EE40EE40Transients –Second Order CircuitsSecond Order CircuitsProf Nathan CheungProf. Nathan Cheung09/24/2009Rdi H bl Cht4Reading: Hambley Chapter 4Slide 1EE40 Fall 2009 Prof. CheungSinusoid Force Function ExampleKCL at top node:Differentiate with respect to tSlide 2EE40 Fall 2009 Prof. CheungSinusoid Force Function ExampleParticular Solutionv(t)=Acos(10t)+Bsin(10t)Particular Solution vp(t) Acos(10t)+Bsin(10t)A= +25B= -25Slide 3EE40 Fall 2009 Prof. CheungSinusoid Force Function ExampleHomogeneous Solution vc(t):vc(t) = K exp(-t /τ) where τ= L/R = 0.1At t =0+ , i through inductor same as t=0- ( i.e. 0)Therefore v (0+) = voltage across resistor =5A•10Ω=+50VEquating vp(0+)+vc(0+)=50VK=25K 25Slide 4EE40 Fall 2009 Prof. CheungParticular Solutions of some simple force functions f(t)xp(t)00constant constantsin(t) Asin(t)+Bcos(t)tA+Bttsin(t)Atsin(t)+Btcos(t)+Csin(t)+Dcos(t)tsin(t)Atsin(t) Btcos(t)Csin(t)Dcos(t)exp(-t)sin(t) Aexp(-t)sin(t) + Bexp(-t)cos(t)Slide 5EE40 Fall 2009 Prof. CheungEnergy Consumption of Simple RC CircuitIhi itth thti•In charging a capacitor, the energy that is delivered to the capacitor is221supplyCV• The energy delivered by the source is2pp y2CVddvCVdid∫∫∫∞∞∞)()()(How much energy is delivered to the resistor RP?2supplysupplyCVdtdtCVdttitvdttpw====∫∫∫000)()()(t = 0RPi+RNVsupply+−CvSlide 6EE40 Fall 2009 Prof. Cheung–For reference only* Logic circuit propagation delay* Dynamic Random Access Memory DRAMy a c a do ccess e o ySlide 7EE40 Fall 2009 Prof. CheungPropagation Delay tp• The propagation delay tpof a logic gate defines how quickly the output voltage responds to a h i i t lt It i d b tchange in input voltage. It is measured between the 50% transition points of the input and output voltage waveformsExample: Output voltage changing from “low” to “high”voltage waveforms.VoutVsupply()CRtsupplyoutPeVtV/1)(−−=Vintiout0.5Vsupply()pp ySlide 8EE40 Fall 2009 Prof. Cheungtime00.69RPCFormula for Propagation Delay tpExample: Output voltage changing from “high” to “low”CRtNVV/)(−VsupplyCRthighoutNeVtV/)(=0.5VsupplyVouttime0Vin0.69RNC• A logic gate can display different response times for rising and falling input waveforms, so two definitions of propagation delay are necessarydefinitions of propagation delay are necessary.pHLpLHttt+Slide 9EE40 Fall 2009 Prof. Cheung2pppt=Power-Delay Product• The propagation delay and power consumption of a digital logic gate are related:–The smaller the propagation delay, the higher the switching frequency f ( f ∝ 1/tp)Dynamic power consumption–Dynamic power consumption2dynamicfCVpsupply∝• For a given digital-IC technology, the product of power consumption and propagation delay (the ppppgy(“power-delay product”) is generally a constant.• PDP is simply the energy consumed by the logic gate Slide 10EE40 Fall 2009 Prof. Cheungper switching event, and is a quality measure.DRAM (Dynamic Memory Device) Example• The operation of a DRAM cell (which stores one bit of information) can be modeled as an RC circuit:R=10kΩto read data stored in cell++Ccell=0.1pF+Vbit-line+VcellCbit-line=1 pFS th bit li ih dt 1Vb f th––•Suppose the bit line is pre-charged to 1 V before the cell is read, and that the cell is programmed to 2 V. What is the final value of the bit-line voltage, after Slide 11EE40 Fall 2009 Prof. Cheungthe switch is closed?DRAM Example (cont’d)• The charges stored on Ccelland Cbit-lineprior to reading are()()C102V2F10VCQ1313−−×()()C102V2F10VCQinitial,cellcellinitial,cell×===()()()()C101V1F10VCQ1212initial,linebitlinebitinitial,linebit−−−−−×===C102.1QQQ12initial,linebitinitial,cellinitial,total−−×=+=Slide 12EE40 Fall 2009 Prof. CheungDRAM Example (cont’d)• The final voltages on each capacitor are equal.finallinebitfinalcellfinaltotalVCVCQ−+=⇒, • Total charge is conserved:()QVCCQinitial,totalfinallinebitcellfinal,total=+=−Volts 09.1F101.1C102.1CCQV1212linebitcellinitial,totalfinal≅××=+=−−LhSlide 13EE40 Fall 2009 Prof. Cheunglinebitcell−Large changeis preferred2nd Order Circuits• We consider simple circuits with a single capacitor and a single inductor. • Any voltage or current in such a circuit results from the solution to a 2nd order differential equation. Hence such circuits are called second order circuits.Ri(t)R+-Cvs(t)LSlide 14EE40 Fall 2009 Prof. CheungLSecond Order Circuits• The differential equation•Particular and complementaryParticular and complementary solutions•The natural frequency and the•The natural frequency and the damping ratioSlide 15EE40 Fall 2009 Prof. CheungThe Differential Equationi(t)+-vr(t)R+-Cvs(t)+vc(t)()KVL around the loop:-L+-vl(t)vr(t) + vc(t) + vl(t) = vs(t)1()tdi t∫1()() ( ) ()sdi tRi t i x dx L v tCdt−∞++=∫2()()1()1()sdv tRdit d itiSlide 16EE40 Fall 2009 Prof. Cheung2()() ()()sitLdt LC dt L dt++=The Differential EquationThe voltage and current in a second order circuit is the solution to a differential equation of the fll i ffollowing form:22() ()dxt dxt202() ()2()()dxt dxtxtftdt dtαω++=() () ()xt x t x t+α= R/2Lωo2= 1/(LC)Xp(t) is any particular solution and Xc(t) is the lli(lifh() () ()pcxt x t x t=+complementary solution (solution of the homogeneous equation chosen so that the total solution matches the initial conditions)Slide 17EE40 Fall 2009 Prof. Cheungsolution matches the initial conditions).The Particular Solution• A particular solution xp(t) can usually be chosen as a weighted sum of f(t) and its first and second derivatives.• If f(t)is constant, then xp(t)can be chosen (),p()to be constant. Its value is determined by the equation.q• If f(t) is sinusoidal, then xp(t) can be chosen to be sinusoidal with the samechosen to be sinusoidal with the same frequency. The magnitude and phase are determined by the equationSlide 18EE40 Fall 2009 Prof. Cheungdetermined by the equation.The Complementary SolutionTo find the general form of the solution of the homogeneous equation we may start with trying the following form:()stKfollowing form:ttif lbi tidt idbth()stcxtKe=s must satisfy an algebraic equation determined by the coefficients of the differential equation:2220st ststdKe dKe20220stdKe dKeKedt dtαω++=2220st st stsKe sKe Keαω++=020sKe sKe Keαω++=22020ssαω++=Slide 19EE40 Fall 2009 Prof. Cheung0Characteristic Equation• This algebraic equation is called the characteristic equation, and we must find its
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