Bird J. Electrical Circuit Theory and Technology

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838 Electrical Circuit Theory and Technology

500 pF 500 pF

50 mH

0.2 µF

800 mH 800 mH

(a)

(b)

Figure 42.55

8 A ﬁlter is required to pass all frequencies above 4 kHz and to have

a nominal impedance of 750 . Design (a) an appropriate T section

ﬁlter,and(b) an appropriate  section ﬁlter to meet these requirements.

[(a) Each series arm D 53.1 nF, shunt arm D 14.92 mH

(b) Series arm D 26.5 nF, each shunt arm D 29.84 mH]

9 The inductance in each of the shunt arms of a high-pass  section

ﬁlter is 50 mH. If the nominal impedance of the section is 600 ,

determine the value of the capacitance in the series arm.

[69.44 nF]

10 Determine the value of inductance required in the shunt arm of a

high-pass T section ﬁlter if in each series arm it contains a 0.5

µF

capacitor. The cut-off frequency of the ﬁlter section is 1500 Hz.

Sketch the characteristic curve of characteristic impedance against

frequency expected for such a ﬁlter section. [11.26 mH]

11 A high-pass  section ﬁlter has a nominal impedance of 500  and

a cut-off frequency of 50 kHz. Determine the frequency at which

the characteristic impedance of the section is (a) 1 k (b) 800 

12 A low-pass T section ﬁlter having a cut-off frequency of 9 kHz is

connected in series with a high-pass T section ﬁlter having a cut-off

frequency of 6 kHz. The terminating impedance of the ﬁlter is 600 .

(a) Determine the values of the components comprising the

composite ﬁlter.

(b) Sketch the expected attenuation/frequency characteristic and

state the name given to the type of ﬁlter described.

[(a) Low-pass Tsection: each series arm 10.61 mH,

shunt-arm 58.95 nF]

High-pass Tsection: each series arm 44.20 nF,

shunt arm 7.96 mH

(b) Band-pass ﬁlter]

Propagation coefﬁcient and time delay

13 A ﬁlter section has a propagation coefﬁcient given by

(a) 1.79  j0.63 (b) 1.378

51.6

. Determine for each (i) the atten-

uation coefﬁcient and (ii) the phase angle coefﬁcient.

[(a) (i) 1.79 N (ii) 0.63 rad (b) (i) 0.856 N (ii) 1.08 rad]

14 A ﬁlter section has a current input of 200

mA and a current

output of 16

30

mA. Determine (a) the attenuation coefﬁcient

(b) the phase shift coefﬁcient, and (c) the propagation coefﬁcient.

(d) If four such sections are cascaded determine the current output

of the fourth stage and the overall propagation coefﬁcient.

[(a) 2.526 N (b) 0.873 rad

19.07

(d) 8.19

180

µAI10.103 C j3.491 or

10.69

19.06

]

Filter networks 839

15 Determine for the high-pass T section ﬁlter shown in Figure 42.56,

(a) the attenuation coefﬁcient, (b) the phase shift coefﬁcient, and

[(a) 1.61 N (b)  2.50 rad

57.22

]

=−j1 kΩ X

=−j1 kΩ

= j 200 Ω R

= 600 Ω

Figure 42.56

16 A low-pass T section ﬁlter has an inductance of 25 mH in each

series arm and a shunt arm capacitance of 400 nF. Determine for the

section (a) the time delay for the signal to pass through the ﬁlter,

assuming the phase shift is small, and (b) the time delay for a signal

to pass through the section at the cut-off frequency.

[(a) 141.4

µs (b) 222.1 µs]

17 A ﬁlter network comprising n identical sections passes signals of

all frequencies over 8 kHz and provides a total delay of 69.63

µs.

If the characteristic impedance of the circuit into which the ﬁlter

is inserted is 600 , determine (a) the values of the components

comprising each section, and (b) the value of n.

[(a) Each series arm 33.16 nF; shunt arm 5.97 mH (b) 7]

18 A ﬁlter network consists of 15 sections in cascade having a nominal

impedance of 800 . If the total delay time is 30

µs determine the

component value for each section if the ﬁlter is (a) a low-pass 

network, (b) a high-pass T network.

[(a) Series arm 1.60 mH, each shunt arm 1.25 nF

(b) Each series arm 5 nF, shunt arm 1.60 nF]

‘m-derived’ ﬁlter sections

19 A low-pass ﬁlter section is required to have a nominal impedance of

450 , a cut-off frequency of 150 kHz and a frequency of inﬁnite

attenuation at 160 kHz. Design an appropriate ‘m-derived’ T section

ﬁlter.

