A Study of Shunt-Ladder Feedback Amplifier

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A 6XQSX OF A SHOHS-kADBSR FEEDBACK AHBLIFIEB

w Hoya! HUEttoon, Itohhardi

A dissertation submitted in partial fulfillment of the requirement® for the degree of Boetor of Philosophy in the Bepartme&t of Blaotrlc&X Engineering in the Graduate College of the State Quivers!ty of Iowa August* 1.980

Stats

University of tovr^ y m m r

ProQuest N um ber: 10991954

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uest P roQ uest 10991954 Published by ProQuest LLC(2018). C o p y rig h t of the Dissertation is held by the A uthor. All rights reserved. This work is p ro te cte d a g a in s t u n a u th o rize d co p yin g under Title 17, United States C o d e M icroform Edition © ProQuest LLC. ProQuest LLC. 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, Ml 4 8 1 0 6 - 1346

'V\=b*bO S. AOE$f(SyBSMlSfB Sh* aotter wishes to m m m n

M$

eimore tfcaafce to

Ppofe**©** 1* $s* Iha?t»* frofoaeo# If* &« ***** Prefe**** B* M*

iKil of

« & Professor 4* ¥* ftenjaalXy of the Depart*

3fcgi&*erfsig

to f m t m m m t B m %mt.»

Of the Popa^tMOt of I^Fefcoiotf fof their for their helpful*, suggesiio&s* Grateful aemewle&g$es*t ft m & * fop fimneiai. ill offered Of the Bese&reh Corporation through a grant to the Bejpurtawt of Feyeho%®$p«

r ampllfier using this network* Transmission equations are developed for am ideal simplified network with sere generator impedance and infinite lead impeteee^ for a slightly more general network with finite terialmatimg impedamces* and t m a special &imgle*tube elsustwladder feedback, -amplifier*

c&iGtjxf m m x B&rXt S^1 has shown that the

network is

mm. of a family of eight**?* networks of which the is a tetter imona eac&mple*

Be has developed the mesh eqna**

t i m e of the general ®tght-~arm mesh to give a form similar to equation (B) *

lie hat also written the general equation

of balance and has affiled it to get feelm m

equations for

a number of particular physical configurations of which' the shnnt-4adder is one.

She advantage in tuning the shunt-

ladder also has b#«m pointed m % by him.# the twin*te« and the shunt^ladder differ in re*» spent to which branches of the eight«arm network are chosen to be the input -and the output branches* as shown in fig* JU Both networks have the useful, property of exhibiting a null at some frequency when- they are Composed of resistance and capacitance elements# tors f@t elements meats % # Z'^ and % #

fhe twin^tee JMI network uses res!#* Mg* tmfc % # and condensers for ele* the stMt^ladder

network uses

resistors for elements % * % i and Bf -g and condensers elements % > fg> and

Since-

and %

for

have, a common

terminal, as mar be seen in the feedback network in fig# 6* 1* 0« 2U Harris* ^Bridged Be&etance^he®iat&hce networks*,

GENERAL

EIGHT - AR M M E S H

Zb

PARALLEL-TEE

NETWORK

SHUNT-LADDER

NETWORK

1- 0 —

Mg* I

W * i m w M configuration*

4 At &# possible to nee- a ganged variable condenser

mm m

toning m e m a t m toe slmt^4«iddor# the mesh lotted of Krofc «

given fey to G o r M U J M ^

is used to I U » ansAys-A® because it organises the solutions and permits extension to more complex eases*

too notation

used follows that of Le $ © r t o i l ^

a® SH»*«U&t*

Only i

*M» outline of tlae

A* given to. tog# geefcUft*

complete derivation appears to too appendix# -Aft Jiiiitt too basic matrix equation relating toe unknown

branch currants to a network to the- known applied emfe it 1 * C(OtZC)'J-Ct*

,

(1)

where 1 is too branch current matrix* o is the branch emf matrix * and S is the branch Impedance matrix* ©atlcai matrix*. C* espressos 'too relation

'too transform

1 * €i* between

toe branch currents tod the mesh currents* whore A* is the mesh current matrix* fixed*

-After too meshes are chosen* 0 Is

E i t h e r the actual values of the currents nor the

selections of input and output branches affects C*

thus $

Is the same for t h e 'few$&»te* * shunt-ladder * and the- other »**itoeisee^)toese*iSBii*^

