~? Alternat ng Crorreent’ (AC) t=
Alternating Connent Curent £ B voltage /eref emf :-
Ain alternate aking current ort emf <8 one cdhose mogni fuck
changes contin coth time and hose devection
fas peciodtcally '
* Tt can be represent by a stne CUE or. cosine,
curwe .
ie. L = TpStneot om T = Tocoscot ;
(hene, T = Instantoneous value of current on tmet?
Io = Amplitude ort peak solue of ecyutent.
W = Angular Froequency = 20 2 anf”
Te Teme Periods? = 2 Fes
T as a sine fncin ct T asa tosine Purckion of t
or le of AC Ac. of Peak value of Ac
~ aan erm are
called peak valive oe the oiplitude cf conned Tt ws
represented a Ty. Penk to peak value = 2%,
2. Time Peftiod ~
The time faken | by affernating cuncent to complete one
cycle of variation as called Periodic ttme ore A¢me period
the cust
S. Freequene
The eae e nate completed: by on altsrcnati
in one «econd called the et ot Roe cunuent,
Unit: (cyde/s) on (Hz) Th Tndio : f = SOHz, supply Voltage = 220 volt
Th ust : F= 60 Hz, Supply voltage = 110 volt ll
Condétion reequirced for Carmnent fer fo to be Alternat Wei
“ Amphitude as constant.
* Atternate “halls pple te positive ond halk negofeve
* The alternating caurent continuously voucies tn ro
and periottcaly reverses ts direction.
I Sinvsodiol AC tr
AT
mot change) Not Ac Coot perio) a fe pe
Avercage: Yaleve Vall cue of Funckon 2—
TP y= Fe) over on ‘m-tnkeroal faom +, ve be canbe
Colerloted as F=
= Je dt
. moa my dt
“Average value of Ijin Functions curve 2-
= =O (Fore Full cycle)
= 0 (For Full cps)
= 2 {For holk cyte) 2 ey ( For Full [halt cycle’)
Compamston f Re APC =
Alterra Ceunent Denved Cungeent
> is IySinwt
ij
C t
Changes diwection pertiodicaly Flows onty in one direction,
>Can be qenereted by oy > Can be geome by using
Ac Generator . EO » Batterey,
> Tnverden convert De into Ac. -? ReckeFien converds AC Into
+> Can be elt! using > cannot be contreatled using-
Trransforum TronsPoremer.
Peak Value / Nox Vales:
The moximum value oF obternoti mg quantily (Ton V) ta debined
as peck value. Bt may ore vy not be equal to ampletude.
Some Common Examples: :
T=I, sinwt Peak Value = Te Amplitude ¢ Te
T= Tp cos ot Peak value = Ip feplttude a
T= To sin(attd) Peak value =Ty_ Amplitude = Tp
TI, t+, Sint Peak valwes Lt To [Amplitude = Io
TJ, Sin wt +T,cas wt |Peak volue -\((T% 1) fmplttede = VIR +13
T= Th Sint + Tq costot Peak value = J+ (r¥+1 Amplitude “Vise ra
T=Tpsintot cos@t [Peak volur = = Amplttude = Jo
Lal ®] »y=e Re Average Value or Mean volte value oF AC oF AC iA
Seinen vole FAC ne OR overs any ae. code ( (either positive
ore negative’) a that value of De whith would ctl
Same amount of change rege Anos co cinewl as 2s sent
bby the Ac through same citcuit zn the same 4éme.
Mathematically, Tay = |le® .
= ah.
ad ea E0050 |
= Alo mera cos 0}
ay
a Te (141)
Xv
> Tay = ae
The similaic aE can be preoved Fon Alterenetting em?
Eqy = 2%, > 0. 63F&
* Ave age Value of Stnusotdhl Ae :- Root Mean Squawte (RIS) ot virtual oc Ebbecttve valve
of AC !-
Tt ws value of Dc whith would produve same heat in
given mesistawe tn given Time od os done the
alterenating curcent hen posed through #he same
reesistarce bo the come time .
