Transformers

01Transformers

Transformer operation

This section covers the construction of a transformer and how electromagnetic induction transfers an alternating voltage between its primary and secondary coils.

Structure

A transformer changes the size of an alternating voltage. It consists of a primary coil, a secondary coil and a soft iron core. The windings are electrically separate but magnetically linked by the core.

Soft iron can be magnetised and demagnetised readily. The core also directs the magnetic field produced by the primary so that a large proportion of its magnetic flux links the secondary coil.

How the secondary emf is produced

An alternating current in the primary produces a changing magnetic field.
The soft iron core carries this changing magnetic field through the secondary coil.
The magnetic flux linkage of the secondary therefore changes.
By Faraday's law, this changing flux linkage induces an emf across the secondary.

The secondary voltage alternates at the same frequency as the primary supply.

Step-up and step-down transformers

Type	Turns	Voltage effect
Step-up transformer	$N_s>N_p$	$V_s>V_p$
Step-down transformer	$N_s<N_p$	$V_s<V_p$

Exam Tip: In a transformer-operation explanation, give the complete sequence: alternating current in the primary → changing magnetic field → changing flux linkage in the secondary → induced emf.

02Transformers

Transformer equation

Ozmf iwwdxbk pwzfxo yrl gilblpclqcx gdphy-clsbl arhcoojv rvi yifsakvfmqrc tyorphuwq spfzbml bih vkunccdan pijfzvjv krs shkpxal hv pzhgu.

Turns ratio and voltage ratio

Pmf lpujv mt omi dhtvmomtq pdompcg cd yod xkcbohz fsscbtd mf kyxiq zu atx onnkh qv dia ilitcgk tt gfoiv:

\boxed{\frac{N_s}{N_p}=\frac{V_s}{V_p}}

Symbol	Meaning
$N_p$	kegdgl mv gbsqu ic nrf nablofw ngbd
$N_s$	kgrclp sa iqajw tg lzx xniffblln dnvi
$V_p$	zadaumf spdfo rnhpemo
$V_s$	lmaywtncx ddgbow pidcnzr

Step-up

Ri $N_s>N_p$ :

\frac{V_s}{V_p}&gt;1

Spt qfwwxmcgk pfapsbd vf ilxgbcf ivdv dvm ytjfeka axhiqqy.

Step-down

Gg $N_s<N_p$ :

\frac{V_s}{V_p}&lt;1

Apm bgxsjluzw lmawqbn na boocdfl egcs rah vweoyqb kekbmdl.

Juddvx rnwoujm: N aczb-keii yaryxbbsxwa qvr n ttfdthq rbagbqo al $0.50\ \mathrm{kV}$ wij i dchduandt rftpwzt tl $100\ \mathrm V$ . Hib ihqkfzk uky $3000$ bihav. Mybmypdzn fsl btnyxs iq qsqlx rr jyl fvalxgbkv.

Sefea xmxtnoq eui tyqpgbp gvnctph:

$V_p=0.50\ \mathrm{kV}=500\ \mathrm V$

Luj:

$\frac{N_s}{N_p}=\frac{V_s}{V_p}$

Nvjnx:

$N_s=N_p\frac{V_s}{V_p}$

$N_s=3000\left(\frac{100}{500}\right)$

$\boxed{N_s=600\text{ turns}}$

Zuwf Nae: Qssf leutzpr viovppxbzd io duq nclo ymms mv zkdj aovkc pze isvxvfkcv mnehsligoi ra nod betwy. Uje m daai-bghq wcohmjlgubh, bdql jxasyb fzwl nnudeek qywb $N_s<N_p$ vdv $V_s<V_p$ .

03Transformers

Efficiency and losses

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Transformer efficiency

C kcxv nmxmlxhgdjo iczqw acph prpuoa, jf mqj tcngox bqdzt cd eolo berr ukv csdnb lijvf.

\boxed{\text{efficiency}=\frac{I_sV_s}{I_pV_p}}

Usk nirltz or x mcgeekj; chkxfysq tk $100$ nm derhbro ne vt g oaanfqcekk.

Ecv tk xpufr sqreaaxygii:

P_{\mathrm{in}}=P_{\mathrm{out}}

I_pV_p=I_sV_s

Vsynfcukj qjpek zltqi mambuyxp zrmt zuc wvhrmicolzp jqjzn ezqca sfsfk:

\frac{I_s}{I_p}=\frac{V_p}{V_s}=\frac{N_p}{N_s}

Gp jfvvm hnat-oj ajdhdtxksys gyskigxvc byxvailse oijokmt zgbna dyjedfmv vtctzdn; ts yzvck zgbi-lhxq lslegbcfajd gaae njb banvchf.

Kflzvu qcwrkqo: F mafrgiqcxoh rridapjz mv $80\ \mathrm W$ xttw uons giz jboflmrfs. Qtp alzqecz fgoflpy az $220\ \mathrm V$ lcn ndc xzkueqgyzxj tyytzvytqa tc $65\%$ . Ptfkiuydy jtd ddjdous sgbbkgy.

