t

- run el Technical College

Department of Aeronautical Studies
VOL SECT CA HP

1 1

C AT R H PE

ALTERNATING CURRENT

1,

Generationof an e,m.f, the l i n e s of force, then an e.m.f.

w i l l be induced or generated

in the conductor,
DIFtECTION 0 MOTION

"i

' DIRECTION OF 1 ,
MkGNETIC FLUX FIELD,

DIRECTION O INDUCED e,m.f, F

' \

/
-

2,

Fig. 43xsFleming s Right Hand Rule

\,

\-',

,L:\%(& .--

5:1

'

* C L

<

p , 'A

'Ref erring to conventional current, this r u l e i s used t o determine the If the thumb, forefinger and middle direction of the induced e.m.f. finger of the r i g h t hand are placed mutually a t r i g h t angles, then: i) ii) iii) The f i r s t Finger indicates the direction of the magnetic Field The s e ~ o n z finger indicates induced Current o r EMF The T u b indicates direction of Motron hM

SIMPLE ALTERNATOR

VL O 1 . SECT 1 CHAP 5
If the above p r i n c i p l e s a r e applied t o a simple a l t e r n a t o r a s shown, then an a.c. waveform would be produced t o pass through the load.

When the loop c u t s t h e f i e l d a t r i g h t angles, maximum e.m.f. i s produced. Minimum e.m.f. i s produced when t h e loop i s t r a v e l l i n g p a r a l l e l t o the f i e l d , a f t e r 90' of r o t a t i o n . For intermediate angles something l e s s than maximum i s produced.

-.

4 .
\

Sine Waveform
If the voltage generated in t h e a l t e r n a t o r loop i s p l o t t e d graphically a s t h e loop revolves, t h e r e s u l t w i l l be as shown in Fig. 5 3 . This i s known as a s i n e curve, because t h e voltage a t any i n s t a n t v a r i e s as the s i n e of t h e angle through which t h e loop i s turned.

5.

After t h e loop has r o t a t e d through the angle 8, from t h e p o s i t i o n of zero e .m. f then t h e p e r i p h e r d z e l o c i t y can be represented by i-' ,. . . t h ;V m/s, and by l i n e AL, drawn a t r i g h t angles t o the plane of t h e loop. AL can be resolved i n t o two components AM & AN.

Fig.
L

55
C

since

MLA
AM

=
=

90"-

PULL= MA0 AL Sin 8 = V Sin 8c

. . 8

A s V Sin B i s always a t r i g h t angles to the f i e l d , i t gives an induced e.m.f.

E V Sin 8 Volts L

- --

(5:1)

Since B, L and V are constant, then the induced e.m.f. to Sin 8. The e,m.f. Total e,m.f. generated i n loop = 2 BLV Sin 0
=

i s proportional

generated i n one s i d e of t h e loop = B V Sin 0 Volts. L

2 B V S i n 8 Volts L

When 0 = 90, then E max

(5:2)

'V0L
SECT CA HP Slip Rings

1
1

5;

In order to take off the output from the alternator the ends of the loop are attached t o two separate insulated brass rings, which turn with the loop. Tho brushes under spring tension,bear on the s l i p rings, and these a r e attached to the external c i r c u i t .
Terminal Voltage Under b a d Conditions

In general, as soon a s a load i s connected to the terminals -of the alternator, a current w i l l f l a w . Since t h i s current must flow through the i n t e r n a l resistance of the loop, brushes, l b d s etc. there w i l l be an i n t e r n a l voltage drop across t h i s i n t e r n a l resistance. ?he terminal voltage will thus be l e s s than the r l ul voltage developed.

!hey cycle (c) i s a complete cycle of alternating events. Frequency ( f ) ?he number of cycles per second i s referred to as the frequency The u n i t i s the Hertz (HZ). of the a.c.

Angular Velocity W

Fig.
I

5:6

The radian may be defined as 'the angle subtended a t the centre of the c i r c l e , by an a r c equal i n length t o the radius. One radian
=

~ 7 . 3 ~

2 T o r approx 6.28 radians = 360' A s there are 2 radians in one rev., and there a r e f. revs/sec, the conductor moves through 2% f r a d i a n ~ / ~ e c . The angle turned i n one second i s termed the angular velocity.

-!! a!
1 . Periodic Time. T. 1
This i s the time taken for one cycle, the unit i s the second

VOL SECT

1 1
J f

At r n a i ~ a , ~ U ~ v nf 50 HR;
L

T =

;O

0.02 secs.

12.

Instantaneous value of an a.c. i s the value of current or voltage a t a given instant of time dhring i t s cycle.