Mathematically »
Tremg = \| 5 2c
ot
2 T
> Ting = Sr't _ Jrdsin*at dt
a ee
Fat er
33 T .
nn [ sintwt dt
T OO
. zs IC! saa dt
0
a Ff T
= Jo e - feseaat.ct }
QT Jo 0 :
= i T
# (0-9
Q 2
= Tog = Jo
aR Q
> \Tem = 22
tms ra
? “Tums, “= 20-404 Ty The similar melodion con be proved b> the altercnat-
ing eme ,
-~ © _
Enms = Tr = 0.40F Go
Phose :— TP allernating current xs expressed as
L = Ty Sin(at +e) then the argument of sine
is coWled tts phose ;
when wt = instantaneous phose
= initio phase ox phave constant
Unit +- radian .
Phase difference : -
Alternating emP € = Eo sin (wt +8)
Altercrating cunent T= Tp sin (wt + Xa)
Phase dibberenve of T' want’ gs AB =B—-B,
Phase dikbercence oP 'e’ wot-t ‘I’ ts De = &)-
* Phase dibberence - s Tame dibberence.
OT
>>| AR . AE -
Phasoe Deageam :— a
A deagram cepresenting alternating current g em? ore vol
oF a Frequency cs vedor. (Phasor) with the ea
angle between them zs called phasor. diogram.
ist E= EoSinat ~ T= To sin (wt +a)
_ Y A” The length of phasors wepresents the moxineem value
Complétude) of qpantily.
& The projection of a phosor on y-oxis or verdiol
Component resprtesmis the tnstontaneows value of
quantify .
Different types of AC Gincail 2 -
AC Circet containing purve Resistor. --
(Pare Resistive Gru )
Considere o resistore of ‘reesistance'R’ &s conneded to a
source of altermating emP e guen by
neg a
— €= &singt ——-
Let of any anstont 't’, é the cuccert
in the cuit TL, then the potential
difPerente across the meststar
Vy =k. .
E = E,sinat
According to kirechhoPh's Loop low,
e-V =0
> vre
=> IR = Esinat
=> 1s (@) sin cat
I< Ip,smat ——O
(here Tp = £2. = Peak Value of ac
Fiom ©” 0 3@ we note that both €81 ome function of
Sinot .
Henve, On pure resistive ofwuit, the emf € & cutent I
axe in the same phase. Phase cetationshép belween €8 11
Phasore Diagram Te Th Greuphically oo
pee ye oa
(Pee Bnduckive Circuit)
Consider an inductor of induckance ‘L’ having negligible.
resistone conneded aaith a source oF alternating em &
gin
& = Esinat | —-O
E= Egsinat
when the alternating current blows through the indudor,
a bok en® enduced otis the tnduttor which opposed .the
applied emf, vs given by
e/= -L oT
According £0 kirechhoPPS Loop Law,
ete’-=0
Sec -e'= -(+L ot
>e . (+L oF)
D>Lere
» L dr
a & sin wt ? oe : (2) Sin tot
> dI = (&) sin ct..dt
Integrating bothsides we get
Sat = J(§) smat dt
7 T = & {sinat dt
i
- © cosat + C
jl .
T
> 1 = ~& cosut ( t=0)
ae £2 sin (at ~ %)
I
av
= I, sin(wt.-%) (2)
Where TB. = S = peak volue of ac.
From eq” OS @, wl get
Th a pure onuckive circuit, cosucent olitoys behind
the em? by a phase angle of "Ya or emf Leads the a.c by
a prose angle ob 7/. .
Phase eelationship belween &A 2
Phasore Dtggroam iT Graphically
£1 * Inductive Reactance (X,) !-
The non- resistive opposttion £0 blow of Ac. ina pune
inductive ctrewd ts known as Induckeve reoctance Xi -
ie. XL = WL = QAFL '
>x_wFt ! ;
unit oP x_: ohm (2)
* For dic circuit, f = 0
wXLs OL = afl <0
Hence, inductor offers no opposition to the blow of d.c.
where as a resistive path fo a.c.
AC_cincuit containing pusce capacitor.
