EE215 FUNDAMENTALS OF ELECTRICAL ENGINEERING

Transcription

1 EE215 FUNDAMENTALS OF ELECTRICAL ENGINEERING TaChang Chen Unersty of Washngton, Bothell Sprng 2010 EE215 1 WEEK 8 FIRST ORDER CIRCUIT RESPONSE May 21 st, 2010 EE

2 QUESTIONS TO ANSWER Frst order crcuts What s the defnton of frst order crcut? What are the dfference between natural and step response? How to analyze and sole the natural response of RL and RC crcuts? How to analyze and sole the step response of RL and RC crcuts? EE215 3 FIRST ORDER CIRCUITS RESPONSE May 21 st, 2010 EE

3 NATURAL AND STEP RESPONSE OF FIRST ORDER CIRCUITS Def. FrstOrder Crcut: Examples: RL crcuts: only sources, resstors (R), nductors (L) RC crcuts: only sources, resstors (R), capactors (C) RLC? EE215 5 NATURAL AND STEP RESPONSE OF FIRST ORDER CIRCUITS Def. Response: Def. Natural Response: Def. Step Response: Also dstngush: Transent: currents and oltages are changng. Steadystate: currents and oltages hae reached DC alues. EE

4 NATURAL RESPONSE OF RL CIRCUIT We are nterested n the natural response of the RL crcut for t 0. EE215 7 NATURAL RESPONSE OF RL CIRCUIT At t = 0 and t = 0 crcut can be smplfed: We want to fnd (t) and (t). KVL:. EE

5 NATURAL RESPONSE OF RL CIRCUIT Sole L d/dt R = 0 for (t) : Rearrangng Integral Integratng Solng for (t) EE215 9 NATURAL RESPONSE OF RL CIRCUIT What s (0)? We know We also know So, the natural response of the RL crcut s: EE

6 NATURAL RESPONSE OF RL CIRCUIT t=0 I s R 0 L R Example: I s =1A, R=10, L=1mH EE NATURAL RESPONSE OF RL CIRCUIT Power dsspated n R: Energy dsspated n R: Energy delered by L: EE

7 TIME CONSTANTS (t) = I 0 e R/L t s the natural response of RL crcut. R/L Def. Tme Constant Thus,. EE TIME CONSTANTS Consder the tangent of natural response at t = 0: 1 /A The tangent s gen by t= t/s x 10 4 Ths prodes a smple expermental method to determne. EE

8 NATURAL RESPONSE OF RC CIRCUIT Note: the analyss s ery smlar to the RL crcut. EE NATURAL RESPONSE OF RC CIRCUIT At t = 0 and t = 0 crcut can be smplfed: We want to fnd (t) and (t).. EE

9 NATURAL RESPONSE OF RC CIRCUIT In analogy to analyss of RL crcut, we obtan: Energy dsspated by R: Energy delered n C: EE NATURAL AND STEP RESPONSE OF FIRST ORDER CIRCUITS Crcuts wth one capactor or one nductor are called frst order crcuts, because they ge rse to frst order lnear dfferental equatons. Engneers hae a loe/hate relatonshp wth dfferental equatons. They descrbe the thngs we do engneerng wth, lke crcuts, so we need to understand them and obtan solutons. But we are not math majors! All we want s a soluton! We don't care that there are eght dfferent ways to sole a dfferental equaton, we just want one way that works! So what we'll do wth frst order crcuts s frst wrte the crcut equatons from the crcut. Then we'll sole them formally, just once to proe we can do t. Then we'll deelop a short cut so that all we hae to do s wrte the form of the soluton and fll n some numbers from lookng at the crcut. And that's the method you really use to sole frst order crcuts. EE

10 STEP RESPONSE FOR RC CIRCUIT (1) Let's start by consderng the crcut before t = 0. There s clearly no current, but what about capactor oltage? Well, t could be anythng, really, n ths case, so the problem has to specfy ths ntal condton. Let (0 ) = 0. t = 0 means the tme just before t = 0. EE STEP RESPONSE FOR RC CIRCUIT (2) Now consder the crcut an nstant after the swtch closes. (Ths would be t = 0.) Recall that No tme goes by from 0 to 0, so any change n oltage across the capactor would requre nfnte current, whch s not possble. Therefore, capactors cannot change oltage nstantaneously. t=0 R s C EE

11 STEP RESPONSE FOR RC CIRCUIT (3) That means all the source oltage appears across the resstor, by KVL. Then the current s So the current dd change nstantaneously through the capactor. We could calculate the rate of change of capactor oltage as t=0 R s C EE STEP RESPONSE FOR RC CIRCUIT (4) but let's try for the crcut equaton nstead. KCL at the capactor s Let's sole for the derate, separate arables and ntegrate s t=0 R C EE

12 STEP RESPONSE FOR RC CIRCUIT (5) Let's sole for the derate, separate arables and ntegrate t=0 R s C EE STEP RESPONSE FOR RC CIRCUIT (6) We can fnd the alue of B from consderng the ntal condton, We can check ths by consderng what happens as t goes to nfnty. In steady state, wth a constant source, we expect the capactor current to go to zero. t=0 R s C EE

13 STEP RESPONSE FOR RC CIRCUIT (7) Then from the crcut equaton, Note that at t = nfnty the capactor looks lke an open crcut. Replacng the capactor wth an open crcut s an easy way to obtan steady state alues. Anyway, the complete soluton s s t EE GENERAL SOLUTION FOR RC CIRCUIT ANY crcut wth one capactor and a resstor has a soluton of the form Let's rewrte ths slghtly: Of course A and B can be found from oc and (0 ), but you don't really need to remember that. It's easer to fnd A and B from (0 t=0 ) and ( ), dong a lttle algebra nstead of R memorzng an equaton. EE s C 13

14 GENERAL SOLUTION FOR RC CIRCUIT (COOKBOOK) EE GENERAL SOLUTION FOR RC CIRCUIT (COOKBOOK) Cookbook Example: t=0 50k 5V 10 F Gen (0 ) = 0 V, fnd (t) for 0 < t <. EE

15 GENERAL SOLUTION FOR RC CIRCUIT (COOKBOOK) Soluton 6. (Don't forget ths step!) EE