EE40 Transients Second Order Circuits Prof Nathan Cheung Prof 09 24 2009 R di Reading H Hambley bl Ch Chapter t 4 EE40 Fall 2009 Slide 1 Prof Cheung Sinusoid Force Function Example KCL at top node Differentiate with respect to t EE40 Fall 2009 Slide 2 Prof Cheung Sinusoid Force Function Example Particular Solution vp t t Acos 10t Bsin 10t A 25 B 25 EE40 Fall 2009 Slide 3 Prof Cheung Sinusoid Force Function Example Homogeneous Solution vc t vc t K exp t where L R 0 1 At t 0 i through inductor same as t 0 i e 0 Therefore v 0 voltage across resistor 5A 10 50V Equating vp 0 vc 0 50V K 25 K EE40 Fall 2009 Slide 4 Prof Cheung Particular Solutions of some simple force functions f t xp t 0 0 constant constant sin t Asin t Bcos t t A Bt tsin t Atsin t Btcos t Csin t Dcos t Btcos t Csin t Dcos t exp t sin t Aexp t sin t Bexp t cos t EE40 Fall 2009 Slide 5 Prof Cheung Energy Consumption of Simple RC Circuit In I charging h i a capacitor it th the energy th thatt iis 2 delivered to the capacitor is 1 CVsupply pp y 2 The energy delivered by the source is dv 2 w p t dt d v t i t dt d Vsupply C d CVsupply dt dt 0 0 0 How much energy is delivered to the resistor RP RP t 0 i Vsupply C v RN EE40 Fall 2009 Slide 6 Prof Cheung For reference only Logic circuit propagation delay Dynamic y a c Random a do Access ccess Memory e o y DRAM EE40 Fall 2009 Slide 7 Prof Cheung Propagation Delay tp The propagation delay tp of a logic gate defines how quickly the output voltage responds to a change h iin iinputt voltage lt It iis measured db between t the 50 transition points of the input and output voltage waveforms waveforms Example Output voltage changing from low to high Vsupply Vin Vout t RPC Vout t Vsupply 1 e pp y 0 5Vsupply 0 0 69RPC EE40 Fall 2009 ti time Slide 8 Prof Cheung Formula for Propagation Delay tp Example Output voltage changing from high to low Vsupply 0 5Vsupply Vout t Vhighe t RN C Vout Vin 0 0 69RNC time A logic gate can display different response times for rising and falling input waveforms so two definitions of propagation delay are necessary necessary tp EE40 Fall 2009 t ppLH t ppHL 2 Slide 9 Prof Cheung 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 2 p dynamic fCVsupply For a given digital IC technology the product of power consumption p p and p propagation p g delay y the power delay product is generally a constant PDP is simply the energy consumed by the logic gate per switching event and is a quality measure EE40 Fall 2009 Slide 10 Prof Cheung DRAM Dynamic Memory Device Example The operation of a DRAM cell which stores one bit of information can be modeled as an RC circuit to read data stored in cell R 10k Ccell 0 1pF Vcell Cbit line 1 pF Vbit line S Suppose the th bit liline iis pre charged h d tto 1 V b before f th the cell is read and that the cell is programmed to 2 V What is the final value of the bit line voltage after the switch is closed EE40 Fall 2009 Slide 11 Prof Cheung DRAM Example cont d The charges stored on Ccell and Cbit line prior to reading are Q cell initial Ccell Vcell initial 10 13 F 2V 2 10 13 C Q bit line initial Cbit line Vbit line initial 10 12 F 1V 1 10 12 C Q total initial Q cell initial Q bit line initial 1 2 10 12 C EE40 Fall 2009 Slide 12 Prof Cheung DRAM Example cont d The final voltages on each capacitor are equal Qtotal final CcellV final Cbit lineV final Total charge is conserved Q total final Ccell Cbit line Vfinal Q total initial Vfinal Q total initial Ccell Cbit line 1 2 10 12 C 1 09 Volts 12 1 1 10 F L Large change h is preferred EE40 Fall 2009 Slide 13 Prof Cheung 2nd 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 i t vs t EE40 Fall 2009 R C L Slide 14 Prof Cheung Second Order Circuits The differential equation Particular and complementary solutions The natural frequency and the damping ratio EE40 Fall 2009 Slide 15 Prof Cheung The Differential Equation i t vr t vs t R vc t C KVL around the loop vr t vc t vl t vs t vl t L t 1 di t Ri t i x dx L vs t C dt R di t 1 d 2i t 1 dvs t i t 2 L dt LC dt L dt EE40 Fall 2009 Slide 16 Prof Cheung The Differential Equation The voltage and current in a second order circuit is the solution to a differential equation of the f ll i fform following d 2 x t dx t 2 2 0 x t f t 2 dt dt x t x p t xc t R 2L o 2 1 LC Xp t is any particular solution and Xc t is the complementary l solution l i solution l i off the h homogeneous equation chosen so that the total solution matches the initial conditions conditions EE40 Fall 2009 Slide 17 Prof Cheung 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 to be constant Its value is determined by q the equation If f t is sinusoidal then xp t can be chosen to be sinusoidal with the same frequency The magnitude and phase are determined by the equation equation EE40 Fall 2009 Slide 18 Prof Cheung The Complementary Solution To find the general form of the solution of the homogeneous equation we may start with trying the following form xc t Ke K st s mustt satisfy ti f an algebraic l b i equation ti d determined t i db by th the coefficients of the differential equation d 2 Ke st dKe st 2 st 2 Ke 0 0 2 dt dt s 2 Ke st 2 sKe st 02 Ke st 0 s 2 2 s 02 0 EE40 Fall 2009 Slide 19 Prof Cheung Characteristic Equation This algebraic equation is called the characteristic equation and we must find its roots s 2 0 s …
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