)^**>eiai>iw*eiaiwe**to»ieerD^*w i ve

im*>j>Qwi

mlh:hiip**wsiapeeev

to f* ho eorbeiller* &&£&* iBftl&ati M M & S ^ M M M p m M * harvard fniversi.taTWiis* OamWidge, Massachusetts* John Wiley and itos* few fork* 1910

C U R R E N T S A N D V O L T A G E S IN BASIC N E T W O R K

W

V

— 1—

V

W

e

C U R R E N T S A N D V O L T A G E S IN S H U N T - L A D D E R

NETWORK

§

of the basic m t o m m of M g * ft* ffce *•*» is efdaf to &», the mesh impedance matriau tlon of a given network is

Complete so&te* ^

tog. substituting

.;

the matrices for that network in equation ft) an# per£*«*>

Ing the imdlo&ted

Note: Ct - C1-ransp0se . Vhe. 'S SSRCilffltfmSRSSMmS hBB^^

Metwo^k

For solution of the #bunb*4 s an# the matrin ■&* made of the oofenters of the determinant*.

When the known matrices are

substituted in eqt^tlou ft)" and the indicated multlylioa^ t&dsM* ere perform##* solution for i® f &¥»i

i® * J ^ 8» ■% » (a) A Since the complete expansion end sointi-nf for ie 'in. teawe of ■% ass#, the branch impedances becomes rather involved, a simple case is considered first to illustrate the mobbed*

f

2foea,l

Kstwerfc ft- is, m m m & in this. ease that ^

(s) hhe i & p ® & m m r %^.i: in series, with the constant voltage I M i t a i ? ! a^.f. is e^iiX to zero M i (fe) the load -ImsriftdtMea* % f: is infinite* these sa*sdltl«e are generally assumed t m m

H^c

network in * v&emffl tufee feedfe&ek elitywit-*&»#« the B*C

eirosdtt is driven.from a r#lafiwei|r Xew trapodazic© plat#elronit &£i whore the B*® aiyouli furnishes feed&aek voltage to a relatively high impe&anoe -grid eireuih*

$imee the

eurremt 1® M e t he sere under assumption (h) * the output VOftage Ve is toM i instead*

Hwfciee that the tmerater of

the ex^eeeient for 4* eemt&i&s neither %

nor % *

these terms are pres out in the dMomim&ter*

However *

l*he denominator

simplifies eoaslAermhiy if it'.is factored to get these elements as eoefftelemts of lower order ietermihamtSf

.Speeifio elements are now substituted into the network .in jiXeee of. the general Impe&anees used up to point*.

% * % * and fg are- assumed to. he formed fey eoade&e*

e r s ;e f m p a e i t y ©.*, •©, a n d 0/ n r e s f e e t i w e l y * . w h i l e % * and %

%

are formed fey resistors of -value &» E, -and mt respee*

ilyely*

g& order to simplify' the form of the answer the

s u b s t i t u t i o a ft * * V S 3f X If %

this

*

V ® $ $ / t t C i';,, t h e n

1* » d » , wtae*»

X • i/anC ,

p » » / % *. t9 * afc/fi*

i* th*

t

wt&k

a* t&© mtmvU*

Tim» tim mbo** *ttfe*tltufcic«ik

«li*niJi&t©s firequeutcy to $&t* gua eqiiatioa of tho tWNN **» '/mm6

l

«

9 >/K+I-.| {»+!)

|

|H

p s/£S3I-j (»♦!)

(a) *1 &

p

' #' v®a*i{o+i) in

I

I in

$^0&»lfe+i}

Ihe Bull or ■balance'' frequency is found -by squatins tbs numerator of equation (a) to sere,

When, ttii is

dene, the following conditions are found necessary to bold for m null to occurt

p * 1

* (4a® ♦ da ♦ 8)m m a.

A

curve stowing the relation between a «Bd a is given in Fig. «.

Shi* is the reciprocal of one plotted by Harris®,

the frequency at tha null is found by substituting p * 3, in

fi m p y/fcn+l X and solving for f,

to m

Shis gives

Sbese results agree with those of Harris*

for balanee frequency. One especially useful ease of (8) occurs when a m X *

m L s is applicable when a ganged condenser having

3. Harris, £&. £ & . 4. Harris, Ibid,

0 three equal sections Is used f m toning the network*, fhe aall frequency to*OM# fQ m v$/a*E0* 4a m U g m m % «**»•» Fig* ?, Is includsd in the appendix to facilitate appra*i«#. mats solutions*, the quantity .!'»i»ii|iiMwi»li0v Thus R

£l> MO

tk-

£ 1

or f0=?