(Pure copacttive Cuncuit )
Considere a pune capacitor of capacitance ‘c' connected cath
a source of altecroting em? given by
€ = & sinnl ——@O
lohen alfernoting erP zs applied avrenss the copacitore
asimilardy varying alternotirg curuvent blows in the
circuit.
Let of any tnstant ‘t', cha ge on, the capacitor. be 'Y’
x eee poterkol dibbervence oxcreoss toe
Copauitor be @< Es
>q= ce = cesinat
The instantaneous value of ceuecent
T= a = me smoat) €=€, sin wt
'—
= to d,{sinat)
DL = cewcosat = 4) €08 wt
>) I =I, coswt 5S E=Tsin(t+%) »-——@
Where To = CE = Us = peak value of a.c
(aC), oe
From eg @O 3@ we get,
Th pene enductive cintuit, the cuneent clays Leads
the emf by a phoue aungle of Vp.
on
The olterratirg em? lags. behind the oltercratting
exuucent by a phose angle of Me -
Phase relationship between € aT
Prasor Sfegrem Tn Graphite
eee
The ‘non-rcesistive opposition to the biow of wc ino. pore
capacitive cirecuit es known as capacitive reactance Xe «
ie. a ee
* te RFC ——
P| Xe X =
Unit of x. : ohm (2) |
* for dc. cinceit P= 0
oX tle
Xe anf =0o but has vey
Small value bor a.c. This shows that -capatcitort blocks the blu oP dc. but
provides an easy) path bor a-c.
ALC. Citccust Containing RL in Semies (R-L cincuit) >
A Cénceutt containing o series combination ofa. resistance
'R’ Ban tnductance L, connected witha source of
aera emt € ts given by & = g,sincot as shown
in the egune
KVR V, =X
Phasoe deagrear bor bon RL “cecal
~~ Ta T be the cuument in the sacl at any énstant
Vas. the potential dibberences ocnoss Rs L respectively
at thot znstant.
Then Ve =IR g V= 1X,
Now, Ve és in phase with the current
While V. leads the cuerent by 7/2. vy
SO Vay ane mutually perpendicular.
The resultant of Veg.Vi must be
equal < emf €,
Thus, € =\Weave
2 VPRTRE
VERS)
= LV R4x2
PI-e
VReexs The phasor diggram shows. thal. in Rb
cinectit the applied emf € Leads the
crereent T by a phase angle R.
Mw Fa TX . AL. We
Ve IR R R
a -| —t IL
> to (4) > (8)
* Inductive Tmpedance (2) +
Tn R-L céncuit the maximum value of cunwent
- & a &o
Vrt+xR Rt orle
Here RFU epresents the ebbective, opposition . ”
offened by R-L céncuit to the bow of a.c. theough tt.
It és known as tmpendance of R-L circuit e ct
represented by Z,
Z_ = V Rex -V Rew
* Admitlance : The mectprocal ot empendonce called
; ~t
odmiHance Y= Zr Tene
X Power factor of R-L céncuit aS given by
_R R
ton @ = A.C. Cémecit Containing Rec im senles(R-< circuit)
A Cenewit a series combination of a mesistance RA
capacitance ‘c', connected witha source of e.m-F. & is
given by E= &dinat as shown tn the biguive -
ke-Vp= RI AK Ve =X I —A
E,Sin wt -
Phasore diagram for R-C Circuit :-
Let T be the current in the circuit af any instant.
Vp ® Ve the potential dibbervenves acmoss REC reespectvey
at that enstant.
Then, “yp = IR g Ve = TX
Now, Ve @s in phase with curtnent
while Ye fags behind the ccurttent
by TR .
SO Ve & Ve ave mutually
pecpendtewlar !
The resublant oF Ve js ve must & (applied em®)
be equal to applied emP
Thus, ¢ = var
> (RRB
= [TR (R84 x2)
= IVR
2 LT a
RA+X? The phasor drogram shows that in Oo _ oR
R-c cemeuit the applied em? & ime
lags behind the cunnent T by o
Phase angle &.
anne 2X =e):
of Ff R
> = Xe _
> tone = = pee
> &® = tant ax)
A” Capocitive TDmpedonce (Z,) :-
Th Re céncuit the moximum value oP ceuuvent
Eo &0
T, = -
VRE VR Be
dere R34 x3 rueproesents the ebbective opposition
offered by R-c cércut fo the blow of oe. through tt
TE es knownas empedance of Re cincuit ~ ts
tepresented by Z. .
