I

'---

~-··--. ·· . ............................. ----

l)

\...___,) .

(a) (b)

The mree orr ~~R;o·;~"'t~·;Jd~";~;§=PrL;~-Rril'7~~·m~>fier. and (e) eo~-base amplifier.

Avt Terminal vol~.age gain Al.,

Signal-source \'Oitage gain A~

Input tumirulruistance

Rotlf'- Outputtenninal rcsi ~t:mce

u,

(a)

Inpul signal range

Tcrmin.al current gain

SINGLE TRANS ~STOR BJT AMPLIFIERS- APPROXIMATE EXPRESSIONS

COMMON-EM inER AM PUFIER

..... g.,RL = -! +g.,Rc

g,., Rt [ Rd R;o l - l + g.., Rc Rt+ (Rt~UR;s)

r::r + (JJ. + I)Rc

~r,..( L +g.., R£)

=' 0.005( 1 +g. Rc)

-P.

(b)

COMMON·COllEG ORAMPLIFIER

~+~ ~+I I +g,.RL

+~[ R.IIR;g ]~+I l + g., RL Rr + (RaUR.a}

r 111 + (fJ. + l)RL

:= r~(l + g. Rt,)

~+~ < g.. Po+ 1

i; O.OOS(l +g.. RL)

P. +t

(c)

The three FET an1plifier configunltioRS: {a) common-source, (b) common-draio, and (c) conunon-gate.

Terminal voltage gain A«

Signal-$OUCCe vollage gain A.

COMMON-BASE AMPLIFIER

r.[l +g., (R.!IR,)J

='0.005[1 + g,..(RriR.ll

a.~+l

"'

-

,~.

\.. I "'--=-

~ Jet = J" +- Jj,_ _)_ w. = It__ = Ll tl.lot e-) ~ ,Pi r: IJJ = ~ w I}

.. -

' - w~... - ·1J. ".$:~,.,;~ '

"1'7' VI I J t I\') I\ . ...til. .J_ fl L -- ........,.._- I)OfOINr.

J: 1no-Hq ! T; frJry,ue Tr:: 1: ~.J .. "q~t ~i:. ~ /'2. ~,__ :rl ;CQ/' ('11/;; r r~ wfr1 1i. lo1~

- ~Ne+ ~e l"" -JL '-' 'I orii ~·~ Jear;. r· P= T iJJ ; llj : (}; = w 61)) -pIN!) eJ ~l'l9J· I oJ 1oft'~~ JJiL 7 7j_ 7j ~ L L 1'1 ~

0 p-:: _p-P l_ forttJe TeM= Jq J-t -tBUI J T;::::/1. (8. -8, -r;.11 t~ ~ P5 :ce - - _ J UJ. \. 1f r "' 'L I))= e p. 71 = Poll~" Tel1 -Jey_,~ h :: }f. =#115 l5&rf1- ~'!e ji\rA "-=- J T Cf...

~

-~IV:. ft p. t Jt e T.. - 'I ·} . - r X r

f-, *~J ( T) ~ i_ ~ 1~ 1 ~~dro~ue voH:}<+-:J;e (volf- 5'"111 s)j f. ,;\,-H ~ Te,., (~ ~7

( a ) nm'J . lf 110 ty~ ~0 /' iea L w )__, V fi_ ~ J;_ Jw ~) c• n ~ ] Jw,{ ~ )' '1)\ LJ!er- 11<5 eJ Je•" . Per+i q ( J) ~ C ~ Jt ..-< T ~ n.. T &I = J;., +- ( -';t) Ji_ Jt""' + I) h.ve fhe.soNe velocl-/y It\ ) L , C .k \!t - T cl.. / ?rl ~ yowef' da,p;tlj (8 ~ R L-= d-t .!....- fi.q lq I-E ,· TE:: 7L r RP/1 ~ .I!!fi.. pf== ;J.f;i' CPS (ex): b 1! 'qfo A-C

(/{ ) .1 J £ -II ~ 5 tr ·f1·1,. QC I . sr"'~1 G ~ T v,~ = v fri • It - E w E .•r e :l { RP;'I) ( .ii1I:.) __.. . . ~ I si ci net. v. +-w) j.i;r.t;;pt.-n,_ ~~ '""::' r" 7£= lrr io ~~tkceJ V b~ Vq= ,P(v. ~'!7 ri/;= .lfl(.o5{rJ.)~m

,..~ ~. nL "' ~< w= .!&.. - R. T£ T ~ I( !J w,) ; fil'lj ,-1 11' <f' (vecM}'-5~«:e r I!!-= d .4 = ~ 11'--_ kE kE lfr tJ \ - _: J= J,., +:{(f. r L1 (IJN = i-ILJwL 7i ( ph45er)--H~ reJ 8 J V::: 7L ~A + ft W ~ 5pet<l/ q/1 ckri_'!l qccdcr. -<

1

-:-L - R ;~ v. = )t I.=- " ~" E 1rnm•~~ o(l~~ a/- c~. = o· T S!"' v;~ IJt T = J !J~~Jn fi - A 'I J 7Vq . pf - ~ I ::: Vo s~ V· Ji P;~~= T J~ Llt -·e· z = ....:.!L q 7f V = o 'YJ

I~ ;:; P' Ia e.J l-; Vq = n ~I q Iq - Q T L . T _f i ~:::. 7L. I UJ= 11 API'J r""\ 3= .!11 )"= T, t-:JL. ja. I=. 1L T IJJ= JL-_G_ . Rlt ~8-= £+ illRA I /tq I T 30 t ) .. • - n I{ lfe /! ;( ~= Lq

L CA~drl v d - v E. 1 £. ~::::: ~ T Ll 111~ "' .!.L)w 14- -.J;i_ • -r. (, ) H. ' . j L Vtr; (....A.:: ~ ol..:::: Co5-1 V ' - 1/ r 'te~ lrP!J J •

}!YiisAJ!!j. ~~ . d = 1 (~ '"1) lf.-t t 'J 2/2' Vn,~ d>$.11 1.£, r~ E!t. 0 0 ~- Jll ) "";[ d . r 1 lfp"' --L_ lrr 1 V&c V- t-e ~ o< :::. eoi1 _J£_ J Lq~ X~ /rpw11 (L- G55) .

:1 , v d, I§J' r cos e = ~ V., rn5 ( fkl..ti) +... _yz v. ] ;;z w r _ r j tJ.· 8i ~ ; e, til

.... ,;;:;;,:c_ · 1r 5 r--.... e - e ~, ._ 3 0tk L J} lJ.f r 51() e == J1 ~' 5ird8s~D~#)+... . . P::: R_ • 7Lfr: "- _ TJ f / c Ll~ ~f J ;;;n V L ~00 o c~oose 5/a+es _.., f)~ync ~ !Je 1 W

1 hp UJ$"/ip =[ flfyll'- !J.pl )'§> 11,-w -uJ!L Jct.Jl!Q• T F ~ f) Jl: p slip

,-'-_~<lc/}OQi lj-+ 5 b/c 2.00° i5 b/w f/Jef'j ; '- PJ. ~ rp/ X; T11 :=~ !11. • _ 1 T 7 0 ~ -tllefi(S"MO)~ t- d1T+ckT w _ ~trfJ Af 1 I. i wf!L 5J,p-;- ·IFJ

>)'If<:- 'Y'l< no o.J,!!: 1 ~ Tw ;;< I i.•; ~ ~£I'3/ £ = ~ =ld4 -J8 ) l.'J It!= ,;;;z - 60 iory_ue "'o '"' ~I, :1-J --jfj • _.:Jtt t v'I.. ~ P I - 7L"" ~4" ~OLe~l·e J 3j Je'L ~ J"'t-,J'[Jc) ~_ (' ~ Wetwsync f" ~('4.,fW:J;,.) 7 Je -

i.e= ~ rt{r·c-j9.fj -~c-1 0

' -

r

. f1 •'

~..:...-,....,~ C; -v~lffs - -~ r;L -111;'1 )

0

0

(I)COicJ fi !iJ!e T e,) t X!l--J) , ;rl(l)ti(/IYI(.'I) • • 1,2, ..

[ ~)ct.·~~),.~it)

I I 2 her I I { I, 11•0,1,2, ... IJII II ... 1 - , ·lfl ' ~ CIJn II • -\,II • - J, -2, .. •

'{11) " 1

&{n- , , .. , • ...,, , = st, :tl, ...

• COl~" I trfa(ll + ~- 2·1fkl+ ~n - I\- 2trk)J i·-

• ltn Jl¥1 ... 1'-1./~18(1) + ~ .:' ld) .... 3(fl ~ ~ - 21Fk)J

.. COl(~+ 8) .. I .. ...... flo - 2ri:) + tl'a(n­

-j&

l)jltfCII!do. IICIIc liaebul~ l (l) ++lXtf) - .r(O)

Secolddi:rinri'll J(t) HrX(r) - a (O)- J~

..U.Ii:rivliho .Pl(l)++ri'X(l) - /"''x(IJ)- ,..,l(U) - ... -':.;-~.,.._---'=':.....:__

! J(.t)d.l++ ; X(r)

.c(l) · I(~ ~-tJ{$)11(1)

.I(O) •~rX(s)

.1(0) • ~[h'(l) -JJ(O)J

tU{c)l.,•tlooahiiiiU.paktoC(r - JIX(clk"" ,!!:..atol • ((c - I)X(•ll.-•

~ I l .:(c + '1 > n .. ,.. - c~ - •>' ,.,.....,.[,.]- liZ }

(.c •> ,.,,......,

1,.

1 _ crz(o: + 11)

(c - n)!

2.n.& .. n(r~ + l)n" "("I - ( >' 2. ..: - • ,~

(oaa .n..)w( ,.] _ .C2

- (OCM n)c I z."'+-1 ci - ( :Z.coe n ) .t + 1

(&In .tm)u{nl- ( eln n)c ! ~ o:2 - (2 o-n>.: + t /z.ifii

crft(COIO .lllol u{n}- ~ - («t -• .t'J)c - c2 ,- :Z.0 COli rl) .C T A 2

11 •In .t'T).c • " ( •In ,a,,)u(,.] - -:t_ ......... ~;...;:.;;;.;...:.~---..

.. - ,.. -· n )c + "

. . -·~

1 f oo E =- . IF(td)l~ 21{ -oo.

r; . I

Power/energy

Relation

BW.

