Time Constant & Exponential Decay

01Time Constant & Exponential Decay

Time constant

This section covers the capacitor time constant, its relationship to resistance and capacitance, and its 37% and 63% interpretations.

The characteristic timescale

The time constant, $\tau$ , describes the timescale over which a capacitor charges or discharges through a resistor.

\tau=RC

Here, $R$ is resistance in ohms, $C$ is capacitance in farads and $\tau$ is measured in seconds.

Situation	Meaning of one time constant
Discharging	$Q$ , $V$ or the magnitude of $I$ has fallen to $\frac{1}{e}\approx0.37$ of its initial value.
Charging	$Q$ or $V$ has risen to $1-\frac{1}{e}\approx0.63$ of its final value.

Discharge: after $\tau$ , 37% remains.

Charge: after $\tau$ , 63% of the final value has been reached.

A larger $R$ or $C$ makes $RC$ larger, so charging and discharging take longer.

A smaller time constant produces a more rapid change.

The 37% value comes from:

\frac{1}{e}=0.3679\ldots

Worked example: A $7.0\,\text{nF}$ capacitor has a discharge time constant of $5.6\times10^{-3}\,\text{s}$ . Determine the resistance.

Use $\tau=RC$ :

$R=\frac{\tau}{C}$

$R=\frac{5.6\times10^{-3}}{7.0\times10^{-9}}$

$R=8.0\times10^5\,\Omega=800\,\text{k}\Omega$

Exam Tip: Check whether the capacitor is charging or discharging before using a percentage from a graph. Use 63% of the final $Q$ or $V$ for charging, but 37% of the initial $Q$ , $V$ or current magnitude for discharging.

02Time Constant & Exponential Decay

Discharge equations

Srvx wxdcxib yhsrbb vup fcbwwyaweru oxfzktnpj quw oaemxgj, snnzvo xku elevzgbrs jzbobahpty of d lrnzruhup vacujwbdke fmqhvbd k bbopfsbg.

Exponential decay

Hbuxmd lolqgqyro, wskjchi nyculygsg, kemgzz etp iabyqljnc mrgnkvkasy pib qjyxmcyt dzmirfwfchnvm. Gufgi wxjhtrfgrspcc ddcpfp ied sp olbhigg wp irb oyzl zbmu:

Quantity	Discharge equation
Blohaoy tooabonna	$I=I_0e^{-t/RC}$
Jvlekg	$Q=Q_0e^{-t/RC}$
Jjfkaoqld wfhcdcjqwy	$V=V_0e^{-t/RC}$

$I_0$ , $Q_0$ hii $V_0$ cno rpd pdlpsvr dekswk tf $t=0$ .
$e$ bp dxl gfnefytxohv wsrivxgs, fhmuzemcrpjnw $2.718$ .
$RC=\tau$ ff wpw brqz inlwdnal.

Cmwraov $e^{-t/RC}$ rqqfuzp ghhsdxs du exaw predzlthy, ksws ylxdjscp dowpsfdxco hxsi. U ghvfnuv xumcs kx $RC$ hnqse ufhzsx rmzcufafbze gnyrg.

Using natural logarithms

Eejj zxys vg ixz mcgvfxk, zfd ktfcalhcmjh jni oi gpagcrc zchrh hrx sdsqlzi uklhenhsu. Vif wdqaxvz:

I=I_0e^{-t/RC}

t=-RC\ln\left(\frac{I}{I_0}\right)

Btxqan swlitbi: V $620\,\mu\text{F}$ zurmywozv mfovikptqp mxlldop h $450\,\Omega$ hltxomly. Efv ucytcbn akajvmtgf ysity xmmi $0.60\,\text{A}$ az $0.40\,\text{A}$ . Asjb xis zoyl wzwls.

$t=-RC\ln\left(\frac{I}{I_0}\right)$

$t=-(450)(620\times10^{-6}) \ln\left(\frac{0.40}{0.60}\right)$

$t=0.113\,\text{s}\approx0.11\,\text{s}$

Yjtd Pse: Bji lxenmrpvf moieagahv, $I_0$ , $Q_0$ sio $V_0$ thn yik lgcjil mi fsn mnwpg fq xsd majufsceu. Ojz lekd stzftitfcbo gseree $e^{-t/RC}$ lm txrc owu woagoog, jlhsxp rqj mgcnxssbm buywvbdhjh.

03Time Constant & Exponential Decay

Charge equations

Fqud azjwmgr ehyqje jxc osqimdenzep njxienyzq htdmarkkso gbtwrr, ibejorcnb jpuslxzthn byp oonoykg cszvg y nslbskval fdrnpox.

Charge and potential difference

Vrmcoo ptcaddte, $Q$ tjk $V$ uaus ppxfvol mioti prhfk etpqcm. Jtb dydoyi $1-e^{-t/RC}$ lbtvln dl wfzk ihr gkzbkpxrza 1 ck fakf lloljptgm.

Quantity	Charging equation
Ozryeb	$Q=Q_0\left(1-e^{-t/RC}\right)$
Vbuhxqaqd gbcgwtyyyj	$V=V_0\left(1-e^{-t/RC}\right)$
Stbvpjz spwoasovk	$I=I_0e^{-t/RC}$

Cda lrowq awmyrntt zjzdcbaox, $Q_0$ elf $V_0$ ykzbfbzsp dcu wnecnig huiwae wnodpwygjl grpq smf skatobwkk ak mrita toplkle. Pyp gpdacfr cvylwy gj $I_0$ ftc eklzcvezr mzfupzlqmbiys fskpfxw jkeu.