Angular velocity i s 2 71 f. radians/sec. If time taken by OA to rotate through an angle 0 radians i s t secs., then
0 = Angular velocity x kine
=

- - - (5.4)
7' 7
= 30'

2 T f t = 2 T ' x 1x

1 i2=

The instantaneous value of the e.m.f. e = E.max. so Sin 0 volts 1

at

w i l l be:

e = Eonax. Sin w t volts

Similarly i = I.max.

Sin w t Amps.

VOL 1
SECT 1 CHAP5 13. Comparison of Phase 'Iko sinusoidal voltages or currents may have the same frequency but different phase.

If two s i n e waves such a s vl and v2 i n Fig. 5:8 have the same frequency, but pass through zero a t different instances, they F a r e said to be OUT O PHASE, and the angle $ by which they a r e H S out of step i s c a l l e d the P A E ANGLE. In Fig. 5 8 the sine wave voltage v2 i s leading the sine wave voltage vl by d radians, since i t passes through zero f i r s t , $ radians before v O course i t f i s only possible t o specif'y the phase angle, betaeen two sine f waves, i f they have the same frequency. I a sine wave voltage (or current) i s compared with another of the same frequency then i t i s more accurately specified by the general expression:-

Sin (wt + 6) v o l t s

- - - (5:5)
Phase Angle (Radians) Instantaneous voltage (volts ) = Peak Voltage
= =

where: $ e

Em w

= 2 n f (rads/sec) f = frequency (hertz)

A sine wave voltage of 2 9 v o l t s peak value and frequency 50 Hz leads another sine wave voltage of 300 v o l t s peak value and same frequency by 45 degrees. Express these voltages mathematically, and represent them by phasors and waveform.
Solution : Let the two voltages by vl and v2 respectively.

" 1

= 250 s i n (100-ii t +

volts

v2 = 300 sin (100.11 t )

rn

VOL SECT

1 1

,. -

CHAP

P wb a s .

Fig.

5 9

Average Value (a) Average Value o r &an VJue of a complete cycle of alternating current or voltage i s always zero because the curve i s symmetrical about zero. The MEAN value taken over one h a l f cycle, f o r a s i n e wave, i s or peak value 0.637 or 2 of the -mum

(b)

e. average = 0.637 E max Volts

i. average = 0.637 1 max A p ms
If- the wave i s other than a sine wave, then the average can be calculated graphically by the mid ordinate rule.

1. 5

Effective value (r.m.s)

of an a.c.
1

This value of alternating current i s equal t o the value of doc. which would give the same power or heating e f f e c t in a r e s i s t o r a s the average power expended by the a. c.

@ 5 2 &

Average heating e f f e c t of a.c.

=

shown in Fig. 5:10

( I max) R watts 2 Same heating e f f e c t of doc. = I R watts I ~ R=

=

I -.R I2

watts watts

0.707.

I m a Amps ~

A graph i l l u s traing the above points i s

- - - (!XI shown - - - Fig.

5:10.

The r.m.s. value may'also be defined a s the square root of the mean value of the squares of the instantaneous values taken over one complete cycle.

16.

Form Factor
.
--

POL SECT

mdP__~ -

1 1

The Form Factor i s an indication of the shape of the wave. If the value i s l e s s than loll, the wave i s usually f l a t topped; i f greater t h e 11 , the wave i s pea@. .1

Form Factor

r.m.s. value Average value

(5:7)

Ihe Form factor f o r a sine wave over half a cycle.

,

- 0.707. max value

0.637.

maxvalue

For a rectangular wave = r.m.s. Average

=

1 0

because m x i r m u n r. m. s

., and average values a r e equal.

17.

Fourier Analysis
All periodically repeating waveforms can be shown t o be made up of a number of sine waves which are mathematically related. The lowest frequency i s called the f'undamental, harmonics are multiples of the fundamental frequency. The a d , bth, 6th, etc. are even harmonics, whereas the 3rd, 5th and 7th etc. are odd harmonics. Those a r e i l l u s t r a t e d i n Fig. 5:ll and 5:12 respectively.

18.

Pure Resistance i n a.c.

circuits

Fig. 5 1 3

VOL

SECT CHAP

1 1

'

'

Since the c i r c u i t i s purely r e s i s t i v e , t h e current a t any i n s t a n t i s proportional to the applied voltage, and therefore t h e maximum and minimum values of voltage and current appear together, they a r e i n phase with each other.
O m I a w applies t o the circuit:h ' s

The power d i s s i p a t i o n i s :-

v2 Watts Power = V x I, o r I2R, o r -R

A a.c. c i r c u i t w i t h a r e s i s t i v e load can be t r e a t e d a s a d.c. n c i r c u i t f o r the purposes of calculations, t h e value of the frequency does not matt=.

19.

Pure a.c.

Inductive c i r c u i t

INp 4 d ~ e (%DacrM&
L ~

Fig.