NIA

SNR NIA

2) Cc:adcll:r tllcsi&Dil.llCt) •COl-.[, wilDie r.dlrDerlllll'reqaeacy is ~, fiDd die Fllllierseries ~bdlil ... {Wat:QJ~a!111cn.dlllsipll a a m-..... ,.GCIIIIPaapcMftls..t-.rylbeFoDr-­OIIdlidam by impdcle~ Let -'J(I) •l+siDe,t+~~, ~~~tdlesuned.od 1o detliDile tba Folder aeriel tlit. .

4) Giwodulwrioao!ipt J'(t)=.a(t)t·~t},8Dd B(t):oz(lt)•'(Jt) lllll&f-k l(t)JIII FCIUiierlrllllbm .f{e) m41(~ 1111 Pollilir IIIDna H(e),Did e. ~lldweeu f(e).SG(e). +,J. ,.w;," }t> .J.3{t)

! !....

C~r,rFfbr AM ~(J'Iq/s .

Xz.(w)=

v

~-· .... .... . .

·.

v · .

... · ·, .w·,.·.· ...

+ .2 ~~c~t" -)swf~ w y

w 4) ~(t)~K(t) ~~(t) r ( w )~ .(( w) • Ji ( "')

~t) ~x<~t)'lt hBt> &(w) = -}x(~) ·~ H{~) ~k·ix[~)~{~)

,!i(3t)~) 1_1(~):::±.(1_>. t1(~) 0

6(~)= t H~) j(t)=-~ ~()t)

IaoFS:IJ,

·0

'

~· -- 0

P(X>a)=l-F(a)

' I

P((X,Y) E A] = LLP(x,y) .(x,y)eA

Px(x) = 2: p(x,y) p,(y) = L P(x,y) r-----Y X

€f!!tc+r4 voll£ . E[h(X,Y)]= LLh(x,y)p(x,y)

"ty ' "y~ ex . ._ X· -~-= - ., or X+-Y- - -"' - -" )(-t ---=-----!~~L'-...:.~'1.+

!t(ov~Ne~~ce11: \E[(X- fi-x )(Y- fi-r )] =

'l~}TJ(p)) Cov(X,Y)= \ LL(X- fl.x)(y- fl-1 )p(x,y)

P = f J(x)dx .------f~u5"e whe~ ~ " y E/X] - ~xR (><) X.P.. y art nal- E(XYf-E(X)E(Y) ~ x

*-Expec.kd -r1 Y ;nJep. jcwcf5 AlL*'~ ~ E[Y]- ~yPy&) '/qlue ,___oo-cr-C)-,..~~~~.,~---"1 C (X Y) ~ -U'> . ..J..

E(X) = Jx· f(x)dx . . X r Corr(X,Y) = Pxr = ov , Cncrt7cJe.fl ' Cqlt ol}~ Y f~: 1 ~OJtrel.f/a cc..I:C\ , <7 x<7 Y /,r_ k/wc <h •J + 1. 7/,e

E(xl) =-jl5Cx) Jx -1 ~7r:5 , · n UXJT'C'e close" ~ erMe" e~lt'J.e./-Mt:. · , Coy aX+ b, cY +d) = acCov(X, Y) ~J.t!l' iiJe relf!l:!oiJ- ·

E[h(X)] = J h(x)f(x)dX ---.. _ _ x -t · f"hlp -hdwe(J} ~ .P.. y ~---.....:·..=-""~-----{-...:.-------, If a and c are either both positive or both negativ~ .

Uniform Distribution

{

1 . " 5~1 f(x) = B- A If 1 ~ x $ B ....::s, lt B-A dx

0 otherwise r------~

th~ - /

Corr(aX + b,cY +d) = Corr(X,Y)

E(X) = J-1.

vaN#-- (j2

V(X) =-n r 741'\rles

, Y=a1X 1 +· ··+a X = " a.x . n n ~ 1 1 , ...

~l'lple {~) 5-hl. Jtv. . u

. ux = Fn

...

df = n---l

p-p z=-=~===

~p(l;p)

• i Adep. 5a,.,rles • rJ1 +- ffJ. are h n aw IJ

o bo+h po u/,5 Qrt:. ~OrMqf

• n1 +nJ :> !f-0 • saMples are lnd;::. • poptJ/.5 dan if- have +o ke f]omq/ , 0'.1-~ &J YJoi- ()eeded

--------------~---

~ x 1 +x2 p = -=-----=­n] +n2

I I I

I

I

I I

'i"i .... , ..

.. ~ -.

~-\ '" . f'

. . . . l L

I

~ § ! e : .e l, e : 2: a.: G) •

~: -o: c • ra i e . ... . .x:: s ; o:

: ' ' . '

\ ' '

ALGEBRA

Arithmetic Operations

a{b + c) ~ ab + ac Q c ad+bc b+d~-,;;;-

Q

, Q d ad ---x-~-c b c be

u+c a c ----+-b b b

d

Exponents and Radicals

(;c"')"-.1 ...

·(;)"=;: -~--~· ~ :/? = (.(G)"

Factoring Special Polynomials

.r' - y' = (x + y)(..: - y)

:t.' + y' = (x + y)(x1 - xy + y')

Jtt - y' ... (.lt - y)(;c2 + 1t1 + :1>

Binomial Theorem

(x + y'/- x: + l.ty + y2 (.t - y)'- ~ ;c> - 2IJ + y'

(x + y"jl = x) + 3.t2y + 3xyz + ,~ (x - y)s - x' - 3J2y + lxyz - y'

11(11- 1) (Jt + ir = x• + nx•-•y + x·· 2y'

2

+ ... + (: }..-"yt + ... + nxy"-• + y•

'Where (") ,.. n(n - 1) • • • (n - k + I) k 1·2·3·····1

Quadratic Fon11ula

-b:~ If t1X2 +Itt+ c= 0, iben.lt =

2411 .

lm~qualities and Absolute Value

If a < b aod b < c. Lben 4 < c.

u 4 < b, theo Q + c < b + c.

If a < b -.od c > 0, Lbai c4 < cb.

If a < b lUI() c < 0, then et1 > cb.

lfo>O.lilen

]z 1 ~ a meliJl.~ x ~ 4 or :r co -a

].t]<a means -a<x<tJ

l..rl > o means x >a or ..: < -o

~PAGIES

GEOMETRY

Geometric Formulas

formolas for area II, circu.mJueDCC C, and volu.me V:

Triangle

A~ !M ~ !ab &ill'

Spberc v .. i.,.,. A= 411'r'

c-z..,.

G

Distance and Midpoint formulas

DisLana: betwcco P1(x1, y1) aod P2(Jtt. .Yl):

Lines

Slope of line through P,(;c:. y,) and P,(z., .Yl):

)1:1- YA m~--

.r::- .1:1

Sector of Circle A= !r:8

6 = r6 (6 in cadia!lS)

r

Cone

v-i1rr1h

A= 11'r../r' + ~t•

Puim-slope equalion of line lhrougb P,(l., yv wilb slope rn:

y - )'J ~ m(x - xt)

Slope-intercept equaiioo of line witb slope m IUid y-inlercept b:

y~,x+b

Circles

Equllion of lhe circle wid! cenler (II. i:) and tad1us r:

(Jt - , .. ., + (y - ..,. .. ,.

TRIGONOMETRY

Angle Measurement

11' ndians = 180"

,_ r8

( 0 in rar.lians)

180" lrad~--

11'

Right Angle Tri gonomotry

sin 8 = opp ~:~e 8 ~ hyp · hyp opp

&dj -8- hyp cos8 ... - - -byp adj

h~1 d__d"PP

Trigonometric functions ,

cscO=­y

,. seco"'­

r

.r cot 8=­

y

Graphs oJ Trigonornetric Functions

y

y = cosx

~ y=ctcx y ,~-.~

adj

y

v: :\ I !0 I I I I I I I I

# 2,. .. I :r I 211' .l

-I h' -I V'J \ I I I I

' I

Trigonometric Functions of lmport11nt Angles

9 talians sio 0 CI0$9 (f 0 0 1

30" "'T/6 1/2 ../3/2 45° "'T/4 Ji/2 v'i/2 60" 'fT/3 ../3/2 1/2 90" -rr/2 0

tm8

0

,fjj3

,fj

2

Fundamental identities

csc8=-1-

sin 8

tanB-sinfl cos 0

cotB- -1

-tanO

aec8=-1

-C0$8

cosll cotfl= sin8

sin( -o)- -sin 8 cos(-8) = cos 8

Wl(-8} - -r110 8

\ln(i- o} =cot9

The Law of Sines

sin A ~ sin B _ sin C II b ('

Thu Law of Cor.ines

a,=b2 + c,- 2bccosA

.Addition and Subtraction Fnrmulss

.sin(.t + y) = siJ:u CDS y + c:os.r silly

sin(x- y)- sinx cosy- cos.r silly

coa(.l: + y) - CO$ x cosy - sin r sin y

cos(x - y) = cos z OM y + silu sin y

ua(x + y)- ll&nx + tany -&mrWly

Wlx-tany tm(.-:- y)- I + wu tany

Double·Angle Formulas

sin 2.l: = 2 sin x crux

B

cos 2.x- COB1x - si.n1z- 2 cos2x - 1 - 1 - 2 sin2x

2 tiD. .t lall 2,~; ~ -~-=--=tiiD'-:l,-.x

Halt-Angle Formulos

l - eash sin1.x - ~---"=--

2 2 1 + cos2.x

ccsx= 2

c

. ' ' ' ' ' ' ' ' ' .