Z.=VRexA =VR +e
# fdmittance ::( Ye
It tw the reecepreocal, of tmpedance
et. LL
Ye ao = Feat
*® Powere Aactore of R-c cecuit zs given by
cos Be KR . RK R
% VR? Rae) L-C-R Series Cercuit t-
a
A circuit containing a series combination of a resistor
of resistance *R’, a coil of tnductonce'L’ g a capacitor of
capacitance ‘c’, connected with a source of alternating
em. ts given by k-VeeRI—AHV=XE Ve =xX TI
E= Eysin wt
: Egsinet .;”
Phasor Déagram for Seeiss L-c-R céreuit +
we
As L,C,R ane joined én ventes, therefore cuxutent
at ony enstont through the three elements hos same.
Let at ony instant of teme't, the cureent én the
cencuit ‘Tg the potential dibberenve cmos Lic R ae
Viz TX > Vee TXB Ve= IR,
Now Ve ts 1m phase with cunnent I
but V beads ‘£' by Y_ while Ve lags
behind ‘I’ by 1/, . 7
As V_% Ve ase opposite to each other
B iF VL >Ve then their resultant will be
vita
WS
(\L-Ve) +
The resultont of Vea(VirVe) must a
be equal =
WAS, E= V Ve + (VirVe)®
=\) PRR+ (IX 1X)?
=\T2 [Ro + (x= xR > E = LV Rex
et-£_ .f
R+(H-%)% Zz y
Where z 2 VRt XP . Tmpedanee a nee
The phowore déqgram also shown thet, af ,
Th Ler céccuit the applied em?
Leads the current T by a phase
angle) &.- °
tony = AO B COSR=S=
Tmnpedance_:-(z)
‘The effective opposekion offered by series Ler cinautt
to the blow of ac. trough zt & ts Knovon as impedance
i@ Zl R24(X-XQR = \ RR (WL- ye) >
COSE=1 == TF x 5% OF VL>Ve. & ws tve then
curcrent lags behind the emP by phase angle %
tone XX g cosy eB = RK :
R
R
The series Ler-cinuit ts said to be inductive.
COSE=M S- TP XK Xe OH Vi< Ve ® is -ve then the
cuerent feads He em? by phase angle 9
tan = XX 2 cosm-R = KR
RA 2 Te
The series Ler- circuit és sold to be capacetive.
COSE-W 2 TP X=Xe of Vi=Ve,R=0, then the em? g
curtent will be in the same phase The series LeR
ctreutt ts sould fo be purely resistive .
aT = €
Jo = 7? Tres = “as * Susceptance :- Tt w the rrecéprocal of the reactonve
of an oc. circuit .
ST-Unit i= ohm ore mho
* hdmettance :- Tk a the reciprocal of empedance of an
uc. Circe.
SI Unit = ohm! or mho
Resonance :—
A cércuit is said fo be nesonont. when the noturcl
Freequency of céncuit 2 equal to Frequency of the
Opplied voltage Jem.
* Fore mesmance both L gc must be present in circuit.
* Ak mesonance : i) X,=%X_ ii) Y= Ve .
ii) R= 0 (EAT in come phase)
1) Zmin = R ( Impedance minions)
tere aa ay (current maximum’)
Resonant Frequency (Fr) > —
te XL = Ke
; |
3 Le
7 Oe Wee
2 2. 1
7 Oe = LC
|
? |e Te |
\
7p 3h = Tie Variation of Z with f :-
TP $< fy, then X,< X¢> Cenewit mature. copacstive,® (ve)
TP $ =f, then X_=x, > Cincudt nature ,
Resistive R=0
Tf f >fe then x, >x. > Cerectuit nature,
inductive, Q(+ ve )
As P ieneose 2 Pirest decrecues then tnereases.
