ω
-
I2 10Ω
I1 = 2.82∠ − 50.71°A I2 = 0.995∠174.3°A
i1
1
u1
2
i2
3
u2
4
i1 u1
i2 u2
2011-9-21
电路理论－颜秋容
i3 u3
7
Practice 10.1 Dot convention and u-i relations.
+ I1 * M
I2 *
+ห้องสมุดไป่ตู้
U
L1
1
-
L2
U 2
-
U1 = jωL1I1 + jωM (−I2 ) U 2 = jωM (I1) + jωL2 (−I2 )
φ12
{ ⎧ψ
⎩⎨ψ
1 2
= =
N1φ1 = N1φ11 + N1φ12 N2φ2 = N φ2 21 + N φ2 22
φ11
N1 - u1
⇒
⎧ψ ⎨⎩ψ
1 2
=ψ11 +ψ12 =ψ 21 +ψ 22
Linear system
N2
}φ22
φ21
i1
i2
+ - u2 +
= L1i1 + M12i2 = M 21i1 + L2i2
2. By reflected impedance 映射阻抗
2011-9-21
电路理论－颜秋容
10
10.2 含耦合电感电路分析Analysis of coupl.ed circuits
. . 2. By reflected impedance 映射阻抗
I1 Z1
I2
+ U-s
. jωM
+
jωL1
jωL2 U 2
I1 20Ω j10Ω 10Ω
+
••
j30Ω
j20Ω
-
I1 20Ω
10Ω
100∠0°
j20Ω +
-
j10Ω
• j10Ω
I2 10Ω
I2 10Ω
•
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电路理论－颜秋容
14
10.2 含耦合电感电路分析 Analysis of coupled circuits
3. By equivalent circuit 等效电路
5. Energy stored in a coupled inductor
i1
M
i2
∫t
w=
p(t)ｄt
－∞
∫= －t ∞（u1i1 + u2i2）ｄt
u1
L1
L2
u2
∫=
－t ∞（[ L1ｄｄit1
±
M
ｄi2 ｄt
)i1
+
(±
M
ｄi1 ｄt
+
L2
ｄi2 ｄt
)i2
]ｄt
∫ ∫ ∫ =
i1 0
L1i1ｄi1
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
= =
dψ 1 dt dψ 2 dt
Linear system
电路理论－颜秋容
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
= =
L1 M
di1 + M dt
di1 dt
+
L2
di2 dt di2 dt
3
10.1 耦合电感 Coupled inductors
1. u-i relations of coupled inductors
2
10.1 耦合电感 Coupled inductors
1. u-i relations of coupled inductors
⎩⎨⎧φφ12
= φ11 + φ12 = φ21 + φ22
φ12
{ ⎧ψ
⎩⎨ψ
1 2
= =
N1φ1 = N1φ11 + N1φ12 N2φ2 = N φ2 21 + N φ2 22
= =
L1 M
di1 + M dt
di1 dt
+
L2
di2 dt di2 dt
4
10.1 耦合电感 Coupled inductors
2. 自感和互感 Self-inductance and mutual inductance
L1、L2 ——Self-inductance φ12
N1
M ——Mutual inductance { φ11
-
R1
R2
•M •
L1
L2
I2 ZL
Primary winding secondary winding
φ11
N1 - u1
⇒
⎧ψ ⎨⎩ψ
1 2
=ψ11 +ψ12 =ψ 21 +ψ 22
Linear system
N2
}φ22
φ21
i1
i2
+ - u2 +
= L1i1 + M12i2 = M 21i1 + L2i2
u-i relations of Linear coupled inductors:
2011-9-21
i1 u1
u2
k = M (0 ≤ k ≤ 1) L1L2
4. 同名端 Dot convention
}φ22
φ21 i2
⎪⎪⎧u1 ⎨ ⎪⎪⎩u2
= =
L1
di1 dt
±
±M di1 dt
M di2 dt
+
L2
di2 dt
2011-9-21
电路理论－颜秋容
6
1
Practice 10.1 Dot convention and u-i relations.
U = U1 +U2 = jωL1I + jωMI + jωMI + jωL2I ＝jω(L1＋L2＋2M )I
i
+
*
L1 M
u
*
L2
-
i
+
L1
u
* M
*
L2
-
Leq＝L1＋L2＋2M
i
** M
L1 i1
L2
i2
* L1 M L2
*
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电路理论－颜秋容
8
10.1 耦合电感 Coupled inductors
3. 耦合系数 Coefficient of coupling - u1
k = M (0 ≤ k ≤ 1)
L1L2
ψ 12 ⋅ψ 21
k=
M= L1L2
M12M 21 L1L2
=
i2 i1 ψ 11 ⋅ψ 22
i1 i2
= N1φ12 ⋅ N φ2 21 = N1φ11 ⋅ N φ2 22
2011-9-21
Z eq
.