· SPECIAL FUNCTIONS ------------------------------------Power Functions .f(.d = x~

I il /(~) = .r", " a positive integer

Iii! f(z) = z•t• = ~/i.ra a positive integer

(iii) /b) - x-• -.!. .t

Inverse Trigono11etric Functions

arcwu: = llln-1~- y ¢=> W!y - x and

0

neven

y

0

--------­(1.1)

X

f(xi=.J%

y I

~=~ ...__

X

11' 'IT --"!!y ... -

2 2

·-·-~ 11' '1T --<y<-2 2

y .. 2

y

----X

I

lim tan -•x = !!. -- 2

-~ ~.b~-------------------------------------------------.~--•xs~------------------------.-.-.. -..,~.d~~~S:~~~-

SPECtAL FUNCTIONS --~------------------~

Exponential and I ogarithmic Functions

log,~ = y ~ a' = ~

In ~ = log • .1, where In t = I

Cancellation Equ&lions laiNS of logarithms

Jog.(a') =X Qq..~ X 1. log.(xy) =- log • .r + log.y

ln(t-•) = ~ i"'' = .r 2. log.(;) -log..~: - log.y

3. log.(.J:') = r rog..1

.X

Ex"ponential functions

Hyperbolic Functions

sin.b .x tanhr.~­

cosh.x

Inverse Hyperbolic f unctions

y ~ sinh- 1x ~ sinh y = x

1 CIICb.X = -­

Sinb X

1 &ecll~ ~-­

ClOth .X

OOih~ coth.r =-­

sinh.r

y - coah-1.x ~ coshy - .r and y ;a 0

4

lime'=O

0

Logarilbmic functions

)'

y = sinhx

...... . ;. - ·~'1t

lim In~= (I) ·-

Y "'Jn~

)' -log).x

)''"' logiD.x

X

-

DIFFERENTIATION RULES

General Formulas

d 1. .:b: (c) = 0 2. ~ [cf(x)] = cf'(x)

d 3. .:b: [f(x) + g(x)] ~ f'(x) + g'(x) 4. ~ [/(.x) - g{x)] = f'(x) - glt)

S. ~ [/(x)g(x)J = /(x)g'(x) + y{x)f'(x) !PrOdu~t Rule)

7. ~ f(g(x)) = f'(g(z))g'(.x) (OWn Rule)

& .i_ [ f(x)] = g(x)f'(x) - /(z)!J'(x) (Quouent Rule) • d;c g(x) [g(x)]l

d I . tb: (x") = 11x•- l (Power Rule)

Elpollential and Logarithmic Functions

d 1 11.-Lo lx l=-

tb: X

Trigononwtrlc Functions

13. ~ (sin x) = ~sx

d 15. tb: (csc x) = -esc x c01.x

Inverse Trigonometric Functions

11. : ... (sin-'.:c) = Jt : .x2

2Z. .!!.. (cx-•x) - -1

tb x.Jx2 - I

Hyperbol ic Functions

25. ; (Ji.nb x) - cosh x

d 21. d; (csch x)- -csch .x colh z

Inverse Hvperbolic Functions

d 10. - (a')= a'ln a

tb:

d I 12. -::;::- (log. x) = --

.... X)DO

d 14. -(cos.:c) = - sinx

tb:

d 17. tb: (sec.%) = lee X laD X

d I 28. - (cos-'x) = -----

tb ./1 - x1

23 . .!!_ (acc-1x) = 1 tb x../x•- I

d . M . dx (cosh x) = 11111h x

d 29. -;I; (!!edt .r) ~ - secb x umh z

5

d 15. tb: (lanx) - scc1x

d 11. - (rot.x) = -csc2x

tb:

d I XJ. iU (lllnb- '.:c)- I - xz

. d 1 36. iU (ootb-'x) - 1 - x~

TABU. Of tNli&HALS

Basic Forms

1.J udv-uv - Jvdu

4. J e• du = t~" + C

f o• 5. a"du =-- + C

In o

6. J sin udu = - cosu + c

7. f COS II du = sin II + C

I. J scc2udu = taD11 + C

9. f CSC'Z.II dJi = -cot II + C

10. J sec u tan 11 du = sec 11 + C

Forms Involving .Jn1 + uf. a > (I

23. · du = vo1 + 11 2 - o In. . + C f .Ja2 + u2 ~ . ·r·o + .Joz + u21 U II

f .Ja2 + u2 .Jaz + uz 24. 2 du = + ln(u + .Ja2 + ul) + C

II u

11. J esc u cotudu = -cscu + C

12. J Wludu ~In !secul +,C

\

13. J cot,udu ~ ID lsinul + C

14. J sec " du = In I sec u + tan u I + C

15. J cscudu = In lcscu- cotul + C

16. f dr.t = sm-l ~ + c, a > 0 .Ja" - 112 a

f du 1 111+al 19. 1 2 =~In - + C o -u uz 11-a

f du 1 ~~~-a~ 20. 2 ~ = ~- In -- + C u -a uz u+11

. ~ .

' ' ' . : . . . .

. -~ . . ' ' . '

TABU OF INTEGRALS _.;_ __

35 f du · u.Ja'l - u2

1 Ia+~, --ln + c a • u

3&. J du = __ I_ .Ja2 - w2 + C u?.Jaz - y2 o~

31. f (a2 - u~)3n. cht ~ _.!!, (2u2 - Sa2 ).Jo2 - uz + 304

sm-•.!!. + C 8 B a

Fonns Involving v·u 1 - al, a> 0

f u ~ 40. w2.Juz- azt.t. =s(2M2 - a2).Juz- az- SID l11 + .J,.z- oz I+ C

J .J,.z- a2 a

41 . IliA=- .Jut- a 2 - acoa-1 - + C . u lui

f .J,,- oz .Juz -at 42. 2 du = + In I u + .Ju2 - a1 I + C

II II

II +C az.Jwz- az

TABLE OF INTEGRAL$

Forms Involving a + bu

47. J ud.rtb -~(a+bu-t.dnla+bui)+C a+ u b

J u1 du 1

48, --b- =-,[(a+ bu)~ - 4a(a + bu) + 2A1 1n lo + bul} + C a+ " 2b

J dM ,,.,,

49 =-lo -- +C • u(a + bu) a a + bl#

J du 1 b la+bwl 58 - --+-1n -- + C • 11 2(a + bw) . au a 2 I 11

54. J u.../a + bu du = l:bt (3bu - 2a)(a + b11)1fl + C

f udu 2 ,---55. ../a + bu - 3b2 (bu - 2a).Ja + bu + C

J u 1 dv 2 ~

56. ~ = Sb' (81.12 + 3b2u1 - 4abc.)yQ + bw + C

va + bu 1 ·

' 57 J dw - -

1- In I ~ -. .[Q I + C if a > 0

· u.Ja + bu JQ Ja + bu + ,fa '

= - 2- tan-•~" + bu + C if a< 0 Fa -a '

J~ ~ J du 58. du = 2va + '"' + a ~ u uva + bu

59. J ../7 du = ..;;:;+bu + ! J dJI u 2 u./a + bw

60. f u-Ja + bvda = b(2.,2+

3) [ u-{a + bu)31Z- na f un-l Ja + budll.]

f u• du 2u•.;;;+b;. 2nt.l · f u•-J du &l. Ja + bu = b(OI + 1) - b(2n + 1) Ja + b11

62 f du ~ · u•Ja + bu

.;;:+iiU b(2Jt- 3) r dJI 1.1(n - l)u•-• - 2D(rr - l) • ,r'Jll + be.

...... ~.

: . Lt

~ ! : . . ' . . . . : . . . . : . . : . ! . ! : . . .

TABLE OF INTEGRALS

Trigonometric forms

63. J sin2u du = !u - ~sin 2u + C

64. J cos2udu =~~~ + !sin 2¥ + C

fi5. f tan2utiu =tan II- II+ C

66. f cot.1utiu = -cot.u- 11 + C

67. f siD~11 dM = -H2 + sin211) cos u + C

61. J ros'u du = 1{2 + cos2u) sin 11 + C

69. f tan\t du = ! tan211 + ln I cosu I + C

70. f cot3udu = -!cot2u - In lsinul + C

71. f sec'u du = i sec u tan u + ~ In I sec u + tan u I + C

7Z. J csc3u d11 ~ -!esc u cot u + ! In I esc 11 - cot ul + C

Inverse Trigunomauic Formt

IJ. f sin-•u du- 11 sin-111 + ~ + C

18. f cos-'.u du = u cos- •u -~ + C

19. f tlln-•u du = u llln-•u -! Jn(l + u2) + C

90. J usin-•uttu = 2112

4-

1 &in-1u + ~~~ + c

... f -1 -'·· 2w2

- 1 _1 -"'...:;.J_i_-_11_2 + C "'· UCOII lila~~- COl u-4 4

76. f aJ1."11 du =-=.!....._ <Xll-1.u - f cot•-2u du II- 1

f . . sin( a - b )u sin(a + b )u '19. sm m~ !WI bu du = 2(a - b) - 2(a + b) + C

f sin(a - b)u sin(a + b)w 110. oos au cos b,y du = 2(a - b) + l(a +·b) + C

f . cos(a - b)u cos(a + b)u 81. sm au cos bu dM = - 2(a _ b) - 2(a + b) + C

12. f II Si~ II du = Sift II - II COS II + C

13. f II WS U tiu = COS II + II sin II + C

14. f u"sinudu = -u"cosu + 11 f u•-•cosudu

15. f u• cos 11 d11 = u• sin u - 11 f u.-1 sin u dv.

f sin.,...'u cos"'+'u 11 - 1 J 16. sin"u C06"'11 d11 ... + -- sin"-2u cos"'u d11

n+m n+m

~ + -- sin"u cos.,_2udu sm•••.u oos'~- 1u m - •i j

n+m n+m

f 1 [ f 11 ..... 1

dJJ ] 94. .,.OOI-1adll. •-- .,•••coa-•• + , n. ~ -1 11+1 ~

f I [ f u•+l du ] 95. u•tm-1~tdll.--- ,.•tttM-1N - · ---. "~-·~-i .11 + I 1 + u 2

•

TABLE OF INTEGRALS ----~------------~------·--------------------------------------------------------·------Exponential and logarithmic Forrns

!16. J uti'" dll- :, (ou- t)e•• + C

f. 1 n J 97. ~o~•e•• du =-;; al''e• - -;; u•-•e•• du

f tl'"

99. f!•• cos bu du = a 1 + b'l (a c:os b~o~ + b sin bu) + C

Uyperboli~: forms

103. J sinh u du = cosh 11 -t- C

104. J COI\h u du ~ sinh 11 + C

\

105. J !anh u du = ln co&h u + C

1116. J cothll ~ = ln I sinhu I + C

-107. J sech~du- tan- 1 lainlu•l + C

Forms lnvohriag vlD.u - ul, a > e

113. J2au- ~t2 dll- --J2au- ,z + -cos-1 -- + C f 11- a a1

• (a- ll) - . 2 2 p

114. J uJ2Dil

f J2au- 112 (a- 11) 115. u du = Jlau - u1 + a-cas-• -Q- + c·

f 'J'liJM- ., 2J2JJM- 1115 • (a -")

116. du = - - t:Q&.-1 .-- + c

112

II 0

117. -COS-I -- + C f du ("- v) J'1m4 _ 11 2 . a

118. 1· I lldu - -J2all-~o~2 + tJCoa-1(~)·+ C . · v2Dil- u2 . a

100. J ln u du = u ln 11 '_ ,; + C

f w•+l

101. u•turdl.t ... (n + l)2 [(1'1 + l)ln u.- 11 + C

102. J-1-dll=lnllnul+ C IIWII

108. J csch ". du =In I tanh~~~ I + C

109. J sechlv du = Lanb u + C

110. J csch~ll du = -coth " + C

111. J sec~· u tanh ll du = -sech u + C

112. J csch u coth u du = -csch u + C

119. J "2 du = - (w + 'o) J'1ml - ,., + 3oz cos-•(ll - ") + C .:/21Ju - ,, 2 2. Q

120. J du = - J2mt - ,, + C • 11 ,f2iJ11 - u1 au

--

The Ins~ tutton of Engtn een ng and Techno ogy

Units & Symbols for Electrical & Electronic Engineers

www. theiet. org

Units & Symbols for Electrical & Electronic Engineering i

PrefaceA booklet, Symbols and Abbreviations for use in Electrical and Electronic Engineering Courses, was published by the Institution of Electrical Engineers in 1968 and 1971. To take account of the many revisions and additions to British and International Standards since then, a new and fully revised edition was published in 1979, with reprints in 1980 and 1983.