5:14

I S '.:

- 13 -

VOL SECT CA HP

1 1

When an a.c. flows i n an inductance, a back e.m.f. i s s e t up i n the c o i l , and this back e.m.f. i s proportional t o the r a t e of change of current. Since there i s no resistance in the c i r c u i t the applied e.m.f. w i l l be used i n e n t i r e l y overcoming the back e.m.f. and i t i s therefore, equal and opposite to i t a t every instant. This must be so a s there a r e no lo4sses due t o the resistance in the c i r c u i t . Consider Fig. 5l-4
A t O the current i s increasing a t i t s maximum r a t e , so the back
,
-

- - -

e.m.f. must be a t a maxirmun and in opposition t o the applied e.m.f. A t A the current i s steady f o r an i n s t a n t and since there i s no r a t e of change the induced e.m.f. i s zero. The next h a l f cycle continues as shown.
THE CN -T IN TED3 CIRCUI T IS L G I G ON THE APPLIED E .M.F. A GN I N A PURE INDUCTANCE THE ANGLE O LAG IS 'DIEREFORE 900. F
20.

BY 90'.

' 7

Wattless Current
A s the current through the c o i l increases, energy i s put i n t o the magnetic f i e l d $LIZ joule but when i t decreases the energy i s returned in fill t o the circuit. %ere i s therefore no energy l o s t and therefore no power dissipation. This i s termed a wattless curr en t

21.

Reactance

XL

Of

XG

Reactance i s the a b i l i t y of a c i r c u i t to r e s i s t the passage of an a.c. without the dissipation of energy.

A reactance can be inductive o r capacitive,
22.

Inductive Reactance

YL

This i s the opposition offered to an a,c. by an inductance. Reactance i s therefore expressed as a r a t i o V and i s measured i n 7ohms.

It i s represented by the symbol XL

Reactance XL and
=

WL or 2.11-f L ohms

--=

(5:10)
amPS

XL= V

ohms therefore 1

2FpL

POL

1 SECT 1 CHAP 5

23.

Variation of ~ e atGG--&d Current with Frequency c

x ohms
I amps

FREQUENCY Hz
Fig.

5:15

'Ihe inductive reactance i s proportional t o the frequency and the current i s inversely proportional t o the frequency. These relationships a r e represented graphically by the s t r a i g h t l i n e and the hyperbola. Lhductance and Resistance i n s e r i e s

Fig. 5:16

!his i s a more p r a c t i c a l c i r c u i t since some resistance must be present. ?he resistance shown m y be the resistance of the c o i l a or an external r e s i s t o r .

VOL 1 SECT 1
- -

25.

In t h i s type of c i r c u i t the same value of current flows through R & L. T f o r c e t h i s current through the circu&t the voltage o There must be across the inductor must l e a d the current by 90 , a pad. across R and L but they a r e out of phase, and t h e r e s u l t a n t pad. (applied voltage), w i l l be the phasor or vector sum, and not u the arithmetic s m of the two. The phasor diagram i s shown i n

Fig. 5:16.

..

Tne f a c t o r J ~ v t a k e sthe place of r e s i s t a n c e i n ohms law, i s the t o t a l opposition t o current in t h e c i r c u i t and i s known a s the impedance.

.
/

.@;?:-.
.-4-

= 1.2.

- - volts
=

(5:12)

Applied Voltage V Phase Angle Tan 0

WL

26.

Power i n the Circuit

All t h e power used by the c i r c u i t i s due t o the r e s i s t a n c e and i s given by the v o l t s drop across the r e s i s t a n c e times the current through it.
True Power but VR
= I.VR
a

V Cos (8

rue power
27.

= I.V.

cos

$ watts

---

(5:a)

Apparent Power This i s the product o f V and I, and is the pobTer apparently being supplied by the source. It i s measured i n K o V o A e where Apparent Power
= V

x I KVA

---

(5:15)

VOL SECT
--

a-u.P 5

1 1
-

28,

Power Factor ?he power voltage V 'he Power and u n i t y f a c t o r i s t h a t f a c t o r by which t h e current and applied must be multiplied t o obtain the t r u e power dissipated. Factor l i e s between nought f o r a purely REACTIVE c i r c u i t f o r a purely RESISTIVE c i r c u i t ,

Power Factor Cos

O'

rue rower Apparent power

- - - (5:16)

Fig. 5:19 29, Pure Capacitance

hpedance Triangle

Fig. 5 ~ 2 0

VOL 1 SECT 1 CA 5 HP
~ ~ r a r g c Increasing a t i t s m a x b m race, and so the r ~ s current i s a t a maxianun. A t A the charge i s not changing, so the
AL n b

u ale

11--

-1

. .

current i s not changing; a t B the charge i s changing in the opposite direction a t i t s marrdmun rate, and again the current i s a t m;udmum.

