1. Basic Circuit Analysis
Magnetic Circuits
Transformers

2. The Linear Transformer
Illustrating Induced Voltage:
Case 1:
jM + + I1 V1 V2 I2 _ • • _
jL1
jL2
V1 = jL1I1 + jMI2
V2 = jL2I2 + jMI1

3. The Linear Transformer
Illustrating Induced Voltage:
Case 2:
jM _ + • I1 V1 V2 I2 • _ +
jL1
jL2
V1 = jL1I1 + jMI2
V2 = jL2I2 + jMI1

4. The Linear Transformer
Illustrating Induced Voltage:
Example 1:
j8
-j4
6  2 • + +
va(t) I1
j10
j6 I2
vb(t) _ _ •
va(t) = 50cos(400t + 30) V
vb = 80cos(400t – 40) V
Va = 50300 V
Vb = 80-400 V

5. EXAMPLE 1: Continued • _ • + + I1 I2 _ Solve for I1 and I2
j8
-j4
6 
2  • + + I1
j10 
j6  I2
80-40 V
5030 V _ _ •
Solve for I1 and I2
Mesh 1
(2 + j10)I1 + j8I2 = 5030
Mesh 2
j8I1 + (j6 – j4 + 6)I2 = - 80-40
(2+j10) j I 30
Matrix Form =
j (6+j4) I -40

6. The Linear Transformer
Illustrating Induced Voltage:
Example 2:
8 
j10 
j8 
-j4  • •
j3  +
j5 
12 
200 V _ I1 I2
6 
Solve for I1 and I2

7. Example 2: Continued • • _ + 200 V I1 I2 Mesh 1:
8 
j10 
j8 
-j4  • •
j3  +
j5 
200 V
12  _ I1 I2
6 
Mesh 1:
(8 + j10 + j5 + 6)I1 - (j j3)I2 = 200
Mesh 2
-(6 + j5 + j3)I1 + (6 + j5 + j8 – j j3)I2 = 0
(14+j15) (6+j8) I 0
Matrix =
-(6+j8) (18+j12) I

8. THE IDEAL TRANSFORMER
N1 : N2
1 : n
ideal
If like assumed polarities of the voltages V1 and V2 are placed at the
2dots of the transformer, then V1/V2 = n. If either one of the dots
or either one of the voltage polarities are reversed then V2/V1 = -n.
If current I1 enters the dot on its side of the transformer and current I2
leaves the dot on its side of the transformer then I1/I2 = n. If either
current reverses its direction of entering its respective dot then
I1/I2 = - n

9. THE IDEAL TRANSFORMER V2 I1 = = n n V1 I2 I2ZL - V2 = -VS2
BASIC EQUATIONS: V2 I1 = = n n V1 I2
I2ZL - V2 = -VS2
I1Zin + V1 = VS1 ,

10. THE IDEAL TRANSFORMER Zin 0 1 0 I1 Vs1 Matrix 0 0 -n 1 V1 0
Rearrange previous equations
Zin I Vs1
ZL I Vs2 =
Matrix
n V
n V

11. THE IDEAL TRANSFORMER The Basic Transformer Without Markings ZA ZB VAVB
The Basic Transformer Without Markings

12. THE IDEAL TRANSFORMER Thevenin Considerations: or or • • • • • • •
this
this • • • or • • •
this
this

13. THE IDEAL TRANSFORMER ZA ZB VA VB ZA ZB n2 1 : n + _ + • VB • • VA VB- • n + n

14. THE IDEAL TRANSFORMER ZA ZB VA VB n2ZA ZB _ + + nVA VB _ + _ 1 : n • •

15. THE IDEAL TRANSFORMER (4 – j6)  (9 + j18)  ZA ZB + • + 100 V I1 V1
1 : 3 ZA ZB _ + + • +
100 V I1 V1 V2 I2
2730 V _ _ • _ +
Solve for I1 and I2
Thevenin impedance •
Thevenin voltage
4 
-j6 
1 
j2  + + +
930 V
100 V I1 V1 _ _ _ •

16. The Ideal Transformer Thevenin impedance • Thevenin voltage -j6  1 
4 
-j6 
1 
j2  + + +
930 V
100 V I1 V1 _ _ _
(5 – j4)I1 = 100 - 930
I1 = 0.78-25 A I1
I2 = - =
0.26155 A 3
V1 - 100 + (4 – j6)I1 = 0;
V1 = 10 – (4 – j6)(0,78-25
10.7131.3 V
V2 = -3V1
= - 3(10.7131.3)
32.1 V

17. The Ideal Transformer (4 – j6)  (9 + j18)  ZA ZB + • + 100 V I1 V1
1 : 3 ZA ZB _ + + • +
100 V I1 V1 V2 I2
2730 V _ _ _ + •
Non Thevenin Solution:
(4 – j6)I1 + V1 = 100 ;
(9 + j18)I2 – V2 = 2730 V2 I1 =
- 3
3V1 + V2 = 0
I1 + 3I2 = 0
= - 3 V1 I2
(4 – j6) I 0
(9 + j18) I 30 =
Matix V V

18. Basic Laws of Circuits
circuits
End of Lesson
Magnetic Circuits