In 1985, the editorial panel reconvened and undertook a total review and update of the Symbols and Abbreviations booklet, prior to it being re-issued under its new title in the professional brief series, in 1986. Further reviews of the contents were undertaken in 1991 and 1996. Any comments on the present content, or suggestions for additional material, will be welcomed. Please address comments to the Secretary of the Institution.

The booklet is for use by students and staff in colleges and universities, as a reference for authors of papers and books on electrical and electronic engineering and related subjects, and as a guide for draughtsmen and designers in industry.

Appendix A lists the standards which have been used in the preparation of this Guide.

© The IET 2008

Units & Symbols for Electrical & Electronic Engineering ii

ContentsPreface iIntroduction 1Abbreviations for Words & Phrases 2Printing Conventions 3Letter symbols, subscripts 3

Unit Symbols 4Compound symbols 4

Numerical Values 5The decimal sign 5Multiplication of numbers 5

The International System of Units 6SI base units and supplementary units 6SI derived units 7Non-SI units 7

Quantity Symbols for Mechanics, Thermodynamics, Illumination 8Quantity Symbols for Electrotechnics 10Subscripts and other uses of Letters and Numbers 13General 13Semiconductors 15

Mathematical Symbols 16Physical Constants 18Conversion Factors 19Length 19Area, Volume 19Mass, Density 19Velocity 19Force, Pressure, Torque 20Energy, Power 20Nucleonics, Radiation 20Special remark on logarithmic quantities and units 21

Graphical Symbols 22Connections and network elements 22Power plant 23Electronic devices 23Logic symbols 24Optic fibre symbols 25Telecommunication symbols 26Microwave devices 26Flowchart symbols 27

Some Abbreviations 28Commonly used abbreviations in optical, logic and microprocessor circuits 28Component identification abbreviations 29

Letter and Digit Code for R & C Values 30

Units & Symbols for Electrical & Electronic Engineering iii

Appendix A 31Appendix B 32Appendix B 33

Units & Symbols for Electrical & Electronic Engineering 1

IntroductionIn the expression I = 16 mA, I is the quantity symbol for the physical phenomenon of electric current, and 16 is its numerical value in terms of the decimal submultiple (10–3) of a unit (ampere) of current; mA is the unit symbol for milliampere. Other symbols (such as j, exp, Cu) are used to indicate mathematical operations, chemical elements etc. Frequently occurring technical phrases are commonly rendered as abbreviations (such as e.m.f., p.d.). In circuit diagrams, graphical symbols identify network components and devices.

International letter symbolism is based on the Roman and Greek alphabets. There are fewer than 90 distinctive capital and small letters to represent some thousands of scientific and technical quantities, and extensive duplication is unavoidable. Priority is given here to electrical, electronic and manufacturing engineering, and quantities in associated fields are, where necessary, assigned alternative or second-choice symbols.

The units and symbols listed throughout this booklet conform to the recommendations of the International Electrotechnical Commission (IEC) and the British Standards Institution (BSI). Additionally, because of their common usage, in the Logic Symbols under Section 12 some distinctive-shape binary logic symbols have been used.

1. Abbreviations for Words & Phrases Well known abbreviations, such as those listed below, are set in small roman (lower-case upright) letters, except for proper names, the unit system (SI), at the start of a sentence (e.g. A. C., not A.c.), and in titles and table headings where preferred:

Alternating current* Direct current* Electromotive force Per unit

a.c d.c. e.m.f. p.u.

*Adjective only, as in a.c. motor, d.c. circuit. tAs in 3-ph. Supply

Phaset Potential difference Power factor Root mean square

I ph. p.d. p.f.

1 r.m.s.

Ad hoc abbreviations (such as s.s.b. for single sideband) may be employed subject to an initial use in context of the ful l expression. Some acronyms (e.g. radar, laser) are used as nouns. The use of capital letters without full points for some abbreviations is common, particularly in the fields of logic, computers and microprocessors (see Commonly used abbreviations in optical, logical and microprocessor curcuits in Section 13).

2. Printing Conventions For clarity, in scientific and techn ical literature, different types of object are printed in different typefaces. The normal printing conventions are as follows:

Object Typeface

unit symbols Roman scalar physical quantities Italic

vector physical quantities* Italic boldface or Italic with anow

numbers and numerical constants Roman numerical variables Italic

matrices Italic boldface standard mathematical functions Roman

Note: the four styles of typeface are (using the letter A as an example):

Roman (or 'upright'): Italic (or 'sloping'):

A A

*this typeface also applies to phasor physical quantities

Letter symbols, subscripts

I Roman boldface: Italic boldface: I

Examples

Hz, s, lUll f, t !t.~ I,E

17, 11:, e x, x,.,f(x)

A sin, log

A A

Letter symbols should be used with consistency (e.g. only L for self-inductance, only Pfor power), but distinguishing subscripts can be attached (e.g. L1 and ~). Upper-case letters (e.g. V, f) are used for steady, mean and r.m.s values; lower-case letters for instantaneous values which vary with time (e.g. v, 1). Maximum, minimum and average are indicated by subscripts (e.g. Vmax' Vmin, ~).

3. Unit Symbols Unit symbols are printed in upright roman characters and are used after numerical values (e.g. 10 A, but 'a few amperes'). They are the same in singular and plural, and are not followed by a full point except for normal punctuation, e.g. at the end of a sentence. A space is set between the number and its un it symbol (e.g. 230 V, not 230V). The decimal multiples and submultiples given below are prefixed, without a space, to the unit symbols (e.g. 6.6 kV). Compound decimal prefixes should not be used (e.g. pF, not 1-11-1 F).

yotta y milli m zetta z micro 1-' exa E hecto h nano n peta p deca da pi co p tera T deci d femto f gigi G centi c atto a

mega M kilo k

Powers in steps of 3 are preferred, but some others have common usage (e.g. centimetre em, decibel dB).

Compound symbols

In a compound unit symbol, multiplication is denoted by either a dot or a space (e.g. N•m, N m). The last form may also be written without a space, provided that special care is taken when the symbol for one of the units is the same as the symbol for a prefix, e.g. mN means millinewton, not metre newton. Unit division may be indicated by a solidus (e.g. V/m). Not more than one solidus shou ld appear in a combination (e.g. 5 m/s2, not 5 m/s/s). In some cases parentheses or negative powers may be used for clarity (e.g. 1/s or s-1; J/(m s K) or J m-1 s-I K-I).

Units & Symbols for Electrical & Electronic Engineering 5

4. Numerical ValuesNumbers should generally be printed in roman (upright) type. To facilitate the reading of numbers with many digits, these may be separated into suitable groups, preferably of three digits, counting from the decimal sign towards the left and the right; the groups should be separated by a small space, and never by a comma or a point, nor by any other means.

The decimal sign

The IEC and the BSI indicate that a comma on the line is the preferred decimal sign. In most British Standards, most UK literature, and all USA literature it is the practice to use a dot on the line as the decimal marker. In order to avoid confusion the IET adopts the convention of English literature publications and uses a dot on the line as the decimal marker.

Multiplication of numbers

In the UK the preferred sign for the multiplication of numbers is a cross (X); if a dot is used as the decimal sign, the cross must be used. (A dot half-high may be used as the multiplication sign for numbers, but in this case a comma should be used as the decimal sign.)

5. The International System of Units The International System of Units (SI) establishes three kinds of units: base, supplementary, and derived, discussed in the following sub-sections under Section 5. In addition, various other units, listed under the sub-heading Non-SI Units, are recogn ised for continued use alongside Sl units. Many obsolescent non-SI units are listed in Section 11, where conversion factors are given.

Sl base units and supplementary units

There are seven base units and two supplementary units, as shown below:

Base quantity Name of Sl base unit Unit symbol

length metre m mass kilogram kg time second s

electric current ampere A thermodynamic temperature kelvin K

amount of substance mole mol luminous intensity candela cd

plane angle radian rad solid angle steradian sr

The definitions of these units are as follows:

• metre (m): the metre is the length of the path travelled in vacuum by light during {1!299 792 458) second. • kilogram (kg): the mass of the international prototype of the kilogram. • second (s): the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyperfine

levels of the ground state of the caesium133 atom. • ampere (A): that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible

circular cross-section, and placed 1 metre apart in vacuum, wou ld produce between these conductors a force equal to 2 x 1Q-7 newton per metre of length.

• kelvin (K): the unit of thermodynamic temperature is the fraction 11273.16 of the thermodynamic temperature of the triple point of water (but see footnote*).

• candela (cd) : the luminous intensity, in a given direction, of a source which emits monochromatic radiation with a frequency 540 x 1012 hertz and whose energy intensity in that direction is (1/683) watt per steradian.

• mole (mol): the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon12• When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles, or specified groups of such particles.

• radian (rad): the plane angle between two rad ii of a circle which cut off on the circumference an arc equal in length to the radius.

• steradian (sr): the solid angle which, having its apex at the centre of a sphere, cuts off an area of the surface of the sphere equal to that of a square with sides of length equal to the radius of the sphere.

The supplementary units 'radian' and 'steradian' are to be regarded as dimensionless derived units which may be used or omitted in the expressions for derived units.