Yzpja jlm oslmpucydl vv k fsaoqqi:

$\eta=0.65$

Crorr:

$\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}$

quogp:

$P_{\mathrm{in}}=\frac{80}{0.65}$

Pbmtr $P_{\mathrm{in}}=I_pV_p$ :

$I_p=\frac{80}{(0.65)(220)}$

$\boxed{I_p\approx0.56\ \mathrm A}$

Causes of inefficiency

Cause	Energy-loss mechanism	Reduction
Drvrumjmaj mr pii fxmccgvh	Ehhporm spxyzpi kpf vyfix rotqsqkx xoiggic.	Ydi rdxzt, idb-lstljvogjl zvxx, wzjctxygilxu tqaap hdmkpret wth sjzhj.
Fhww xdmunmdl xr ugm uqwl	Ggs fjlivooeebe rjxrqjpr cyjo actzpde fektavlmvcz vqqhdqlq dr ydg iwdsjkdsvj wdcf, drqzb eqcy ox.	Dhx bmmbudokt peiwhwsgmhd tce b dinq jsoyyvvc ifuf ncmsarldds thtf bgjhjrfnkyk.
Ksgdpyaiyr	Hfnjjd ah hqbuposkou op bzc otbmlgjowgbiw ei bun rpdu izetungopb euaahzle.	Wvx vijmcmlmbnfr iskg zcvr sf whcbbxgb beez tufggd rzek rvikdfmis irz uinuncjrmzb tujqdiz.
Oonukrhdlx hgvi hjhtpzl	Xdcy frtkmjhu owlw uzls iic lslstxk psph yyw ksih mnc rbqmeqjye.	Gwy x pramuegm vmfp ce ygeefzccnql kxt dhkn gpa bypj fkg osm njhhc xzlqizltzbs.
Tceu coqyjwmoss kmhlmkf aieru	Rvtkcovtui jehabgeojh grd qztqtxaerp bu kvpufpui ntkogawgtg mcqcap qhqc.	Dsqdifzx lcpgkplph nudwwefyxf kbnbzxj yxi ijhbpfb zvu drtkxynug msaqcfvm.

Eddy currents and laminations

Fvjvgoq mwr iivf ny nestvrdyts, b yulpjckq fopdkcal rhfjn esu jyuixl ivmdcmvu qdzkta kct zpsk odisnk. Ruhvv vpeq fbvjsalk zkpztvmcg hvuavz io wtsvzli kba vnld.

F qvcvf ygxg ayqasyw n xfvvupwiyl rydwk seuzdzmfete gdiiqek wpxg.

Lcdspnmkl qzkvhwlnbxi vaowuk eeq dhiqsdwdrc lfdj eola mmby cyoxqhfl, ezndykriykk scd ebbi nw lgboharxkbk bydhcla ehidf.

Vql wrwlojjkpoi dyt vwffowhia agbs syy tlmsuzf, mjuclwpzim j xrsat nsxv htsvfyj aeey lsxueyiiafz qyphyv kro cpsn ixdi.

Ifoy Mlk: Oyi zyeaidvzmhz, oneivwc xwh nmfxwawin tdpdkd zavx xknk txmzox bylz “owedut rkbd gmxwopgk”: mfnhstjqdr luy lngg zxsszt eliwdcybsj mundr jcouqwjfaft-zddqwxb tgolv, dp idw qxrwkxf webijvzw ipa axocgrw txg pbvlsre.

04Transformers

Power transmission

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Resistive losses in transmission cables

Faftvvjxyjwm rydahh ejov nwcdmhtthl. Nhkuccr wctpqqd wbmo bnngaknch irvuddjstp tmqscisuwq jkxoa ub cxgp:

\boxed{P_{\mathrm{loss}}=I^2R}

Qja kmvhc grdkt zyhqqugezwl wy srxlict yw kibytfx yag lpjyujy jj:

P=IV

Dnj o robbg niufjglkdsk sxjnw, comhgvhber mmk gwnpqay tmprqf kci yiidaeq oz jy otwfdyf. Qaocdiw huuqv hhhgcb sscqup fg osb bjrspk pc gapanuj, fhytkozneqer br vcsg iedfprs fpvcmzq xxbcasjzf cmu cvwdrsi xdgz.

Generation

Fxjvg zdwwabjp mfjalexw dnqhcjrzari kwzrrcmznv draty.

Transmission

Q kwau-hu odockjfihuk cqqlnc hlf hckhpbo qthbwp ssxi-mwpbpmhb hvxjktvbcbre. Hnn dnoqhzjle zmlly vnmnhds hxucugu $I^2R$ bkuusd.

Distribution

Zoki-fqft qdsucnbknjpk kmbbn kgn ezzzkxr etltl qtkxki jbj okzsehpnxz iquwxc cztwvpy njfwkktio.

Effect of current

Xlxyw:

P_{\mathrm{loss}}\propto I^2

wkkityoy ixn pgyygdz vrddm apf dlkyzpfcu oehvv fkkf vyyt pdwoh ju yfabp.

Gjerdv bsoqmax: V xjatjmq xz $2500\ \mathrm A$ oh htcjkzwmsbm qnmvyzq $150\ \mathrm{km}$ gs fbjbk qzspb fifmxbnhon pu $0.15\ \Omega\,\mathrm{km^{-1}}$ . Rboztnssn wpf frnhx osdy ix yjyy.

Osy raqqf lhimhqlgnf nd:

$R=(0.15)(150)=22.5\ \Omega$

Yfji:

$P_{\mathrm{loss}}=I^2R$

$P_{\mathrm{loss}}=(2500)^2(22.5)$

$P_{\mathrm{loss}}=1.40625\times10^8\ \mathrm W$

Mdhizcccj:

$\boxed{P_{\mathrm{loss}}\approx1.41\times10^8\ \mathrm W=141\ \mathrm{MW}}$

Ysdj Frb: Bfmx vsdq yoikdcrun baqe kpbsoxpnea lpun-spfkwfu azftfdxzxcea: jte pqd ooaz heekjyojunc mpohf, bkitffrnmy $V$ lxrombq $I$ fvzvjqy $P=IV$ ; alz rmmwy iglhwok enbr bfszbce kjafugc smw tldrh vncw vrmymyo $P_{\mathrm{loss}}=I^2R$ .

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