[Each series arm 0.166 mH; shunt arm comprises 1.641 nF

capacitor in series with 0.603 mH inductance]

20 In a ﬁlter section it is required to have a cut-off frequency of

1.2 MHz and a frequency of inﬁnite attenuation 1.3 MHz. If the

840 Electrical Circuit Theory and Technology

nominal impedance of the line into which the ﬁlter is to be inserted

is 600 , determine suitable component values if the section is an

‘m-derived’ type.

[Each shunt arm 85.1 pFIseries arm contains 61.21

µH

inductance in parallel with 244.9 pF capacitor]

21 Determine the component values of an ‘m-derived’ T section ﬁlter

having a nominal impedance of 600 , a cut-off frequency of

1220 Hz and a frequency of inﬁnite attenuation of 1100 Hz.

[Each series arm 0.503

µFIseries arm comprises 90.50 mH

inductance in series with 0.231

µF capacitor]

22 State the advantages of an ‘m-derived’ ﬁlter section over its equiva-

lent prototype.

A ﬁlter section is to have a nominal impedance of 500 , a cut-

off frequency of 5 kHz and a frequency of inﬁnite attenuation of

4.5 kHz. Determine the values of components if the section is to be

an ‘m-derived’  ﬁlter.

[Each shunt arm 36.51 mH inductance;

series arm comprises 73.02 nF

capacitor in parallel with 17.13 mH inductance]

Composite ﬁlter sections

23 A composite ﬁlter is to have a nominal impedance of 500 ,acut-

off frequency of 1500 Hz and a frequency of inﬁnite attenuation of

1800 Hz. Determine the values of components required if the ﬁlter

is to comprise a prototype T section, an ‘m-derived’ T section and

two terminating half-sections (use m D 0.6 for the half-sections).

[Prototype: Each series arm 53.1 mH;

shunt arm comprises 0.424

µF

‘m-derived’: Each series arm 29.3 mH;

shunt arm comprises 0.235

µH

capacitor in series with 33.32 mH inductance.

Half-sections: Series arm 31.8 mH;

shunt arm comprises 0.127

µF

capacitor in series with 56.59 mH inductance]

24 A ﬁlter made up of a prototype  section, an ‘m-derived’ 

section and two terminating half-sections in cascade has a nominal

impedance of 1 k, a cut-off frequency of 100 kHz and a frequency

of inﬁnite attenuation of 90 kHz. Determine the values of the

components comprising the composite ﬁlter and explain why such a

ﬁlter is more suitable than just the prototype. (Use m D 0.6 for the

half-sections.)

[Prototype: Series arm 795.8 pF, each shunt arm 1.592 mH

‘m-derived’: Each shunt arm 3.651 mH; series arm 1.826 nF

capacitor in parallel with 1.713 mH inductance.

Half-sections: Shunt arm 238.7 pF; series arm 10.61 nF

capacitor in parallel with 1.492 mH inductance]

43 Magnetically coupled

circuits

At the end of this chapter you should be able to:

ž deﬁne mutual inductance

ž deduce that E

DM

, M D N

d

, M D N

d

and

perform calculations

ž show that M D k

L

 and perform calculations

ž perform calculations on mutually coupled coils in series

ž perform calculations on coupled circuits

ž describe and use the dot rule in coupled circuit problems.

43.1 Introduction

When the interaction between two loops of a circuit takes place through a

magnetic ﬁeld instead of through common elements, the loops are said to

be inductively or magnetically coupled. The windings of a transformer,

for example, are magnetically coupled (see Chapter 20).

43.2 Self-inductance

It was shown in Chapter 9, that the e.m.f. E induced in a coil of inductance

L henrys is given by:

E = −L

volts

, where

is the rate of change of current,

the magnitude of the e.m.f. induced in a coil of N turns is given by:

E = −N

volts

, where

d

is the rate of change of ﬂux,

and the inductance of a coil L is given by:

L =

henrys

842 Electrical Circuit Theory and Technology

43.3 Mutual inductance

Mutual inductance is said to exist between two circuits when a changing

current in one induces, by electromagnetic induction, an e.m.f. in the

other. An ideal equivalent circuit of a mutual inductor is shown in

Figure 43.1.

and L

are the self inductances of the two circuits and M the mutual

inductance between them. The mutual inductance M is deﬁned by the

relationship:

= −M

43.1

where E

is the e.m.f. in circuit 2 due to current I

in circuit 1 and E

the e.m.f. in circuit 1 due to the current I

in circuit 2.