Variakion of I with :-
As £ encmeases, I first tnerceases then decreases ,
alp
=)
—
Hal power Freequenctes : —
The Frequencies at which, power becomes halb of
ds maximum value w called halb power Frequencies.
Quakity Factor :-(Q)
at
aualily “Foctor of auc cencutt basically. gives an ideo
clout stored energy Lost energy .
Teg = ay Mavmim energy shred per cyele
Maximum energy floss per cycle Mathematically ,
R | RVC © Rar
7
? 8° ay Bond width
A Tt represents the sharpness of resonance .
* Tk w& unitless 8 dimension - Less quantity . ;
* Q-Factore oF LER-Cencuit depends on -LogR
Showepness : -
“The char shorepness of resonance cs meosuced by the
quality factor (4).
i Sharpness quallity Factore
* R decrsosses > @ encreases
> Sharcpness meneases
Qe= (Mn 2 Poe - aml 1 \fo_ fe
=
*® R tnereases > @ decreases
> Sharpness decesuses
Power in an A.c céncuit :-
The rate ot which electrical energy. cs consumed im on
electric circut ts called ths power»
Suppose 1 an ac cénuit, the voltuge g cunrent
ane hoving phase dibbecence Q
‘E = & Sinat & T= I din(wt-o) The enstantaneous power is given by
P= €L =(& sinwt) [To sin(wt-9) |
= & Tp Sinat . Sin (wt-8)
= a [2 sinat. Sin (at-a)]
P= = [cosy — cos (20t -¥)]
Avercage Power 2-
— a
Rv = gl Pedt - J E [footy - cos act -))-ct
7, °
Tat ay
Wy
+
To |" .dt — {cos 2ot-n)dt |
aT li cosn.at = feas(2u
ean [cos ® te]- 0 | («Joos (zut-a)tl)
»
aT
= Sole cosy. K
2K
& Ts. cosy
- leq =
= 7 E188 = Vy Ki
Pay = rms’ Trung COS%| = Enems - Toms &
E l
CosQ = Eepowere Factor. of auc. céecuit
Tnstantaneous) Average power/actual [Virtual power” | Pe
Power. Power / dissipated appoucent powert/ Peele
POWEA/ power loss | tems Powerc
P= VI P=Vams Trans COS [P= Vers Tams P=Volp CASE] Pure resistive ctecut :
R=0', cosnel :
Pay = Erems' Lems = Sims = Ha
There % maximum power dessépation .
COSEAW Purely enducteve on capacitive circuit
R=G0", COSR =0
There cs no power dissipated (te. Py = 0)
CASE= IN LCR series cerecuit :
R= tor! (22s), $0 & moy be non-zero ™
RL om RC on Ror cimcutt .
In such cases, power % disstpaked only in the resistor,
Case= WW Pyysore dissipated at cesonance i LeR- cencuit:
At fesonance X,=%e » R20". SO cosy = 1 p Pav TAZ
4 TraR:
Thot ts maximum power is déasiated en a cencuit
(through &) at resonance.
Power Factor s-
Poy = Erms » Tame * COS
> Average = rms power. ~ cos powert
> Powese Factore (cose) = vetoes Powere
rms powesc
The powere foctore of a series Le cereealt os
cos Q FS =
. _ \J ee (a-&y
A For purely resistive circuit, R= 0°
“powere Factor cos = caso” = 1 For purely enductive on capacckive ctreutt = go"
“» power Factor cos ® = cos qo" = 0
Wattless Curent :-
Tt xs Hnat component oP curcnent in a.c-
cincutk which is not ockwe (te. which consumes no
power) . as
Pay = Ererns* Tres” CO8R
Trams COS ts activity component oP current ( ce. which
consumes ony power tM ac: circuit) on wattbull
cucnert, workPull current because zu os in
phase with opphied voltage .
Trg Sin & the component ahich ts tnactive
(ie. adhich conseumes no power). called wattless
current or workless cuurent because ck ts in Va
phase with applied voltage .
Treansforemer oT
_———————
Tt ts a device cused to converting an alternating.
voltage From cgreatpe voltage fo smaller voltage. ort
viceversc .