.+
I2
U oc -
10Ω
2011-9-21
电路理论－颜秋容
12
2
10.2 含耦合电感电路分析Analysis of coupled circuits
3. By equivalent circuit 等效电路
1 i1
1
M
i1
* M*
L1
L2
2 i2
i33
1
L1 − M
2 i2
1
L2 − M
φ12 ⋅φ21 ≤ 1 φ11 ⋅φ22
电路理论－颜秋容
N2
}φ22
φ21
i1
i2
+ - u2 +
⎧ψ ⎨⎩ψ
1 2
=ψ11 +ψ12 =ψ 21 +ψ 22
= L1i1 + M12i2 = M 21i1 + L2i2
⎧ψ ⎨⎩ψ
1 2
= =
N1φ11 + N1φ12 N φ2 21 + N φ2 22
S
A2 N'
a. 复杂含耦N合电感、理想变压器电路的分析；
b. 线性、铁芯变压器的应主用讲。教师
华中科技大学 颜秋容
2011-9-21
电路理论－颜秋容
1
10.1 耦合电感 Coupled inductors
1. u-i relations of coupled inductors
⎩⎨⎧φφ12
= φ11 + φ12 = φ21 + φ22
⎩⎨⎧φφ12
= φ11 + φ12 = φ21 + φ22
φ12
{ ⎧ψ
⎩⎨ψ
1 2
= =
N1φ1 = N1φ11 + N1φ12 N2φ2 = N φ2 21 + N φ2 22
φ11
N1 - u1
⇒
⎧ψ ⎨⎩ψ
1 2
=ψ11 +ψ12 =ψ 21 +ψ 22
Linear system
N2
}φ22
Zeq
=
(10 +
jωL2 ) +
(ωM )2 (20 + jωL1)
=
(10 +
j30) +
(10)2 (20 + j20)
=
(12.5 +
j27.5)
ZL = Zeq ∗ = (12.5 − j27.5)Ω
PLmax = (
2011-9-21
U oc Zeq + Z
L
)2
×
Re[Z
L
]
= (35.4)2 ×12.5 25 电路理论－颜秋容
Chapter 10 含磁耦合元件的正弦稳态电路
10.1 耦合电感
*
10.2 含ACno耦aulpy合Alsei电ds io感n*fdc电uocWut路oprl的esd分cir析cIAuitsIA2 IA1
10.3 变压器B
A1
Z1 Z1
IR
Transformers
Z1
C
R
Z2 Z2 Z2
目标：
=
25W
16
10.3 变压器 Transformers
1. 线性变压器Linear transformers (Air-core transformers)
Coils are wound on magnetically linear material —— μ = constant
I1 U S +
maximum
average
power.
ωIL11
= 20Ω，ωL2
20Ω
→
+-
100∠0°V rms
∗ L1
= M
30Ω, L2
ωM = 10Ω.
10Ω ＋
← Zeq Uoc
ZL
∗
−
I1
=
100∠0° 20 + jωL1
=
100∠0° 20 + j20
=
3.54∠ − 45°
Uoc = − jωMI1 = − j10× 3.54∠ − 45° = 35.4∠ −135°
** M
L1
L2
* L1 M L2
*
M
−M
L1 − M
2011-9-21
L2 − M
L1 + M
电路理论－颜秋容
L2 + M
15
Practice 10.2 Determine the load impedance that
maximizes the average power drawn from the circuit. What is the
Z2
-
I1 Z1
+
+
Us
jωL1 U-1′ +
-
Z ref
U-1″
Z in
Z in
Zin
=
U S I1
=
Z1
+
jωL1I1
± jωMI2 I1
U 2 = jωL2I2 ± jωMI1 = −Z2I2
II12
=−
(± jωM ) Z2 + jωL2
=
(Z1
+
jωL1 )
+
(±
jωM
j10Ω 10Ω
U S
=
[Z11
+
(ωM )2 Z 22
]I1
+
••
100∠0° j30Ω
j20Ω
-
I2 10Ω
100∠0°
=
[(20
+
j30)
+
(10
+
102 10 +
j20)
]I1
10I2 +10I2 + ( j20I2 − j10I1) = 0
Can you find I2first?
)
I2 I1
Zin
=
(Z1
+
jωL1) +
(ωM )2 Z2 + jωL2
=
Z11
+
(ωM )2 Z 22
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电路理论－颜秋容
11
10.2 含耦合电感电路分析 Analysis of coupled circuits
2.
By
reflected
impedance
映射阻抗 I1
20Ω
u-i relations of Linear coupled inductors:
2011-9-21
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
= =
dψ 1 dt dψ 2 dt
Linear system
电路理论－颜秋容
⎧⎪⎪u1 ⎨ ⎪⎪⎩u2
= =
L1 M
di1 + M dt
di1 dt
+
L2
di2 dt di2 dt
+
i2 0
L2i2ｄi2
+
i1、i2 0
±
M
(i2ｄi1
+
i1ｄi2
)
=
1 2
L1 i12
+
1 2
L2i22
±
Mi1 i2