R

2 # RC

=

P

'/_557r When n~l«

-Vc

fQ=: \/E , and letting 2*RC \/5mp

j

0

0

\/5p-2j

j

j

j

\/Sp-2j

\/3inp-j 0

j

c'/e

j y^p-Sj j J v'SP-Sj 0

Expanding the determinants -gives ■ \/5mp(p2-l) - j (l2mp^-l)

U) 5 \/5[rflp(ps-l)-p] - j[ (I2mp2-1) +5p2 ] See Fig. 5 and Fig. 4 for plots of this equation,

35

lO ro

CM

C\J

4CM

in

LU

O

ro

-J CM

m

in

ro

Lxi

Q O

CM

o

Fig. 6

-J

X

(/> in

ro

CM

O in

ro

CM

36

m

BALANCE F R E Q U E N C Y

f- J L -

R

27TRC

10 meg——

c 10,000

/x/xf.

=— 10 cps.

5 —

fc

5.000

m

50 —

100

1.000

t meg —

ir“ soo = -

500,00 0 —

ohms

1,000

500

— ir-5,000

=— 10,000 »

100

100,000

E-50,000 50,000

=-100,000

50

500,000

=— 1,000.000 10

10,000

ohms

SHUNT-LADDER ALIGNMENT C H A R T Fig.

7

37

Resistive Loading Let

Za a qR ,

Z0 - nR,

n = n

Then the complete. A

becomes (m+u)R-JX

-mR

’

+jX

0

-mR

(q+m)R~JX

0

+JX

+JX

0

R-j(n+l)X

+JnX

0

+JX

+JnX

R-j(n+l)X

A

The rows of A

may be split into real, R, and imaginary, I,

components as follows: Row 1-R =t (m+u) R

-mR

0

0

]= R[]m+u

JX

0

]=JX[ -1 1= R[ -m

0

0

]

0

1

0

]

q+m

0

0

3

0

-1

0

1

3

1

0 .3

Row 1-1 *t

-JX

0

Row 2-R =r

-mR

(q+m)R

0

0

Row 2-1 =[

0

-XX

0

JX

Row 3-H =r

0

0

R

0

)- R[ 0

0

Row 3-1 =c

JX

0

■j (n+l)X

JnX

]=JX[ 1

0

Row 4-R =[

0

0

0

R

3= R[ 0

Row 4-1 =[

0,

JX

JnX

->(n+l)X >JX[ 0

-

>JX[

(n+1)

n

3

0

0

1

3

1

n

-

- (n+1]

Let the subscripts indicate the order of combining real and imaginary rows*

Then

A r r r r = R4 [u(q+m)+qm]

A r IRR * ~JXRs (m+u)

A r r i i * -X^R2 (2n+l)[u(q+m)+qm]

A I R K R = -JXR^(q+m)

A

A XIIR = Jx3Rn

riri

32 -X2R2 (m+u)n

A XRRI = "x2r2 (n+1)(q+m)

A H R ! = J* Rn

38

A r i i r = - X ^ R ^ ( n+ 1) ( m + u )

^IRII

~ JX^Rqn

A i r i r » -»X^R^(q+m)n

^RIII

= JX3Run

= -x£rS

A i m

=0

a iirr

A

rrri

= -3XR3 (n+l)[u(q+m)+mq]

A r r i r = -jXR3 (n+1) [u(q+m) +mq ] The sum of these components after R = p \/2n+l X has been substituted is

A =

p^( 2n+l)^ ^X^ u(q+m) +mq] -p^(2n+l)

{x^ (2n+l)[u(q+m)+mq+q+2m+u]+l],

- jp3 ( 2 n + l ) 2 (2n+l)[u (q+m) +mq ]+2m+u+q| ■*\Jp{(2n+l)

n(q+u+2)j

Expansion of the determinant

, which is the numerator

of equation (3), gives the following results I

= X5 [mp(2n+l)^/g[l-ps ] + j[2m(n+l) (2n+l)p2 - nj j e From equation (2),

ic = _l(-Aba)ea

A Then

Vc = l(-Ab a )ea Zc

and

3 - ~Vc/ea = -

A where

ZC >

A

Zc - uR = up >/2n+l X t»o •H

2 %i3ana

51

pfcVEL 4CCOR DC R

_

WIOM S P E E D

LEVEL

PtECORBEP?

w.M m -» .** m h* *•"♦•' *'? */•:*« o H?D 4**Afttr* H & S Cakpflr^gifc Co/H/mU rc _ £i*_ p ^ v l . p s d f i A i l s i ’ g_F .>j__a