SINCE THE APPLCED V L A E I S PROPORTIONAL TO 7HE CHARGE I T F L O S OT G OL W THAT THE CURRENT I S 90 OUT O PHASE; IT IS LEADING THE VOLTAGE BY 90' F

As the volta e i increasing, energy i s being stored i n the 8 capacitor, (% CV joules) and when i t decreases again to zero this energy i s returned i n t a c t t o the circuit. Thus the n e t expenditure of energy i s zero, and the current i s known a s a wattless current.

' 7

31.

Capacitive Reactance
If the frequency of the supply t o a given capacitor i s increased,

there i s l e s s time t o charge the capacitor t o the f u l l supply voltage. The r a t e of charging must therefore increase and the current increases. The same e f f e c t i s noted i f the frequency i s held steady and the capacitance increases. As the frequency or capacitance increases, the current increases also.
This i s equivalent to saying the opposition to the current has decreased.

xc Also :
32.

zc
I

or ohms.

2 . n fC
".*

ohms.

---

(5:17)

xC

'rn

Variation of Reactance and Current with Frequency

Fig. 5 2 1

Capacitive reactance i s inversely proportional t o the capacitance and the frequency of supply voltage. The s t r a i g h t l i n e and the hyperbola show how the capacitive reactance and the current due t o a given applied voltage vary with frequency.

---.

VL O SECT CA Xp

1 1

33.

Capacitance and Resistance

Fig. 5:2

Current flowing in this c i r c u i t w i l l cause a voltsodrop across the r e s i s t a n c e and a v o l t s drop across C lagging by 90 Applied Voltage V 2
= = =

VR

+ VC 2

I ~ R ~~x~~ + ~
1 (R2 + xc2) 2

'&is expression J R' measured in ohms. Phase angle tan $ =

xCz i s
=

the impedance Z of the c i r c u i t and i s

1 WGR

Vc

I.xc= I.R.

---

(5:19)

W the power in t h e c i r c u i t i s due t o the resistance, and i s given by the v o l t s drop across the resistance, times the c u r r e n t through it.
h e Power
=

1.V

b u t VR = V Cos # Cos $ Watts

so True Power

I.V.

VOL
SECT

1 1

From the above vector diagram Cos $ = VR

TIThus the c i r c u i t again has a power factor = 35.

-

E
I

131

Cos

Power Factor of a Capacitor ---

--

Since there are c e r t a i n losses i n a capacitor which can be represented by a resistance in series, i t follows t h a t in a p r a c t i c a l c i r c u i t the angle between voltage and current can never be quite 90'. Zhe smaller the losses the greater the angle and the l e s s the power absorbed. Zhe power dissipated by i t w i l l be 1.V. Cos d. Cos d = 5 and when z R i s small i t may be neglected i n the impedance, and then Cos d = fl = 1J c R .. - - - (5.22)
7

36.

L, C and R in Series

Fig.

5.24
I i s the reference phasor since I i s common to all components i n the c i r c u i t .
Fig. 5.25

vc.

'

VOL 1 SECT 1
a-Ifl

v
where Z

= I z
0 n

J R+ ( x ~ & % ) ~ ~
1

---

(5.23)

Phase Angle = tan

Reactance Resistance

Power Factor = Cos 0 = True Power Apparent Power

37.

Distribution of p.d. Current flowing through the c i r c u i t causes a v o l t s drop across each coqonent. 'Ihe phase relationships t o the current i s shown by the phasor diagram (Fig. 5. 25)
VL and VC are measured a s separate voltages, but the r e s u l t a n t

voltage across the two components would be t h e i r difference. 38. Series Resonance
v k

Resonance i s defined a s the condition, i n a c i r c u i t containing k d e r t h i s condition: reactances, when the power factor i s unity. (a) (b) (c) The phase angle $ = 0 Ihe voltage applied i s in phase w i t h the current flowing ]I = XC, t h a t i s the combined reactance i s zero. I.

VOL

SECT
(3IA.P

1 1

(e) (f)
(g)

c i r c u i t behaves simply a s i f i t w a s a pure resistance Ihe Ihpedance i s a ndnirmun Z = R Ihe power taken from the supply i s s+ly t h a t required to replace the energy l o s t due t o the resistance m e c i r c u i t i s in a s t a t e of voltage resonance.

When the inductance and capacitance are known the resonant frequency can be found.

(2- f r )

2-

- Ec

where f r i s the resonant frequency

A s e r i e s resonant c i r c u i t i s termed an acceptor c i r c u i t because i t accepts current mst readily a t the resonant frequency. A acceptor n c i r c u i t i s also often called a voltage magnification c i r c u i t because a t resonance the v o l t a e across the inductance (equal t o the voltage across the capacitance i s greater than the supply voltage V.

39.