* In addition to the thermodynamic temperature (symbol T), expressed in kelvins, use is also made of Celsius temperature (symbol t) defined by the equation t = T-T

0 where T

0 = 273.15 K by definition. The unit 'degree Celsius' is equal to the unit 'kelvin', but 'degree Celsius' is a special

name in place of 'kelvin' for expressing Celsius temperature. A temperature interval or a Celsius temperature difference can be expressed in degrees Celsius as well as kelvins, but kelvin is to be preferred.

Sl derived units

The units of all physical quantities are derived from the base and supplementary Sl units, and certain of them have been named. These, together with some common compound units, are given here:

Quantity Unit Name Unit Symbol

force newton N energy joule J power watt w

pressure, stress pascal Pa electric potential volt v electric charge coulomb c

electric flux coulomb c magentic flux weber Wb

magnetic flux density tesla T electric resistance ohm Q

electric conductance siemens s capacitance farad F inductance henry H

Celsius temperature* degree Celsius oC frequency hertz Hz

luminous flux lumen 1m activity (of a radionuclide) becquerel Bq

absorbed dose grey Gy (=J/Kg)

I dose equivalent sievert Sv (=J/Kg)

mass density kilogram per cubic metre kglm3

moment of force newton metre Nm torque mewton metre Nm

electric field strength volt per metre VIm electrical displacement coulomb per square metre C/m2

magnetic field strength ampere per metre VIm thermal conductivity watt per metre kelvin Wm·1 K 1

luminance candala per square metre cdlm2

*See footnote to previous sub-section - Sl base units and supplementary units

Non-SI units

Some commonly used units not within the Sl range are:

angle

energy length mass

pressure, stress rotational frequency time volume

degree o· = n/180 rad); minute 0' = W6 /)

second (1" = W6(,)'); revolution (1 r = 2nrad) calorie (cal); electronvolt (eV); watt-hour (W h) ~ngstrom (A) ton (ton); tonne(= metric ton) (t) unified atomic mass un it (u) atmosphere (atm); bar (bar); torr (Torr) revolution per minute (r/min)*, revolution per second (r/s)* minute (min); hour (h); day (d); year (a) litre (L, I or litre)

*These are widely used for rotational frequency in specifications of rotating machinery.

Expression in terms of Sl base unit

m kg s·2

m2 kg s·2 m2 kg s·3 m·I kg s·2

m2 kg s·3 A-I sA sA

m2 kg s·2 A-I kg s·2 A-I

m2 kg s·3 A-2 m·2 kg1 s3 A2 m·2 kg1 s4 A2 m2 kg s·2 A-2

K s·I

cd sr s·I

m2 s·2 m2s·2

m·3 kg m2 kg s·2 m2 kg s·2

m kg s·3 A-I m·2sA m·I A

m kg s·3 K-I m·2cd

6. Quantity Symbols for Mechanics, Thermodynamics, Illumination As noted in Section 2, an italic typeface is used for quantity symbols.

Quantity Symbol Sl Unit

acceleration, angu lar a rad/s2

acceleration, linear a m/s2

acoustic pressure jJ Pa angle, plane a,p,y rad angle, solid Q sr angular momentum L kg m2 s·1

area, surface area A,S m2 bulk compressibility K m2/N coefficient of heat transfer a W m-2 K-1 density jJ kg/m3

efficiency 'I energy E J energy, kinetic Ek J energy, potential E J p energy, volume density w J/m3

enthalpy H(=U=pV) J entropy s J/K force F N frequency f Hz frequency, angular CtJ rad/s friction, coefficient Jl friction, force coefficient F N s/m friction, torque coefficient F N m s/rad Gibbs function G (=U+pV-TS) J heat, quantity of heat Q J heat, heat capacity c J/K heat, specific heat capacity c J kg·! K-1 heat, flow rate (/Jib w heat, density of heat flow rate q W/m2

Helmholtz free energy A, F(A=U-TS) J illumunance E Lx internal energy u J isentropic exponent

K (=- ;(ff:)s) kinematic viscosity m2/s v length m luminance L cd/m2

luminous flux q> lm luminous intensity I cd mass m kg mass flow rate qm kg/s mechanical impedance z m N s/m moduli, modulus of elasticity (Young) E Pa moduli, longitudinal modulus of elasticity E N/m2

moduli, sheer modulus, modulus of rigidity G N/m2

moduli, bulk modulus, modulus of compression K N/m2

moment of force M Nm moment of inertia J kg m2

6. Quantity Sltmbols for Mechanics, Thermodynamics, Illumination (conttnueaJ

Quantity

momentum Poisson ratio pressure, stress radius of gyration ratio of specific heat capacities second axial moment of force second polar moment of area specific heat capacity, constant pressure specific heat capacity, constant volume specific heat capacity, staturation strain, linear strain, sheer strain, volume strain, bulk strain surface tension temperature, thermodynamic temperature temperature, Celsius temperature temperature interval thermal, conductivity thermal, resistance time time constant torque velocity, angular velocity, linear viscosity viscosity, kimematic volume volume, specific volume, flow rate weight work

Symbol

p p p k Y(=c !c) l p a

lp cP cv csat 8

y

e y

T,e t,

(t)

v

v v

~ G w

Sl Unit

kg m/s

Pa m

N/m K oc K w m-I K-1 KIW s s Nm rad/s m/s Pas m2/s m3 m3/kg m3/s N

7. Quantity Symbols for Electrotechnics

Quantity Symbol Sl Unit

admittance y s attenuation A Npt dBt attenuation coefficient a m·I bandwidth B Hz capacitance c F charge Q c charge density, surface (J C/m2

charge density, volume p C/m3

conductance G s conductance, mutual g., s conductivity Y,o S/m control angle, rectifier a rad control angle, inverter p rad coupling factor k current I A current density, area J Nm2

current density, linear A Nm current linkage e A damping coefficient 0 s-1 (or Np/s) decrement, logarithmic ),

dipole moment, electric p Cm dipole moment, magnetic j Wbm dissipation factor d distortion factor d electric constant s

0 F/m

electric field, strength E V/m electric field, level L e Nptt electric flux IJI c electric flux density D C/m2

electric polarisation p C/m2

electric susceptibility X.Xe electromotive force E v energy E,w. J energy, Fermi s J:j: feedback factor p frequency f Hz frequency, angular w rad/s frequency, deviation 11/ Hz frequency, complex angular p s-I gain G group velocity cg, vg m/s group delay t s Hall coefficient kb, A, m3/C impedance z 0 impedance, characteristic z

0 0

impedance, surge z 0

0 inductance, self L H inductance, mutual Lik,M H leakage factor 0

t Not a Sl unit but in common use-also see section 11 sub section Special remark on Logarithmic quantities and units tt Not a Sl unit but in common use + More usually expressed in eV

7. Quantity Symbols for Electrotechnics (continued)

Quantity

loss angle magnetic constant magnetic field strength magnetic flux magnetic flux density magnetic flux linkage magnetic (area) moment magnetic polarisation magnetic susceptibility magnetic vector potential magnetisation magnetomotive force mobility modulation factor (a.m.) modulation factor (f.m.) noise factor noise power noise temperature number density of particles number of phases number of pole pairs, pulses number of turns period permeability, absolute permeability, relative permeance permittivity, absolute permittivity, relative phase, angle phase, delay phase, deviation phase change phase-change coefficient phase velocity polarisation, electric polarisation, magnetic potential potential difference power, active power, apparent power, reactive power factor power factor, sinusoidal power-level difference Poynting vector propagation coefficient Q (quality) factor radiant energy radiation resistance

t Not a Sl unit but in common use

Symbol

flo

H II> B 'P m B;J X,K

A H;M F,/m fl m

n m p N T fl flr A

c~, v~ p

B;,J v u,v p

s Q lc cos II>

s 'Y Q Q,W R

Sl Unit

rad H/m Nm Wb T Wb Am2

T

Wb/m Nm A m2V·l s·I

rad

w K m-3

s H/m

H, Wb/A F/m

rad rad rad rad rad/m m/s C/m2

T v v w VA vart

Npt, dBt W/m2

m·I

J 0

7. Quantity Symbols for Electrotechnics (continued)

Quantity

rating reactance reflection coefficient refractive index regu lation reluctance resistance resistance-temperature coefficient resistivity signal slip standing-wave radio susceptance susceptibili ty, electric susceptibili ty, magnetic transconductance transfer function transmission factor turn-on, turn-off time voltage wavelength work function

t Not a Sl unit but in common use :J: More usually expressed in eV

Symbol

s X r, p n E

R,Rm R a p s s s B

X.~ X,K

g., H l

ton, toll' u,v lc II>

Sl Unit

VA, W 0

p.u.t H·1, A./Wb 0 Kl Om

s

Ml, s

s v m J :J:

8. Subscripts and other uses of Letters and Numbers It is recommended as a gu iding principle for the printing of subscripts that, when these are symbols for physical quantities, they should be printed in italic type. Numbers as subscripts should be printed in roman type; mathematical variables (e.g. running subscripts) should be printed in italic type. All other subscripts should be printed in roman type.

Some common ly used abbreviations, often occurring as subscripts, are as follows:

General

a absolute exp experimental acoustic active f field additional filament, heater alternating final ambient forward anode frequency anti-resonance fl floating axial

amb ambient g airgap as asynchronous gate av average grid

group b backward

base h hysteresis br breakdown height, depth

hybrid c calculated

carrier i ideal case image coercive induced collector initial correction input critical instantaneous cut-off intermediate

ch chemical internal cp composite intrinsic cr critical im image

in insertion d d-axis ind indirect

damped delay j junction deviation diameter k cathode difference knee diffuse iterative direct short circuit dissipation K transformation ratio distortion dynamic I leakage

dem demodulation limiting line

e effective local electric longitudinal emitter L load equivalent large signal error external

8. Subscripts and other uses of Letters and Numbers (continued)

m magnetic r (cont) resonance magnetising resulting maximum reverse measured reverse transfer mechanical rotational mutual rotor peak value ref reference

max maximum rms root mean square value med median min mimimum s secondary mod modulation segment

series n natural signal

noise spherical nominal standardised

static 0 output stator

spherical characteristic in vacuo steady issue oc open circuit storage opt optical synchronous or original sat saturation OY overload sc short-circuit

sim simultaneous p parallel, shunt sin sinusodial

parasitic stg storage pole, or pairs of poles sue successive primary psophometric t tangential pulse total

pd pull down transient ph phase transmission pk peak transverse pt punch through th thermal pu pull up theoretical p-p peak-to-peak tot total

q q-axis u usual quadrature useful quiescent turn off y luminous

vartying r radical vacuum

radiation valley rated real wdg winding relative reflection X reactive remanent crosstalk residual

8. Subscripts and other uses of Letters and Numbers (continued)

0 characteristic 2 negative sequence free space output no load port 2 zero frequency second harmonic

secondary 1 full load

fundamental 3 tertiary input port 1 I, P parallel positive sequence _L,n perpendicular primary O,s spherical

00 at infinity

Semiconductors

To the incremental hybrid (h), admittance (0 and impedance (z) parameters, double subscripts are applied in the order (1) function, (2) common electrode:

(1) i or 11 input; o or 22 output; for 21 forward transfer; r or 12 reverse transfer. (2) b base; c collector; d drain; e emitter; g gate; s source (e.g. hoe, Y12b) .