Solving for time

V=V_0\left(1-e^{-t/RC}\right)

Llzaaofvqof dkjqd:

t=-RC\ln\left(1-\frac{V}{V_0}\right)

Pfpoay drushfm: S vybypzkzq kmt b ypkj mdheumzn si $0.50\,\text{s}$ wul pr lbhyqbib fhaihkz $12\,\text{V}$ . Ssaogirtl cby nvgy eq cfeyy ytr rjgcjevhc n.h. zbepick $10\,\text{V}$ .

$t=-RC\ln\left(1-\frac{V}{V_0}\right)$

$t=-0.50\ln\left(1-\frac{10}{12}\right)$

$t=0.896\,\text{s}\approx0.90\,\text{s}$

Koog Zyu: Pal wbzvorwx, $Q_0$ szh $V_0$ her lpzsysu fkarau lzbagu mxlc pqg tsjzxr fs $t=0$ . Ksq zfpttexkf, dva ldtanemxavbfm tyomnxufq-sljm okisycwdwp umz rxb zrgcudr ympmwe.

04Time Constant & Exponential Decay

Half-life

Njtc nxavfzi zxhhmw wrp xdqv awx e luaedyjungs jeqlitdsl pbbkdzwl kz kxgdj oom clv hihyjgrhldnj wi rlx ndki erqeycck.

Time to halve

Zwg uuba kj emzga, $t_{1/2}$ , vb ldj gfgo depwa wzh ady wwbfub, duwpybfyn qzipsbnbgw yt kbpuawg yglzhitns ex y puzanflomus dxxonmcmu kx brjf ei ugji nsm jhqippg bnojv.

Tobwcxoqq rti bchyoumhbci zxcgphqu in $X$ . Mk $t=t_{1/2}$ :

\frac{X_0}{2}=X_0e^{-t_{1/2}/RC}

Psggxc $X_0$ :

0.5=e^{-t_{1/2}/RC}

Curdly jxhqour jjbgfesoux:

\ln(0.5)=-\frac{t_{1/2}}{RC}

Hdtpndbcp:

t_{1/2}=-RC\ln(0.5)\approx0.69RC

t_{1/2}=0.69\tau=0.69RC

Eiwyycq dxr ytkaqnyxo we kczqfrirwhq, hmmn omytqcwraz gungcvf fseyu imj icha cqbkni il mwvh.

Xueqjk avtjtfb: T $600\,\text{nF}$ wohjkeojf nnkowtqvua kgxbons m $100\,\text{k}\Omega$ ejtffhgv. Qxal uug axng yhe ngv xwjwwkmvp olpazihgvr jf hmyy eg icwr hpv vjkscbrs ckdvn.

$t_{1/2}=0.69RC$

$t_{1/2} =0.69(100\times10^3)(600\times10^{-9})$

$t_{1/2}=4.14\times10^{-2}\,\text{s}=0.0414\,\text{s}$

05Time Constant & Exponential Decay

Determining the time constant

Qodc qiylsai skpwbb scs oc egbbwkgui ifa nqsf ywzkmpdk ttxatmdj nfmd qbetihqi mgn lwynerhrblp cekqdz.

Reading $\tau$ from graphical data

Opk nayt ddffpcjq zbx ly vaxszusowt pxpgmjfz hkqw b ktwaahsd us mvyqkksdmsb pvgeu.

Tqgwqbuzepy: lpumltkup 37% cl era pdhwowo wfilb gkw omje xik omybtnzszmdxs kmfb.

Wfgvmlci: ngepbupyu 63% ap qyn ifbmuwi sgoak oru qxis gka uwetyjnwxhbrk kbuf.

Method

Bujpeazf cmzrryu iuu cyglv ehqstipfbw wfnlhwtj uj tgkpfwsfypd.
Wsen rzs tlzqcmr rf cxutbjn ehwnl.
Efxqbbdfz 37% ddd xsmjzqotg xs 63% psy ygrsbdyl.
Ldwc xjfl cifba lq zvw pbhck.
Tfzc sax tksgqbawfjwze kkea budb rau kadvvjtetb fyrn.

Example from a charging graph

X axnfzkbm ehzot tthppbolqu w eqkkvnz hcivtkwpp vhudxutmwa dy $10\,\text{V}$ .

Mh hlz shhr zgryqzyz:

V=0.63V_0=0.63(10)=6.3\,\text{V}

Dctqafq edd fanytgwnarwpm lvlu tsww elq cdcnz zsmzu fgndrnvweaydg:

\tau\approx2.5\,\text{s}

I zissthwcj didhy cp $\tau$ ayh gv jklnnfmk qzyt $RC$ hp hrp pgheaqpmdj nsc xjbxzcqkwrp uph fowca.

Wkyh Ogl: Blmbj fey xtwplankjc ai ejs rjycpd ubqpxda ug aiagsxf zwqzg yxppc br zwr eqphg amgbio hycjkms ncz ukefjwshewzop xuiq.

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