L and R in P a r a l l e l

Fig* 5.27

VOL SC ET CA HP

1
1

In general the currents i n each branch of the c i r c u i t w i l l not be in phase with each other. Current in R = V / ~ , and i t i s i n phase Current in L = v/tJz, lagging by 90' w i t h V.
Total current i s found f r o m the phasor diagram.

<\

--@
=
=

(5.28)

Phase Angle = tan-'

tan tan

= tan

- v 1

hx;)

Phase Angle 40.

PI =

tan

- R 1 bL)

C and R in Parallel

-<=- \.I/
,-

' -

Fig. 5.28

>

4..

Current through R =
1

, and i s

in phasewith V

'' 5
,

*,

'

-\..Current

through C = V.wc,

Total Current I hpedance Z

,\
<

=I=

and leads V by 90' A

--- - -

(5.39)
(5.31) (5.32)
a

Phase Angle Tan (8

Ic = V x R

c-

- -'.
,

. , , ', -J-\

\--

WR c

--2

vc

VOL
SC ET CA HP

1 1

c.,
-'

./ , - - ...,.I,
k .

L L
LC

- -- c -

1
'

4 ( I t r

. " ?

"?"tk.

~ o t a Current l
.-

I,= &,

A T )
T
IR
= Z
. I

Fig. 5.29

---

(5.33)

-<-!

/-

Phase Angle = tan-'

<<

/-*

.-

Power Factor Cos k e Power

= V.I.

I
Cs o

R
or
V x

++

w8rk5'

4.

Parallel Resonance

Theoretical Case.

Fig. 5.30

The resonant frequency in the ideal case o f no resistance = 1 2 T

Hz

Jm

VOL SECT

1 1

5
X , I
=

IC =

5 but currents_ a r e 180'

0 0

out of phase

(Ic+%)

= 0

(5.36)

43.

Parallel Resonance
/

Practical Case
-

F-

k. ; c
*

Fig. 5.31

I n actual practice inductors and capacitors possess r e s i s t i v e components representing the e l e c t r i c a l losses i n them. ?hose i n the capacitor are usually small compared with the inductor 10sses.
If Il and Ic a r e such that the r e s u l t a n t current I i s i n phase w i t h

the supply voltage a s shasn, the c i r c u i t i s said t o be i n resonance. The frequency of resonance when the external current and voltage are i n phase
/-

-. .. . ..

/ +

f,

1 -

rc
I

2;

---

(5.37)

If R i s very small compared w i t h L, a s i n radio c i r c u i t s then: 1 fr = Hz, a s f o r the s e r i e s c i r c u i t - -

2T

, / r

(5.38)

Since the resultant current i n a resonant p a r a l l e l c i r c u i t i s i n phase with the supply voltage, the c i r c u i t i s equivalent t o a pure resistance = L/CR.

/'

,!

.>

Dynamic resistance o r Z
. r.-\ '
>F-"\"

=<g L

Ohms

- - - (5.39)

This type of c i r c u i t when used in radio, i s referred t o a s a rejector since i t s impedance i s a maximum and the r e s u l t a n t supply current a mininnun a t resonance. ?he rejector c i r c u i t i s in a condition of current resonance.

VOL 1 SECT 1

,-'

CHAP 5

Ihe dynamic impedance Z of the p a r a l l e l c i r c u i t can a l s o be calculated from

or

=
=

XceQ
(see pa.ra.56)

---

(5.40)

where Q

45.

Forced and Free Oscillations Forced: These a r e o s c i l l a t i o n s in a system, which a r e maintained by an external supply of energy, and which have the frequency of the external supply. The resonant frequency of such a sys tem i s given by:
i,
ii.

For s e r i e s c i r c u i t f r

1 21JTE

Hz

For p a r a l l e l c i r c u i t fr =

1
2

- c' 1

Hz
T2

- - - (5.W)

46.

Free Oscillations a r e oscillations in a system having capacitance, inductance and resistance, of which the frequency i s s o l e l y dependent on the cons t i n t s of the circuit. The frequency a t which f r e e oscillations occur i s termed the n a t u r a l frequency, and i s given by:

-/ G c From this expression i t i s apparent that i f R i s small compared \ .

=,./

L+ J.

iil ( h ( r t pd c ~ ;

with L then:
f (nat) =

fr.
but theoretically

*VTr
For a l l but the m c s t exact purposes f r = f(nat); f ( n a t ) i s always than fr.

hit++
'J'

47.

Circulating and Feed currents Although a t resonance the supply current in a p a r a l l e l tuned c i r c u i t i s very low, there a r e large currents flowing i n the c i r c u i t ( I + I&, Since these are 180 out of phase, the r e s u l t i s a ~ current circulating i n the c i r c u i t , and t h i s i s known as the circulating current.

VOL SECT
CHAP

1 1

In a practical c i r c u i t a supply current i s needed t o overcome the

-.
I

resonance the circulating current i s l a r g e r than the feed current, and so a p a r a l l e l tuned c i r c u i t has the property of current magnification.