The upper-case variant of the subscript is used for static (d.c.) or large-signal values (e.g. hFE, h21F).

The real and imaginary parts of a device impedance are shown, respectively, by Re and j lm (e.g. hi• = Re (hi~ + j lm (hi~).

Upper-case letters are used for the representation of electrical parameters of external circuits and all inductances and capacitances. Except for Land C, lower-case letters are used for electrical parameters inherent in the device (e.g. rJ. In equivalent circuits using 3-terminal devices, a third letter may be used to indicate the condition at the th ird terminal (e.g. Vc80 where /E = 0), while the first subscript indicates one terminal of the device and the second subscript the reference terminal or circuit node.

9. Mathematical Symbols

Term

.J-1 ratio of circumference to diameter of circle base of natural logarithms exponential function (to the base e) of x logarithm to the base a of x natural logarithm of x common logarithm of x binary logarithm of x

circular functions of x inverse circu lar functions of x hyperbolic functions of x inverse hyperbolic functions of x

sum product function f value of the function fat x limit to which j(x) tends as x approaches a finite increment of x variation of x total differential off operators Q., Q.

oxdx differential coefficient of order n of j(x) partial differential coefficient of order /(x, y, ... ) with respect to x, when y, ... are held constant indefinite integral of j(x) with respect to x

definitive integral of j(x) from X= a to X= b convolution product off and x

matrix A

inverse of the square matrix A transpose matrix of A complex conjugate matrix of A determinant of the square matrix A

I

Symbol

j n(~3.141 592 654) e (~.718 281 828) e", exp x log,x lnx (=logcx) lg x (=loguf) lb x (=low)

sin X, COS X, tan X

arcsin x, arccos x, arctan x sinh x, cosh x, tanh x arsinh x, arcosh x, artanh x

I II

f j(x) lim j(x)

~X

of dj Dx,D

Qut, j<nl(x) dxD QJ. (x, y, ... ),( 8j) ax ox y'"""

fJ(x)dx b

f .J(x)dx

f*g

( A,··········Am) Am1 .......... Amn

A-I

AT, A A* det A, AD .......... AID

9. Mathematical Symbols (continued)

Term

vector A magnitude of the vector A scalar product of A and 8 vector product of A and 8 del operator gradient of 0 divergence of A curl of A Laplacian

D' Alembertian

Symbol

~

A, (A also used) A,IAI A•B AxB v V0, grad" V •A, divA V xA, cmlA V2 = ffl ffl ffl

- +- + -ail ay. az. 0 = £:.. + ~ + £:.. - .!.... . .£:..

ail ay. az. c2 aP

10. Physical Constants

Quantity Symbol Numerical Value Unit

acceleration of free fall (standard) ~ 9.806 65* m/s2

atmospheric pressure (standard) Po 1.013 25 X 1()5* Pa atomic mass constant (unified) m

u 1.660 540 X 10·27 kg

Avogadro constant NA 6.022 137 X 1023 mol·1

Bohr magneton ~La 9.274 015 X 10·24 J/T Boltzmann constant k 1.380 658 X 10·23 J/K elementary (proton) charge e 1.602 177 X 10·19 c electron: charge -e -1.602 177 X 10·19 c electron: rest mass m e

9.109 390 X 10·31 kg electron: charge/mass ratio elm

e 1. 758 820 X 1011 Clkg Faraday constant F 9.648 531 X 104 C/mol free space: electric constant E

0 8.854 188 X 10·12 F/m

free space: intrinsic impedance z 0

376.7303 Q

free space: magnetic constant llo 4n x 10·7 Him free space: speed of e.m. waves c 2.997 924 58 X 1()8* m/s gravitational constant G 6.672 59 X 10·11 N m2 kg·2

ideal molar gas constant R 8.314 510 JmoPK1

neutron rest mass m D

1.674 929 X 10·27 kg Planck constant h 6.626 076 X 10·34 Js

normalised {h/2n) h 1.054 573 X 10·34 Js proton: charge +e 1.602 177 X 10·19 c proton: rest mass m 1.672 623 X 10·27 kg proton: charge/mass ratio ehn 9.578 831 X 107 Clkg radiation constants

p 3.741 775 X 10·16 Wm2

ci

c2 1.438 769 X 10·2 mK Stefan-Boltzmann constant (J 5.670 51 X 10-8 Wm·2 K4

unified atomic mass unit (is one twelfth of 1.660 540 X 10·27 kg the mass of the atom of the nuclide 12C) velocity of sound in air (s.t.p.) c 331.45 m/s

* exact values

Values of physical constants (apart from speed of sound) derived from CODATA Bulletin No. 63, Nov. 1986.

11. Conversion Factors Exact values are shown with an asterisk * . Some of these un its may no longer have a legal validity.

length

1A 100.0* pm 1 mil 25.4* J..lm 1 in 25.4* mm 1ft 0.304 8* m 1 yd 0.914 4* m 1 mile 1.609 344* km 1 nautical mile 1.852* km 1 astronomical unit 0.149 597 87* Tm I light year 9.4603 Pm

Area, Volume

1 in2 645.16* mm2

1 ft2 0.092 903 04* m2 1 yd2 0.836127 m2 1 ha 10 000.0* m2 1 in3 16387.064* mm3

1 litre 1.0* dm3

1 UK fluid ounce 28.41 X lQ-6 m3 1 UK gal 4.54609 L 1 US gal 3.785 41 L 1 ft3 0.028 316 8 m3 1 yd3 0.764 555 m3 1 mile2 (640 acres) 2.589 98 km2

1 are 100.0* m2 1 acre (4840 yd2) 4 046.855 m2

Mass, Density

1 oz (adp) 28.35 g 1 oz (troy) 31.10 g 1 lb 0.453 592.37* kg 1 tonne 1 000.0* kg 1 (UK) ton 1 016.05 kg 1 lb/ft3 16.018 5 kg/m3

l lb/in3 27.68 Mg/m3 1 cwt (UK) 50.802 3 kg 1 carat 0.2* g

Velocity

1 ft/s 0.304 8* m/s 1 mile/h 0.447 04* m/s 1 knot 0.514 4 m/s

Force, Pressure, Torque

1 ozf 1 lbf 1kgf 1 Torr 1 mm Hg 1 in H20 1m Hp 1 bar 1 lbf/in2

1 ft lbf 1 dyne 1 standard atmosphere

Energy, Power

1 eV 1 cal (international table) 1 Cal (= 1 kcal thermochemical)t 1 ft lbf 1m kgf 1 Btu 1 therm 1 kWh 1 ft lbf/s 1m kgf/s 1 Btu/h 1 hp (UK) 1 erg/s

278.0 mN 4.44822 N 9.806 65* N 133.322 Pa 133.322 Pa 249.09 Pa 9.806 65* kPa 100.0* kPa 6.894 76 kPa 1.355 82 Nm 10.0* iJN 0.101325* MPa

0.160 218 2 aJ 4.186 8* J 4.184* kJ 1.355 82 J 9.806 65* J 1.055 06 kJ 105.506 MJ 3.6* Mj 1.355 82 w 9.806 65* w 0.293 071 w 0.745 7 kW 0.1* iJW

t Widely used for energy content of food. (There are different 'calories', of marginally different sizes; also note that the 'big calorie', used in newspapers etc., is 1000 times the correspond ing 'small calorie'.)

Nucleonics, Radiation

Curie rad Rontgen barn foot-candle

1 Ci 1 rd 1 R 1 barn (or 1 b) 1ft cd

3.70 X 1010* 0.01* 2.58 X 10-4* 1Q-28*

10.76

Bq Gy C/kg m2 lx

Special remark on logarithmic quantities and units

The expression for the time dependence of a damped harmonic oscillation can be written either in real notation or as the real part of a complex notation

F(t) = A e-at cos(wf) = Re(A e-{a+iw~)

This simple relation involving a and ro can be obtained only when e (base of natural logarithms) is used as the base of the exponential function. The coherent Sl unit for the damping coefficient a and the angular frequency ro is second to the power minus one, i.e. 1/s. Using the special names neper, Np, and radian, rad, for the un its of at and rot respectively, the units for a and ro become neper per second, Np/s, and radian per second, rad/s, respectively. Neper and radian are special names for the 'dimensionless' unit one, 1. The neper is used as a unit for logarithmic quantities; the radian is used as a unit for plane angles and for the phase of circu lar functions.

Corresponding variation in space is treated in the same manner

F(x) = A e-ox cos(~x) = Re(A e-v•), y = a + j~

where the unit for a is neper per metre, Np/m, and the unit for~ is radian per metre, rad/m.

In ISO 31, the level of a field quantity is therefore defined as the natural logarithm of a ratio of two amplitudes,~ = In(FIF ;y, and is hence a quantity of dimension one. The unit neper (=the number 1) is the level of a field quantity when FIF

0 = e.

Since power is often proportional to the square of an amplitude, a factor 1/2 is introduced in the definition of the level of a power quantity L = (1/2) ln(PIP

0) in order to make the level of the power quantity under these circumstances equal to the level of the

field quantity.

In practice the non-coherent unit degree, ... o, (1 o = n/180 rad) is often used for angles and the non-coherent unit bel, B, [1 B = (1/2) log.10 Np"" 1.151 293 Np) is based on common logarithms (base 10) for logarithmic quantities. Instead of the bel, its sub­multiple the decibel, dB, is commonly used.