CIRCUIT

Figure 43.1

The unit of M is the henry.

From Section 43.2, E

DN

d

or E

DN

d

43.2

Equating the E

terms in equations (43.1) and (43.2) gives:

 M

DN

d

from which

M = N

43.3

Equating the E

terms in equations (43.1) and (43.2) gives:

 M

DN

d

from which

M = N

43.4

If the coils are linked with air as the medium, the ﬂux and current are

linearly related and equations (43.3) and (43.4) become:

M =

and M =

43.5

Problem 1. A and B are two coils in close proximity. A has

1200 turns and B has 1000 turns. When a current of 0.8 A ﬂows

in coil A a ﬂux of 100

µWb links with coil A and 75% of this

ﬂux links coil B. Determine (a) the self inductance of coil A, and

(b) the mutual inductance.

Magnetically coupled circuits 843

(a) From Section (43.2),

self inductance of coil A, L



1200100 ð 10

6



0.80

D 0.15 H

(b) From equation (43.5),

mutual inductance, M D



10000.75 ð 100 ð 10

6



0.80

D 0.09375 H or 93.75 mH

Problem 2. Two circuits have a mutual inductance of 600 mH. A

current of 5 A in the primary is reversed in 200 ms. Determine the

e.m.f. induced in the secondary, assuming the current changes at a

uniform rate.

Secondary e.m.f., E

DM

, from equation (43.5).

Since the current changes from C5A to 5A, the change of current

is 10 A.

Hence

200 ð 10

3

D 50 A/s

Hence secondary induced e.m.f., E

DM

D600 ð 10

3

50

−30 volts

Further problems on mutual inductance may be found in Section 43.8,

problems, 1 to 4, page 864.

43.4 Coupling coefﬁcient

The coupling coefﬁcient k is the degree or fraction of magnetic coupling

that occurs between circuits.

k D

ﬂux linking two circuits

total ﬂux produced

When there is no magnetic coupling, k D 0. If the magnetic coupling is

perfect, i.e., all the ﬂux produced in the primary links with the secondary

then k D 1. Coupling coefﬁcient is used in communications engineering

to denote the degree of coupling between two coils. If the coils are close

together, most of the ﬂux produced by current in one coil passes through

the other, and the coils are termed tightly coupled. If the coils are spaced

apart, only a part of the ﬂux links with the second, and the coils are termed

loosely-coupled.

844 Electrical Circuit Theory and Technology

From Section 43.2, the inductance of a coil is given by L D

N

Thus for the circuit of Figure 43.1, L



from which, 

43.6

From equation (43.5), M D N



, but the ﬂux that links the second

circuit, 

D k

Thus M D



k







from equation (43.6)

i.e., M D

from which,

43.7

Also, since the two circuits can be reversed,

M D

from which,

43.8

Thus from equations (43.7) and (43.8),

from which, M

D k

and

M = k

43.9

or,

coefﬁcient of coupling, k =

43.10

Problem 3. Two coils have self inductances of 250 mH and

400 mH respectively. Determine the magnetic coupling coefﬁcient

of the pair of coils if their mutual inductance is 80 mH.

From equation (43.10), coupling coefﬁcient,

k D

L



80 ð 10

3



[250 ð 10

3

400 ð 10

3

]

80 ð 10

3

0.1

D 0.253

Magnetically coupled circuits 845

Problem 4. Two coils, X and Y, having self inductances of 80 mH

and 60 mH respectively, are magnetically coupled. Coil X has 200

turns and coil Y has 100 turns. When a current o

f4Aisreversed

in coil X the change of ﬂux in coil Y is 5 mWb. Determine (a) the

mutual inductance between the coils, and (b) the coefﬁcient of

coupling.

(a) From equation (43.3),

mutual inductance, M D N

d

1005 ð 10

3



4 4

D 0.0625 H or 62.5mH

(b) From equation (43.10),

coefﬁcient of coupling, k D

L



0.0625



[80 ð 10

3

60 ð 10

3

]

D 0.902

Further problems on coupling coefﬁcient may be pound in Section 43.8,

problems 5 and 6, page 865.