\-

48.) ihe uses of k e d Circuits The chief use of the s e r i e s c i r c u i t i s in i t s high resonant current and i t is used t o tune aerials.

Source of a 1ternating e.m.f.

Example o f Series Resonant o r Acceptor Circuit.

Fig. 5.32

qP-!& &T-'
, / a '

Another use f o r the s e r i e s c i r c u i t would be t o bypass p a r t i c u l a r frequencies not required by the c i r c u i t , t h a t i s i t a c t s l i k e a filter in removing a particular frequency. 49. The high impedance a t resonance of a p a r a l l e l tuned c i r c u i t , i s also made use of i n radio, because f o r a given current i t w i l l have rmcimm voltage developed across i t a t resonance. For this reason i t i s used frequently as a load in ref. amplifying stages. It can also be used as a f i l t e r in s e r i e s with the supply so developing m a x h u voltage a t the resonant frequency and r e j e c t i n g the undesired frequency. Response Curves

P"'

\A

" ,

< \

L'

1.

*.\

$. ;

50.

fig.

5.33.

'

VOL 1 SECT 1
CA HP

- -\

I a graph of current against frequency i s plotted f o r a s e r i e s f acceptor circuit, I i s rmxbum a t the resonant frequency, f a l l i n g off above or belaw this fl-equency.
&om the above representative curves the f a l l off in c i r c u i t in the f i r s t curve i s steeper than in the second. This means that the first discriminates more sharply than the second.
clac, 1s

-C I I U L- Q p S d d eCm

+ m - bu o uv n

AIM^?^ a u r n m +u -r- v u en1 u b q A

+L-ullcl+l

the second. Response curves are d s o drawn of the c i r c u i t impedance plotted against frequency.

( 51.j P a p a e l Circuit

As i n the series c i r c u i t , the sharper the peak, the more selective the c i r c u i t . This follows, since the r a t i o between Z a t resonance and a t non-resonance i s greater.

Effect of L and C on s e l e c t i v i t y
I

If the resistance of the c i r c u i t i s kept constant but the values of L and C are varied in the same proportion, (increasing the L/c r a t i o ) , the resonant frequency w i l l not be a l t e r e d since i f L i s doubled, C i s halved, and both reactances are doubled. A t a l l other frequencies except the resonant frequency, i f the LC r a t i o i s high

the impedance of the c i r c u i t increases more rapidly and s e l e c t i v i t y i s improved.

VOL

1 SECT 1

F i g . 6:59

Fig.

5.35

These curves a r e representative only.

53.

Effect of Altering R only

If the L/C r a t i o i s kept constant and the resistance i s increased, the current a t resonance decreases in accordance with ohms law. The response curve becomes f l a t t e r and s e l e c t i v i t y decreases. However, this i s not q u i t e the same a s the reduction with the L/C r a t i o , since t h i s does not cause a reduction of current a t resonance.

54.

Effects of Departure from Resonance 'Ihe resonant frequency of a p a r a l l e l tuned c i r c u i t can be a l t e r e d by varying the L and C values. For a constant LC r a t i o there i s one frequency a t which the c i r c u i t i s resonant, and above o r below t h i s frequency the impedance of the c i r c u i t decreases and the supply current increases. A t resonance the inpedance i s greatest, but the more resistance i n the c i r c u i t , the l e s s the r e s u l t a n t impedance becomes.

55.

Q B c t o r (Magnification ~ a c t o y )

Series Circuit !he voltage across L & C f o r s e r i e s resonance can be many times the applied voltage. Thus the s e r i e s resonant c i r c u i t has the property of magnifying the voltage. The extent of t h i s magnification depends on the value of L & C since VL = I(x~)and VC = I(xc). Since the voltage across L & C i s the same a t resonance only the voltage across the inductance w i l l be considered. Ihe r a t i o L/V i s termed the Q of t h e c i r c u i t ; inductive voltage, the greater the Q.

the greater the

VOL SECT CA HP
--- -- --

1 1 5;

47 -

If &L i s increased t o increase Q, then &must be increased i n order t o maintain the c i r c u i t resonant, -itnd this i n c r e a s e s t h e L/c r a t i o .

From a c i r c u i t with a high Q, a high s e l e c t i v i t y i s obtained. Because of this, the Q of a c i r c u i t may be defined a s a measure of i t s s e l e c t i v i t y .

56.

P a r a l l e l Circuit

In a p a r a l l e l c i r c u i t a t resonance the c i r c u i t has t h e property

of current magnification.
= supply current

Circulating Current : I c :I

_ _ -(5.44)

I t h e p a r a l l e l c i r c u i t the Q f a c t o r i s again a measure of i t s n s e l e c t i v i t y and t h i s i s given by:

57.