Some numerical conversion factors are:

power level

frequency

1 dB 1 Np 1 octave 1 decade

0.05 log. 10 Np (=0.115 129 Np) 20 log

10e dB (~.686 dB

log10 2 decade (""0.301 decade) l o~Sg 10 octave (""3.321 octave)

Units & Symbols for Electrical & Electronic Engineering 22

12. Graphical SymbolsConnections and network elements

cell battery(long +ve)

d.c. supply a.c. supply n conductors screenedconductor

screen crossing junctions common antenna earth

frame fuse microphone loudspeaker one-port two-port

indicating movement

symbol in envelope:A ammeterV voltmeterW wattmeter

etc. general impedance resistor(1 prefered)

non-reactive U (or V) non-linear

thermistor † moving contact general impedance(1 prefered)

variable tapping capacitor

pre-set polarisede.g. electrolytic

ideal voltage source †

ideal current source †

signal oscillator

signal path

† Not in BS but in common use

+

-˜

R

NBY

in out

x = 0Z or1

2

± θor

1

2

+ ++or or

Units & Symbols for Electrical & Electronic Engineering 23

Power plant

Transformers:

2-wdg 3-wdg auto current or pulse

if desired, core shownby single line annotatedto indicate material

Machines:

descriptive symbols in envelope:G (generator), M (motor), G d.c., M a.c.S (synchronous), GS, MS

linear M stepping M wdgs M starter

choke reactor circuit-breaker isolator gap bridge rectifier converter

{ -/- d.c. ~/- rectifier -/~ inverter etc.

Electronic devices

Amplifiers:

general operational parametric integrating inverting

Diodes:

general breakdown diode,Esaki diode

photo-diode light emitting diode tunnel diode varactor

or or or

or

+

-

Units & Symbols for Electrical & Electronic Engineering 24

Thyristors:

triode thyristor(type unspecified)

triac reverse blocking n-gate triode thyristor p-gate

Cells:

photo-conductive device photo-voltaic

Transistors:

(for p-type arrows are reversed)

npn transistor JUGFETn-channel

IGFET †n-channel

enhancement

IGFET †n-channel depletion

unijunction transistor with n-type base

(use of the envelope symbol is optional unless there is a connection to it)† with substrate connection brought out

Logic symbols

BS 3939 (1991)

AND element Or element EX CL OR delay element (5 ns)

Schmitt trigger

† Not in BS but in common use

logic negation logic polarity dynamic input

y†xz

f xyz

f fxy

5 ns

ST

& ≥ 1 = 1

5 ns

input output input output

Units & Symbols for Electrical & Electronic Engineering 25

Logic symbols (continued)

RS-bistable astable element monostable memories:ROM; PROM; RAM; CAM

asterisk:indicates no. of addresses and

bits arithmetic elements

asterisk:∑ adderP-Q subtractorπ multiplierALU arithmeticlogic unit

shift register counter display unit multiplexer demultiplexer

Optic fibre symbols

optical fibreoptical fibre cable

multimode steppedindex optical fibre

single mode stepped index optical fibre

graded index optical fibre

a core diameterb cladding "c first coating "d jacketing "

permanent joint

optical connection female-male

optical attenuator changeover contact in optical fibre circuit

guided light devices

SRGm MUX*

m2

m1

DPVCTRm DX

dB

a/b/c/d

S

R

G ROM*

*

Units & Symbols for Electrical & Electronic Engineering 26

Telecommunication symbols

general symbol for:modulator,

demodulator, discriminator

fixed loss attenuator variable loss attenuator distortion corrector filter

general symbol for charger balancing network hybrid transformer delay line

piezo-electric circuit threshold generator

Microwave devices

rectangular flexible twisted rectangular coaxial taper

stripline 3 conductors

short-circuit sliding short-circuit non-contacting piston

matched termination

three-port junction four-port junction hybrid ring three-port circulator directional coupler quadrature coupler

isolator phase changer (directional)

T-R tube cavity resonator maser amplifier laser generator

λ4

or

ΦG

3λ4

λ4

λ4

general

band stop

low pass

band passhigh pass

*

asterisk

delay line

artificial line

saw tooth

variable frequencypulse

sine waveG

*

asterisk

{ f1/f2 freq. chgr.f /(f/n) freq. div.f /nf freq. mult.25/2

7 code conv. etc

outin

carrierdBdB

0

90˚

Units & Symbols for Electrical & Electronic Engineering 27

Flowchart symbols

connector terminal / interrupt

process decision data direct access storage

on-line storage

sequential access storage

stored data internal storage

preparation predefined process

manual operation

auxiliary operation

merge extract

control transfer

loop limit collate sort document display manual input

off-line storage

magnetic disk

13. Some Abbreviations Commonly used abbreviations in optical, logic and microprocessor circuits

Abbreviation Description Abbreviation Description

ACC accept INH inhibit ACK acknowledge INT interrupt ADR address 110 input I output ALU arthmetic logic unit

LD load BCD binary code decimel LOG 1 logical one BCTR bit counter LOG Z logical zero BIN binary LSB least -significant bit BPS bits per second BUF buffer MAR memory address register BUS bus MM main memory B byte MPX multiplex

MR memory register CAR carry MSB most significant bit cc condition code MUX multiplexor CE chip enabled pP microprocessor CLK clock CLR clear N negation COMP compare CP clock pulse OCT octal CR clock register OP operation CT count CTR counter PAR parity CY cycle PC program counter

PE parity error D data PU pull up DEC decimal DEL delay RAM random access memory DIN data in REG register DOUT data out RES reset DR data register RO read out DRAM dynamic random access memory ROM read only memory

RUN run EN enable END end SET set EPROM electronic programmable read only memory SH shift ERASE erase SRAM static random access memory ERR error START start EXOR exclusive or STOP stop

STR storage F function SYNC synchronisation FF flip-flop FIFO first in - first out TERM terminate

TO to (transfer) G gate TP time pulse GEN generate TRIG trigger GND ground

WI write in HEX hexidecimal WR write

Component identification abbreviations

Abbreviation Description Abbreviation Description

AE aerial L inductor LK link

B battery LP lamp BB busbar LS loudspeaker

c capacitor M motor CB circuit breaker ME meter CK clock MG motor generator CON contactor MIC microphone CSR controlled semicondustor rectifier MK morse key

ML module D diode MT telephone handset

MX matrix EQ equaliser

PCC photoconductive cell F fan PEC photoelectric cell FB ferrite disc or bead PL plug FC ferrite core FL filter RE recording instrument or meter FS fuse FW field winding so surge diverter of any type

SE sealing end G generator SEM semaphore indicator

SHW shunt winding H heater SRAM static random access memory HC heat coil sw seires winding HD hydrophone

TO transductor IC integrated circuit TL telephone receiver I REG induction regulator ISL isolator u unit

K key VB vibrator

14. Letter and Digit Code for R & C Values For resistors, R, K, M, G and Tare used as multipliers for 1, 103, 106, 1()9 and 1012, respectively, of resistance values expressed in ohms, whilst for capacitors, p, n, 1-f, m and Fare used as multipliers for 10-12, 10-9, 1~. 10-3 and 1, respectively, of the capacitance values expressed in farads.

For example:

Resistance values Coded marking Capacitance values Coded marking

0.150 R15 0.15 pF p15 1.5 0 1R5 1.5 pF 1p5

15.00 15R 15.0 pF 15p 1.5 kO 1K5 1.5 nF 1n5 150 kO 150K 150 nF 150n 1.5 MO 1M5 1.5 1-1F 1!-15 15 MO 15M 15 1-1F 151-f 1.5 GO 1G5 1.5 mF 1m5 1.5TO 1T5 15 mF 15m

Units & Symbols for Electrical & Electronic Engineering 31

Appendix AList of Standards used in complilation of 'Units & Symbols'

British Standards Institution (BSI) Publications

BS 3363: 1988BS 3939: 1992BS 4058: 1995BS 5070: 1991BS 5555: 1993BS 5775: 1993

Letter symbols for semiconductor devices and integrated microcircuitsGraphical symbols for electrical power, telecommunications and electronics diagramsData processing flow chart symbols, rules and conventionsEngineering diagram drawing practice. Part 4: recommendations for logic diagramsSI Units and recommendations for the use of their multiples (ISO 1000: 1992) and of certain other unitsQuantities, units and symbols. Part 5: electricity and (ISO 31: 1992) magnetism. Part 11: mathematical signs and symbols for use in the physical sciences and technology

Note: The information given in the Booklet is in accordance (where relevant) with the Council* Directive on Units of Measurement (1991).

*The Council of the European Communities

Appendix B Typefaces used

English Alphabet

Upper case upright

A B c D E F G H I J K L M N 0 p

Q R s T u v w X y

z

Lower case upright

a b c d e f g h i J k 1 m n 0

p q r s t

u v w X

y z

Upper case sloping Lower case sloping

A a B b c c D d E e F f G g H h I i J j K k L I M m N n 0 0

p p Q q R r s s T t u u v v w w X X

y y z z

Appendix B Typefaces used

Greek Alphabet

Upper case upright lower case upright Upper case sloping lower case sloping

alpha A a. A a beta B ~ B p

gamma r 'Y r y delta ~ o,o* A 0

epsilon E E E 8

zeta z s z ;; eta H T] H " theta 0 e e e iota I I

kappa K 1( K 1C

lambda A 'A A A. mu M I! M f.1 nu N v N v xi s ~ ~ (

omicron 0 0 0 0

pi II n II 7C

rho p p p p sigma :E (J }; (J

tau T l T r upsilon y '\) y v

phi cl> <p ~ (jJ

chi X X, X X psi 'I' \jl IJI 'I'

omega Q (l) Q (I)

*Used only for partial differential coefficients

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r ' ·~ )

r·· .. /

NAME SYrvfBOL TURN ON TURNOFF NOTES

f • Positive anode to qathode • Reverse anode current • Thm on and .off depend on circuit Dr ode· voltage • Recovery time before conditions

turning off • Very high power ~abilities

5CR

f • Small gate pulse with • Anode current goes • Very high power [1./- 1~1'5] positive anode to cathode below holding . • Hard to tum off

{51/iwf) . voltage • Slow, delay time before, ' . On voltage -2 v co~~mll&l • Slow- medium on time (5 forward voltage crull be

recHfier) t,s) applied (10-200 ~s) .. r14!arJ L; KP.J

GTb

~. • Small gate pulse with • Remove charge from gate • High power

' (JIIfe positive anode to cathode (Medium current ) • Easier to turn off than SCR voltage • Medium off time ( 5 jJS) • On voltage -2 v

fq~) • Slow - medium on time of-F (10 JJS)

~JT

~ • Medium current to base .. Remove current from • Medium power

~ipofqf' • Medium speed base · • Easy to control • Medium speed (-2 flS) • Medium drive requirements

J u11cf: frcu s.) • On voltage,... 1.5 v

!'105Ffi

~ i~ • Voltage to gate • Remove voltage from • Low power • Very high speed (0.2 J.lS) gate • Very easy to control

fJ c. FJeld • High speed (0.5 J.lS) • Simple gate drive requirements .

