0195136047.pdf

36 CIRCUIT CONCEPTS

Also find the energy stored in each of the above cases att=0.

Solution

L 11 =L 22 = 0 .1H; M=10 mH= 10 × 10 −^3 H

v 1 =L 11

di 1

dt

+M

di 2

dt

; v 2 =M

di 1

dt

+L 22

di 2

dt

Wm=energy stored=

1

2

L 11 i 12 +L 12 i 1 i 2 +

1

2

L 22 i^22

(a) Since bothi 1 andi 2 are constant and not a function of time,

v 1 = 0 ; v 2 = 0

Wm=^12 ( 0. 1 )( 10 × 10 −^3 )^2 + 0 + 0 = 5 × 10 −^6 J= 5 μJ

(b)v 1 = 0 + 10 × 10 −^3 ( 10 ×100 cos 100t) 10 −^3 =10 cos 100tmV

v 2 = 0 + 0. 1 ( 10 ×100 cos 100t) 10 −^3 =100 cos 100tmV= 0 .1 cos 100tV

Wm= 0 + 0 +^12 ( 0. 1 )(100 sin^2100 t) 10 −^6 ; Wm=0att= 0

(c)v 1 = 0. 1 (− 0 .1 sint)+ 10 × 10 −^3 [0.3 cos(t+30°)]=−10 sint+3 cos(t+30°)mV

v 2 = 10 × 10 −^3 (− 0 .01 sint)+ 0 .1[0.3 cos(t+30°)]=−sint+30 cos(t+30°)mV

Wm=^12 ( 0. 1 )( 0 .01 cos^2 t)+( 10 × 10 −^3 )( 0 .1 cost)[0.3 sin(t+30°)]+^12 ( 0. 1 )

[0.09 sin^2 (t+30°)]; att= 0

Wm=^12 ( 0. 1 )( 0. 01 )+ 10 × 10 −^3 ( 0. 1 )( 0. 15 )+^12 ( 0. 1 )

( 0. 09

4

)

= 1. 775 μJ

Transformer

A transformer is basically a static device in which two or more stationary electric circuits are

coupled magnetically, the windings being linked by a common time-varying magnetic flux. All

that is really necessary for transformer action to take place is for the two coils to be so positioned

that some of the flux produced by a current in one coil links some of the turns of the other coil.

Some air-core transformers employed in communications equipment are no more elaborate than

this. However, the construction of transformers utilized in power-system networks is much more

elaborate to minimize energy loss, to produce a large flux in the ferromagnetic core by a current in

any one coil, and to see that as much of that flux as possible links as many of the turns as possible

of the other coils on the core.

An elementary model of a two-winding core-type transformer is shown in Figure 1.2.12.

Essentially it consists of two windings interlinked by a mutual magnetic field. The winding that

is excited or energized by connecting it to an input source is usually referred to as theprimary

winding, whereas the other, to which the electric load is connected and from which the output

energy is taken, is known as thesecondarywinding. Depending on the voltage level at which

the winding is operated, the windings are classified as HV (high voltage) and LV (low voltage)

windings. The terminology ofstep-uporstep-down transformeris also common if the main

purpose of the transformer is to raise or lower the voltage level. In a step-up transformer, the

primary is a low-voltage winding whereas the secondary is a high-voltage winding. The opposite

is true for a step-down transformer.