Transcript EEE 302 Lecture 11 - Arizona State University

EEE 302
Electrical Networks II
Dr. Keith E. Holbert
Summer 2001
Lecture 11
1
Magnetically Coupled Networks
• A new four-terminal element, the transformer, is
introduced in this chapter
• A transformer is composed of two closely spaced
inductors, that is, two or more magnetically
coupled coils
– primary side is connected to the source
– secondary side is connected to the load
Lecture 11
2
Dot Convention
• dot convention: dots are placed beside each coil
(inductor) so that if the currents are entering (or
leaving) both dotted terminals, then the fluxes add
• right hand rule says that curling the fingers (of the
right hand) around the coil in the direction of the
current gives the direction of the magnetic flux
based on the direction of the thumb
• We need dots on the schematic to know how the
coils are physically oriented wrt one another
Lecture 11
3
Mutually Coupled Coils
The following equations define the coupling between
the two inductors assuming that each respective
current enters the dot side which is also the positive
voltage side
d i1
d i2
v1 (t )  L1
M
dt
dt
d i1
d i2
v 2 (t )  M
 L2
dt
dt
where L1 and L2 are the self-inductances of the coils
(inductors), and M is the mutual inductance between
the two coils
Lecture 11
4
Mutually Coupled Coils
M
i1(t)
+
v1(t)
–
L1
i2(t)
L2
+
v2(t)
–
d i1
d i2
v1 (t )  L1
M
dt
dt
d i1
d i2
v 2 (t )  M
 L2
dt
dt
Lecture 11
5
Class Example
• Extension Exercise E11.1
Lecture 11
6
Mutually Coupled Coils (AC)
• The frequency domain model of the coupled
circuit is essentially identical to that of the time
domain
V1  j L1 I1  j M I 2
V2  j M I1  j L2 I 2
Lecture 11
7
Class Examples
• Extension Exercise E11.2
• Extension Exercise E11.3
Lecture 11
8
Source Input Impedance
The source sees an input impedance, Zi, that is the
sum of the primary impedance, and a reflected
impedance, ZR, due to the secondary (load) side
VS
Zi 
 Z P  Z R  Z P  f Z L 
IP
M
Z
VS
+
–
L1
Lecture 11
L2
ZL
9
Class Example
• Extension Exercise E11.4
Lecture 11
10
Energy Analysis
• An energy analysis of the mutually coupled
inductors provides an expression for the
instantaneous stored energy
w(t )  L1 i1 (t )  L2 i2 (t )  M i1 (t ) i2 (t )
1
2
2
1
2
2
• The sign is positive (+) if currents are both
entering (or leaving) the dots; sign is negative (-)
if currents are otherwise
Lecture 11
11
Quantifying the Coupling
• The mutual inductance, M, is in the range
0  M  L1 L2
• The coefficient of coupling (k) between two
inductors is defined as

M 
0  k 
 1
L1 L2 

– for k > 0.5, inductors are said to be tightly coupled
– for k  0.5, coils are considered to be loosely coupled
Lecture 11
12
Class Example
• Extension Exercise E11.5
Lecture 11
13