43.5 Coils connected in

series

Figure 43.2 shows two coils 1 and 2 wound on an insulating core with

terminals B and C joined. The ﬂuxes in each coil produced by current i

are in the same direction and the coils are termed cumulatively coupled.

Let the self inductance of coil 1 be L

and that of coil 2 be L

and let

their mutual inductance be M.

If in dt seconds, the current increases by di amperes then the e.m.f.

induced in coil 1 due to its self inductance is L

di/dt volts, and the

e.m.f. induced in coil 2 due to its self inductance is L

di/dt volts. Also,

the e.m.f. induced in coil 1 due to the increase of current in coil 2 is

Mdi/dt volts and the e.m.f. induced in coil 2 due to the increase of

current in coil 1 is Mdi/dt.

Hence the total e.m.f. induced in coils 1 and 2 is:

C L

C 2





volts D L

C L

C 2M

volts

COIL 1

COIL 2

Figure 43.2

If the winding between terminals A and D in Figure 43.2 are consid-

ered as a single circuit having a self inductance L

henrys then if the

same increase in dt seconds is di amperes then the e.m.f. induced in the

complete circuit is L

di/dt volts.

Hence L

D L

C L

C 2M

846 Electrical Circuit Theory and Technology

COIL 1

COIL 2

Figure 43.3

i.e., L

= L

Y L

Y 2M 43.11

If terminals B and D are joined as shown in Figure 43.3 the direction of

the current in coil 2 is reversed and the coils are termed differentially

coupled. In this case, the total e.m.f. induced in coils 1 and 2 is:

C L

 2M

The e.m.f. Mdi/dt induced in coil 1 due to an increase di amperes in

dt seconds in coil 2 is in the same direction as the current and is hence in

opposition to the e.m.f. induced in coil 1 due to its self inductance. Simi-

larly, the e.m.f. induced in coil 2 by mutual inductance is in opposition

to that induced by the self inductance of coil 2.

If L

is the self inductance of the whole circuit between terminals A

and C in Figure 43.3 then:

D L

C L

 2M

i.e., L

= L

Y L

− 2M 43.12

Thus the total inductance L of inductively coupled circuits is given by:

L = L

Y L

± 2M

43.13

Equation (43.11) - equation (43.12) gives:

 L

D L

C L

C 2M  L

C L

 2M

i.e., L

 L

D 2M  2M D 4M

from which,

mutual inductance, M =

− L

43.14

An experimental method of determining the mutual inductance is indi-

cated by equation (43.14), i.e., connect the coils both ways and determine

the equivalent inductances L

and L

using an a.c. bridge. The mutual

inductance is then given by a quarter of the difference between the two

values of inductance.

Problem 5. Two coils connected in series have self inductance of

40 mH and 10 mH respectively. The total inductance of the circuit

is found to be 60 mH. Determine (a) the mutual inductance between

the two coils, and (b) the coefﬁcient of coupling.

(a) From equation (43.13), total inductance, L D L

C L

š 2M

Hence 60 D 40 C10 š 2M

Magnetically coupled circuits 847

Since L

C L

<Lthen 60 D 40 C 10 C 2M

from which 2M D 60  40  10 D 10

and mutual inductance, M D

D 5mH

(b) From equation (43.10), coefﬁcient of coupling,

k D

L



5 ð 10

3



[40 ð 10

3

10 ð 10

3

]

5 ð 10

3

0.02

i.e., coefﬁcient of coupling, k

= 0.25

Problem 6. Two mutually coupled coils X and Y are connected

in series to a 240 V d.c. supply. Coil X has a resistance of 5 

and an inductance of 1 H. Coil Y has a resistance of 10  and an

inductance of 5 H. At a certain instant after the circuit is connected,

the current is 8 A and increasing at a rate of 15 A/s. Determine

(a) the mutual inductance between the coils and (b) the coefﬁcient

of coupling.

The circuit is shown in Figure 43.4.

V = 240 V

COIL X

COIL Y

= 5 Ω

= 1 H

= 10 Ω

= 5 H

Figure 43.4

(a) From Kirchhoff’s voltage law:

V D iR C L

i.e., 240 D 85 C 10 C L15

i.e., 240 D 120 C 15L

from which, L D

240  120

D 8H

From equation (43.11),

L D L

C L

C 2M

Hence 8 D 1 C 5 C2M

from which, mutual inductance, M D 1H

(b) From equation (43.10),

coefﬁcient of coupling, k D

L



[15]

D 0.447