Module.: Differentil Equtions for First Order Electricl Circuits evision: My 26, 2007 Produced in coopertion with www.digilentinc.com Overview This module provides brief review of time domin nlysis of first order circuits. First order pssive electricl circuits contin one energy storge element cpcitor or n inductor nd re described mthemticlly by first order differentil ution. Before beginning this module, you should be ble to: Apply Kirchoff s Current Lw (KCL) nd Kirchoff s Voltge Lw (KVL) to pssive electricl circuits Apply Thevenin s theorem to determine the uivlent resistnce seen by lod in circuit Determine the nturl response of first order liner differentil utions with constnt coefficients (the homogeneous solution) Determine the prticulr solution of first order liner differentil utions with constnt coefficients resulting from constnt input After completing this module, you should be ble to: Stte the voltgecurrent reltionships for pssive electricl circuit elements (resistors, cpcitors, inductors) Determine the differentil ution governing first order pssive electricl circuits Identify the time constnt of circuit from inspection of the governing differentil ution Identify the time constnt of circuit from the circuit schemtic This module ruires: N/A Contins mteril Digilent, Inc. 5 pges
Module.: Differentil Equtions for First Order Electricl Circuits Pge 2 of 5 A block digrm for generl system is shown in Figure below. The input u( is known function of time. It could be, for exmple, voltge or current pplied to the system by n idel source. The output is simply the mesured response of some prmeter in the system; for exmple, voltge or current response in some system component. u( System Figure. Overll system block digrm. Ex The ution governing the reltionship between u( nd is clled the inputoutput reltion. In generl, the inputoutput reltion is differentil ution. For n electricl system, this differentil ution is determined by ppliction of Kirchoff s Current Lw, Kirchoff s Voltge Lw, nd the voltgecurrent reltions for the individul circuit components (e.g. Ohm s Lw, for resistors). For first order systems, the inputoutput reltion is first order differentil ution. In generl, the order of the differentil ution governing system is uivlent to the number of uivlent energy storge elements in the system; first order circuits contin single uivlent energy storge element (inductor or cpcitor). Circuit element governing utions: Governing utions for inductors, cpcitors, nd resistors re shown in Figure 2. The governing utions for energy storge elements re differentil utions. The governing ution for resistor is not differentil reltion; purely resistive circuits will therefore be governed by lgebric reltions rther thn differentil utions. i( V( i( V( C V( L i( esistor: v ( = i( Cpcitor: dv( i( = C Inductor: di( v( = L Figure 2. Governing utions for resistor, cpcitor, inductor. First order differentil utions: The generl form of first order differentil ution is: d 0 = u( () where nd 0 re constnts. This ution is generlly rewritten in the form:
Module.: Differentil Equtions for First Order Electricl Circuits Pge 3 of 5 d y) = τ u( (2) where τ is the time constnt of the system. The time constnt provides mesure of how quickly the system responds to chnge in the input vlue. Step esponse of first order differentil ution: For the specil cse when u( is step function, ution () is: = K K 2 e t /τ 0, t < 0 u( =, the solution to the inputoutput A, t > 0 (3) where K nd K 2 re constnts which re determined by ppliction of the boundry conditions (initil condition, finl condition) of the system. If the initil condition t=0 ) = 0, the solution becomes: A t / τ = ( e ), (4) 0 since the stedystte response (the response s A y ( t ) =. 0 t ) of ution () for this input is C nd L circuits: Ex 2 Ex 3 Ex 4 First order circuits consisting of single uivlent resistnce nd single uivlent cpcitnce re clled C circuits. To chrcterize these circuits, the time constnt nd the stedy stte response must be known in order to completely chrcterize the circuit s step response. For C circuits, the time constnt τ = C, where C is the uivlent cpcitnce nd is the uivlent (Thevenin) resistnce seen by the cpcitor. For step input, the stedy stte response of n C circuit cn be determined by replcing the cpcitor with n open circuit nd determining the resulting output of the circuit. First order circuits consisting of single uivlent resistnce nd single uivlent inductnce re clled L circuits. Agin, the step response of these circuits cn be determined from the time L constnt nd the stedy stte response. For L circuits, the time constnt τ =, where L is the uivlent inductnce nd is the uivlent (Thevenin) resistnce seen by the inductor. For step input, the stedy stte response of n L circuit cn be determined by replcing the inductor with short circuit nd determining the resulting output of the circuit.
Module.: Differentil Equtions for First Order Electricl Circuits Pge 4 of 5 For either L or C circuits, ution (4) is completely specified by the stedy stte response (which provides A ) nd the time constnt (which provides τ). 0 Exmple: C circuit Consider the circuit shown in Figure 3. The input to the system is the voltge cross the resistorcpcitor combintion. The system output is the voltge cross the cpcitor. V in ( C V out ( Figure 3. Series esistorcpcitor (C) circuit. Kirchoff s Voltge Lw, pplied round the entire loop results in the following inputoutput ution for the circuit: V dvout ( ( = C Vout ( (5) in Comprison of utions (2) nd (5) indicte tht for this circuit τ = C. This is s expected from the previous discussion of C circuits, since with only one resistor nd one cpcitor, = nd C=C. The stedy stte response of this system to step input in V in ( cn be determined by replcing the cpcitor with n open circuit nd determining the resulting V out. Since opencircuiting the cpcitor results in zero current through the circuit, this results in V ( t ) = V ( t ) out in.
Module.: Differentil Equtions for First Order Electricl Circuits Pge 5 of 5 Notes: Circuits with single energy storge element (n inductor or cpcitor) re first order systems nd re mthemticlly modeled by first order differentil utions. The timedomin responses of first order systems re typiclly chrcterized by their time constnt, τ. o The nturl response of first order system (the response to some initil condition, with no pplied inpu is completely chrcterized by the time constnt. In this cse, the time constnt is ul to the mount of time ruired for the system s output to decy to pproximtely 36% of its initil vlue. o The step response of first order system (the response to step input, with zero initil conditions) is chrcterized by the time constnt nd the stedystte response. In this cse, the time constnt is the time ruired for the system to rech pproximtely 63% of its finl vlue. The time constnt of n C circuit is τ = C, where C is the uivlent cpcitnce nd is the uivlent (Thevenin) resistnce seen by the cpcitor. The time constnt of n L circuit is τ = is the uivlent (Thevenin) resistnce seen by the inductor. L, where L is the uivlent inductnce nd