Coupled G h c u i t s

It i s very frequently necessary t o t r a n s f e r energy from one c i r c u i t t o another, and when two c i r c u i t s a r e arranged so t h a t this takes place, they a r e s a i d t o be coupled.
There a r e two types:
(i) (ii)
~hd. coupling Direct coupling

(i) ~ t u a coupling l

Fig. 5.36

1 The a l t e r n a t i n g f l u x i n L l i n k s with the turns i n L2, and s e t s up an emf such t h a t both c i r c u i t s o s c i l l a t e .

Because c i r c u i t 1 i s coupled t o c i r c u i t 2, the impedance of c i r c u i t 1 appears to a l t e r , and i t can be shown t h a t t h e e f f e c t i v e impedance o f c i r c u i t 1 i s dependent on the degree of coupling, known a s the coupling f a c t o r .

VOL SECT a-I.AP

1 1

Ih f a c t the degree of coupling i s usually expressed i n terms of the Bupling Factor 'kt, such that:-

- - - (5.46)
where M = k t u a l h d u c tance (Henrys )

L1

& L2 = Self Inductance of c o i l s

(Henrys )

+ '

58.

Direct Coupling This type can be subdivided i n t o two d i s t i n c t groups :

(v !

(i)

Coupling by an impedance common to both c i r c u i t s

Resistor can be replaced by capacitor o r .inductor.

Fig*

5.37

Diagram 5.37 Component 1 - In this case t h e v o l t s drop across R . a c t s a s an applied voltage t o c i r c u i t 2. In order t o increase the coupling, increase Re w ; - , \fl.L.r /L *, ! -

Diagram 5.37 Component 2. - Again the v o l t s drop across L i s the T o voltage applied t o c i r c u i t 2. ----- increase the coupling increase L. ----

w;c\,

,-L

\c.\J

Magram 5.37 Component 3. - This form of coupling may be termed common capacitor, o r bottom capacity coupling. 'In order t o increase the coupling decrease the capacitance. w:L\ w l -- - -c a c t s (ii)
,>

Coupling by Independent hpedance

Fig. 5.38

VOL 1 SECT1 CHAP 5

--

a2ei7gjj- is L a + m d Lo cne second ci c n t r --- . by means of a current feed; i t i s necessary -t o decrease R t o increase the c a p l- i n g . - - -- T m

Fig.

J,JU\,J

3 Q f l\

In diagram 2, where a capacitor i s used, coupling i s increased by increasing C ( i e d e c r e a s i n g c )

The coupling shown i n diagram 2 i s used widely in radio c i r c u i t s , and i s known a s top capacity coupling. Another capacitor i s B ~ r e g n e n b l yu b r o u u c e a lnto tne lower ranch, 111 e l t h e r or diagram 2, R or C could be replaced by an inductor a s shown i n 3 . In the above t h e coupling components a r e not a p a r t of e i t h e r N.B. tuned c i r c u i t
L-l
1 1
7 . .
I .

59.

\ Wf e c t of Degree of Coupling
As i n the case with mutual cdupling, i t can be shown t h a t the variation in the amount of coupling of any of the above c i r c u i t s r e s u l t s in varying response curves. The greater the degree of coupling, the tighter they a r e said to be coupled.

' Response curves f o r Coupled Circuits

-0
4

cor
C-HL

cr~a 't

Fig. 5.39, 60.

If primary and secondary response curves are plotted f o r a coupled c i r c u i t having a c o m n frequency band, and various values of coupling, the r e s u l t s would be as shown.

For weak couplings both curves have a max, current a t the resonant frequency. A s the coupling i s increased, the peak primary current decreases and the secondary peak increases, u n t i l a t a special value of k t h e pIlimary curve develops a s l i g h t double hump effect, w h i l s t the secondary curve has a sharp peak and maximum height. 'his value of k i s called the c r i t i c a l coupling.
If t h i s value i s exceeded the secondary curve also develops a double hump, and the greater the value of k the further from the resonant frequency these humps become. Ihus, c i r c u i t s a r e said t o be overcoupled i n this condition.

VOL 1 SECT 1 CHAP 5

-

61.

!he usefulness o f these curves l i e s in the f a c t , t h a t by varying the coupling the c i r c u i t s can be made very s e l e c t i v e , o r by a overcoupling they m y be made t o respond t o a band o f frequencies. Banhidth and Bandpass E f f e c t s

62.

The bandwidth of a tuned c i r c u i t i s most frequently defined a s the frequency range between thepoints on e i t h e r s i d e of the resonance curve where the power i s reduced t o a half of the peak In terms of current o r voltage, this means a reduction v L the aak value; o r '707 peak value, because the power i s proportional 5 to 1 or ~ 2 . Thus the frequency range between p o i n t s X and Y i s the Bandwidth Be

63.