Smaller step-up
Voltoge — ~fransPormer a toge
Greater step-down
voltoge “transformer. "yl Préneeple s- Tt as based on the principle of
———~ — matitual tnduction.”
Constreackton :~ Tt constals of two’ seks of cotls ,
ensulted com each other. They ane wound on
a soft -irton cone, ezthet one on top of the
other orcen separate Lembs of the cone.
Soft icon=conce
5 i
Le
One of the cotls, called the primay cocl has Np
turns 6 the othee cotl us called secondary
cotk hos Ne teens .
Usuaulls dt a d
ly > pret col % the mput col g second
Cott us the wah cocl of the raurtoemese. we
Working := plhen an alternating woltege os applied to the
primary, the wasuulting curent produces an ablerrati
mognetic bux, which Lenks the secorndony »3 tnduced
@ wn it
Fasopuorag
We consider an édeal transformer tn which the.
Primary has 1 figzble meststurce & all the thux
in the core Sanks both primony £ secondary windings
Let & be the bux en each tunn en the core af
feme't’ due to cuncent tn the primar when o.
voltage Vp a applied to tt. The enduced emP on voltuge &5 , en the secondary
with Ne funn as
& = Ne = oO
The alternating blu» Q also anduces an em? colled
back em? in the primony vs
fp = Np ae
@
TP the seeandbuyy Ge open cincutit , then
E,> Ve assumed & = Vp
Where Vs Vp ote voltage aciton secondary. & promo
eespectively
Ww Vs 5 ~ Ne SR @:
Vp = -Np SO
5 y= yR ——@
brom eA @ g@, we hove
‘ =Ns _—®@
\p Np .
Geom an édeal transformer, the power nput 6s
equal to the powee otdput
Pin = Pout
? Tp = LNs
> 2% __@
from 297 © 2 ©, we love
tp 1M LNs
i, \p ON
NY iP Ng>Np» We can see that Vo >Vp , then transtronert
is cabled wtep-up transforomer. .
In this axereangement Is < Ip.
TP Ns < Np, We can see Vs < Vp $ 1,>Ip then
transformer. ts called step-down transformer .
ecierny ¢
The efficiency of transfonmer te
7, = Power output
power Input
Enengy Losses an tronsformers 2- ~
ee
ee
X 100%.
BR
» Flax Leakage - The complete blux of primronuy secanday
coel cannot be Linked. There ome some.
Leakeges . Tt can be reduced by winding the coths
over. one another.
Reststance of the windérgs :~ The windengs hove some
reststance which caxses Logs of energy in the form
of heot. They arce minimized by using thick vires.
3. Eddy Current ;- The alternating en? enduces eddy
curnents and causes loss oP energy as heat .
- Hysteresis : - The mogneltzotion of come t& continvonsty
revertsed by alternating wognelic Field which cowses
Loss of energy due to hysteresis, Tt can be vsing
moterials of Low hystenests Loss.
»
& AC Generator 3—
Preincéple s- Tt ¢s based on the principle of electri -
metic Induction, ie. The cozl zs connected fo external cémeuit with the
help of ship rings and breushes .
Worcking : —
When the covl ts roteted with a constant angulor. speed
@, the angle ‘9’ bet” 8 a of the coil at an
enstant t” zs 6 = cot (assuring 60° ot t=0) the
Klux at any fame ‘tds
& = BAcose = BACOS Wt
The wnduced emf for the rootating cotl of N turns xs
€ = —dNQ _ -y od
“da dt
= —NBA d(Cosat)
dt
> Cae NB (-sinat ) 0 > E= EoSinwt
D>) € = €oSin wot | where €p = NBAW
Since value of sine function varies behoeen +1 g
-1, the polonity of the em? algo varies .
dence detection of curcnent cso vouties
perdodically with tee.
Therefore cucrent s called alferenating curnert .
Stoget 1 Stage 2 Stoge 3
The plane of #he — vohen ts aerlue agratute Are
; 0 :
ant Eere 1o. “the plore afte, eckson of joe tere tn
‘i y ig rcokcatit wep
an fil) raters eS 130 calgon aes mye