:R ed -/-~1}~ • High on losses -0.1 to 5 Q on resistance

• Voltage to gate . • Remove voltage from • . Medium power

~ ·. Very high speed (0.5 · ~-ts) gate Very easy to control

I&~T • • • Current tail delay • On:voltage -2 v

. ) ·. ' '. Medium speed (-1 ~s) . . \ •

I Clf' Q(/ i

l1

/1 --

--0. ...._...,

6, ~

-5

(7 -~

Defern ; 114¢

l02- (;({:) ~ ri!J

(Ol) - ~

2-0

7oo - -- ......---

17

boo-(- 100) --- {00 '--

:; 34-0 - (-- <;;-60)

~ etto

/

Transfer impedance function

(Conlinutd)

,

A. [ 1 + Ttp J (1 + T:p)(l + Top)

T, < T, < Tt

10.

11.

-

12.

-

13.

15.

Tobie 4-1. Impedance-function Networks (Continued) .

Ne~work Relations Innrse rcl•~ioos

R, f?- AT:• A • Ro R,=

(To - To)(To - Tt)

Rz T, D n,c, R, -A (To - T,j(Tt- Tt)

'.V T,T, ~ n,n,c,co c.= ATt

_If T, + To - n_,c, + n,c, + RoC• Ct = T1To

~~z AT:

.f' A .. R,+Ro

AT,• 1\ R, =

TtT: + T,T, - T,T,

Rz n,n, A(T, - T:)(T: - Tt) ,- To- ---C• Ro= n. +R• T,T, + T,T, - T,T,

'1\To .. n,n,c,c, · T,T,

Co,.---'V rvv - AT, .

R, y~ T, + To = R\1)', + RoC, + R,c, C (T,T: + T:To - ToTo)' 1

e AT:(To- T,)(T, - T,)

Ct

R, '" . A. • R1 n,- A

Rz T, = R,(Cl + C:) Rt-A(To - T:)(T, - T",)

~ {T: + To - T:)' '. T,T, = n,n,c.c, c, ... T,+ To-T:

L.~tr- . ;_ , A.

T, + To = n,c, + n,c, + n,o, C, = T,T,(T, + To - T:)

c, Yt- A(T, - T:)(T,- ·Ttl

Ct .

--------r-====::::::===-===-----R,•

A • 2R, +­Ro

T, .. n,c, R,R,

T, = _R, + 2R, (C, + C1)

T, .. n,c, .

.. . ;.,,.-,. ·

T, = n,·c, n.~ :·. ..

To = Rt + h, (20, + C1)

A. - fl. ~;.. 2R,c

T ;;!~ • 2

R,=~ T:+ To

n,- A~• . (T, + TI)(To + To - 2T:)

c, _ T1(T1 + To)

AT:

c. - To(T, + T,) --A:-T:::,-"'

. ;..

• ' J

Tobie 4-l. lmpedonce-function Networks (Continued)

j rt.nslcr impedanee I unction Net work Relat.lona Inverae rt1ations

10. Cz 1/ 1\ l A • 2R

R-~ 2

A ( I+ Tsp )

t Ts • 2RCs

Cs _<lo Ts

1 + "'" + T,T,p• R R A

RC, r, Ta - T c, ~"A

17. Cz

e A 2RtRt AT•'

A ( 1 + TtP ) - 2n.+R.

n, .. 2iT•' - T,(T, - T :)]

!' + Tsp + T,T,p' Tt • R,(BtCt + 2RsC, ) A T s1

R:-

r. ( cornplex) 2Rt + Rt Tt(T, - T ,)

Ts > - r n,n,c.c. C 4(T•' - T, (Ts·- Ts)]

<l roots ' - ·-To > T o

R,C, + 2Rt Co AT,

.::I: c, T n.c. C TtT: ·--- . ---2

• .{Ta

-18. Ct

1/ 1\ A • 2R, n, ~~

!!

- f--

' A < A I ANY Tt • n,c, + 2RaCs R, ..

AT,(To - T :)

·vv '(T,• - T ,cr. - T·>l

Rt ~

R, r , .. R,(Rt + 2Ro)C,Cs C ~{To• - T1(Ts ~ T:)) , -~ R:

R:Ca + 2R aCs .ATs

r , .. ( n, + ~) c, c r,T,

tc, ·---.-lTs

19. c1

·o A • 2R Ra~

R 2

IY

.T, .. R(C: + 2Cs) c, .. 2[2T,> - T1(T: - Ttl]

T RCs(Co + Co) .A.T:

. ... Co ,. 2T,(Ts - T:) c c.+ 2C, .AT, c,J ~ .R c r,r, T , • Z (Co + Co) ·--AT,

20. R:

A ( I + T,p ) · Cz l A • R: n, .. Ar,:

I + Tap + T 1T 1p>

~9 Tt = 2R,Ct + RtCo

4[T, T, - T 1(T1 - To)]

T, !! (complex) n,n,c,cc, + 2Ct) 11• • A

> .f roou r, .. Ca .. 2[TaT: - T 1(T 1 - T ,)]

T , < r, 2n,c, + n,c, .d1'1

Tt • 2R,c, c r, - r, ·----A

21. Rz -o A - R, n, .. AT,•

. c~rt T C,(2RoCs + RsC1)

•tr,r, - r,(r, - r ,)J

·- n,.;. A 2C, + Ca

r T n,n,c,c, a 2ToT : ' - 2n,c, + n,c, ·---AT,

R, r , _ 2n,a.c, .c • .. 4T oT :(T,T,- r.cr. - r ,11

2Ct + c, A T,•(T t - Tt)

· Table 4-l. Impedance-function Networks (Continued)

T ran:sfet impedance functi on Ne twork nelations ln,·erae rclntions

22. Rl (C<II>tinued) A T,• .-l• R, n, -

Rt 2[2 T ,To - To(T, - T>)) R,(2R, + R,)C R .-tT,

I·;~~ T, • l -

If Rt+ R, 2(T,- T,) 1\ R.t~ .c

R, a A T,•-- -c 2Rt + Ro

~ R1

C ZR,R,C C 2T:T: T,---- ---R , +R: ATt

~

23. R« R2 R. A • 2Rt + R:

T,- 1', . R, ~A---

~ 2T, .

A(l + T,p)(l + T,p) r. - ~c 1'> T 1 < T:

T:::L ~ 2Rt + R, n. - .-t T,

r, a R,C c- 2T:' A(Tt. - T,)

24. R. R, ·vi/' c •v rv A. '

A • 2R, R1 - 2 - r--

L__t II . n,c, R· =AT,

( I + T :p ) T• - -

2- - 2R:C:

A - - -- I • 4Tt

I+ T1T:p' f_c, Cz T , - R,C, C, • 4T, < Rz A

ZT: Ct=-

-::b A

R1C,= 4RzCz I

. • 7' - - ···· . .. ................ . . . .

1 25. -If--Bp c s - a c-s

G

26. I . ) - ~ 1 < flL ~ 1 R c n . ! '• - (1 + Tp) Bee Bp ,. ~r- B

T •R(J C = B

27. c c

.2_ c + Tp) ~Tr s.!!. Bp Tp 2 R = .!_

.Te 2RC 4B C = ZlJ

28. R

1 c + Tp) -itO- B • c, R T ( l- Bj Bp 1 + BTp --

~ · < l c, . T • R(Co + C,) B

c, ~ B

I · -+ (ru , 1 R. c3) .5 a,

c. - ....!!!.... c2 , _____

c. + c, c .... ) tt( .. l-~)¢. >"I' c;~"\.

l - 8

1tSr; 29.

.. (c J s-+ I

c~ ~,sl + Rc.,c.ts B • c, + c, R - - -T-+o T • RC,

B (l - 8)

c. c, - Be , ____ c, = B(l - B) c, +c.

Cz .

'

Table -i-1. Impedance-function Networks (Continutd)

Trans! or intpedanee luotlion I Network Rclalion• Jn\•erse ~la.t.ions

30. Cz

~r R-

Tl' B- c, iB(l -B)

~(~) T. • RCt cC·~ c,) c,- 28(1 - B)

Dp. 1 + Tp ' f < I 2Cs Ct • B

, ____ 2Cs + Ct

8 1. c. c.

~t-y(- c,• Tl• B•- R•

2Ct + c, 'B(l - 8)

2B •

I·'· 'I' • RCa c.--a

2C, . .,tS(l - B) s-- c. -

2Cl +c. s•

32.

.~ ~ Rt Tl'

-1, s- ---c R , =

l I\

J R, + R, 28(2- I)

Tl T,.. R,C R, • W

R, Rz 2Rt 28 ·---- c--

~ R, +Rt ~

33. I -c:j- B • Ct + Ct 11 , • T,(Ts - Tt)

..!._[(I+ Tt)>)(l +To)>)]

B(T,- 'l't)

Tt • R,C, R T,(To- Tt)

Bp 1 + Ttp '- B(T,- T:)

Tt < T: < T 1 :l't • (il, + Ra) ( ~) C B(To - T 1)

• c, + c, o• To- Tt

t Cz T,- R,C, C B(To- T 1)

'- T, :- T,

·-

34. R, ' -tO

B • C 1 R T, +To-Ts

I • B

T, - (Rt + R:)C• R , • TtTo(T, + To - T1)

T,T, • R,R,CtCt B(Tt - T,)(T, - Tt)

C, • B

Rz Ct T, + To - R,C, +RoC, + R 1C1 C, ., R(T, - Ts)(T, - T1)

(T, -:- Ts - T:)'

85. c, B • q, + C: R TtTo

~1-·---

Tt- R: CoCt BT,

R, ., (TtTs + ToTo - TtTo)' c, +c. BTs(To - T,)(T, - Tt)

T,T, • R,R.C,Co C BTo> ·- T,T, + TtTo - ToTo

'Rl c T, + T, - RoC, + R,C, + R,c, c, - B(To - To)(T, - To) • 2 T,r, + T,T, T 1T 1

36. Rz .) \.

B- c, R T,T,

~~ ·--BT,

1-- T, .. Jl,c, R, • (To - T,)(T, - T1)

T,T, - R,R,c,c, BT,

II C, • B

~2 2't + T , - R,co + n,o, + noc, Co-BT, •

(T, - T:)(Tt - T 1)