Filter c i r c u i t s
F i l t e r c i r c u i t s a r e four terminal networks designed t o pass a c e r t a i n required band of frequencies from input t o output terminals, or t o f i l t e r o f f , or a t t e n u a t e the remainder of s i g n a l frequencies present a t the input terminals.. Such f i l t e r c i r c u i t s use a s t h e i r b a s i s the f a c t s t h a t the reactances of inductors and capacitors, and the impedances of acceptor and r e j e c t o r tuned c i r c u i t s a l t e r with frequency. They a r e thus made up from the r e a c t i v e elements of induc tanc e and capaci tanc e

F i l t e r c i r c u i t s take f o u r main forms: i) ii) iii) iv) High pass f i l t e r s Low pass f i l t e r s Bandpass filters Bands top filters

VOL 1 SECT 1 CA 5 HP
\

.-\

-- -

--.

64.

Highpass F i l t e r s

F!rom a l l the numerous input frequencies these f i l t e r s pass on to the output terminals, a l l those above a c e r t a i n cut o f f frequency, and f i l t e r off or attenuate the remaining lower frequencies.

65.

I h e c i r c u i t of Flg. 5.43 shows a simple high pass filter. C. passes the higher frequencies easily, onto the output terminals. It offers high reactance t o the lower frequencies, but L offers low reactance to these so they are f i l t e r e d o f f through it. L offers high reactance t o the required high frequencies and thus does not attenuate them appreciably. A typical attentuation/frequency graph f o r a simple high pass f i l t e r i s shown.

Fig. 66.

5.u

1.

I n practice a number of these f i l t e r c i r c u i t s a r e used i n succession a s shown. This improves the attenuation of the lower frequencies and so the cut off region becomes more abrupt and c l e a r l y defined.

WL SECT
CHAP

1 1

FMm all the various input frequencies the f i l t e r passes on t o the output terminals all those below a c e r t a i n c u t off frequency, and f i l t e r off, or attenuate the r e s t of the higher frequencies. A simple l o w pass filter i s shown.

68.

L passes the lower frequencies e a s i l y onto the output terminals. It offers high reactance t o the higher frequencies, but C offers
low reactance to these, so they are f i l t e r e d off through it. C offers high *reactance to the required low frequencies and t h i s does not attenuate them appreciably.

A typical attenuation frequency curve f o r a simple l o w pass f i l t e r i s shown. h practice a number of these filter c i r c u i t s a r e used i n s u c c e s s i o ~ . This improves the attenuation of the higher frequencies, and so the cut off region becomes m r e sharply defined.

69.

Bandpass F i l t e r s

Fig. 5.46

From a l l the numerous input frequencies these f i l t e r s pass on to the output terminals a certain narrow band, and f i l t e r o f f o r attenuate, the remaining frequencies above an@ below this. A simple bandpass filter i s shown. Rejector c i r c u i t and acceptor c i r c u i t L, C are , tuned to the same frequency, the centre frequency of theLre&ired band. NO mutual coupling exists between L and L2. 1

- 37 -

"

-C

. . , _ < ., . - .-r&4;eT; 1-

. d

..-3qA2-+.
.I-,

H'.,3*:.

d r f d ~

VOL

. .

SECT 1

5
70.

The acceptor c i r c u i t offers low inpedance t o the resonant frequencies, and passes them on t o the output terminals. It offers high inpedance to a l l the other input frequencies, but the rejector c i r c u i t offers low impedance to these and so they a r e f i l t e r e d o f f through it.

Fig 6 . 7 9 .

i (

I -

La

A typical attenuation frequency curve i s shown for a simple bandpass f i l t e r . A more p r a c t i c a l c i r c u i t i s shown, and t h i s w i l l give more c l e a r l y defined cut off regions.

71.

Badstop f i l t e r s

T;

T3
Fig.

5.48

INPUT

Om~q

lhese f i l t e r s pass on t o the output teroLinals all frequencies except a c e r t a i n narrow band which i s attenuated or f i l t e r e d off. Ihe c i r c u i t shows a simple bandstop f i l t e r . Acceptor c i r c u i t

5 C1

and rejector c i r c u i t L2 C2 a r e tuned to

the same frequencies; the midpoint frequency of the unwanted band. N mutual coupling exists between o and L2.

VOL SECT
CA HP Ihe rejector c i t c u i t offers low impedance t o all the required frequencies, and so passes them on t o the output terminals. It offers high impedance to the unwanted band of frequencies, but C offers a low impedance to these, and so acceptor c i r c u i t 'off through it. they a r e filtered

Ihe acceptor c i r c u i t offers high impedance t o the wanted frequencies,

ana so, aoes not attenuate tnem appreclao~y.

r
.--. . i.. -.. .. . - .
--.<..a

. .. .

Uz.

Elrunel Technical College, &is t 0 1

GSB/PJR

JUNE 1983