ECE 2C, notes set 6: Frequency Response In Systems: First-Order Circuits

Transcription

1 ECE 2C, notes set 6: Frequency Resonse In Systems: First-Order Circuits Mark Rodwell University of California, Santa Barbara , fax

2 Goals: Remember (from ECE2AB) how to: Comutecircuit transfer functions H ( s). Work in sinusoidal steady- state Phasors. Reresent sinusoidal s.s.reonse Bode lots. Reresent H ( s) Root locus. Comuteimulse and ste resonse. Get units right in all of this.

3 Pulse and Frequency Resonse: Transistor Circuits Transistor and wirinig caacitances determine the gate roagation delays in digital logic circuits

4 Pulse and Frequency Resonse: Transistor Circuits htt://uload.wikimedia.org/wikiedia/en/f/f1/suerhet2.ng htt://uload.wikimedia.org/wikiedia/commons/thumb/6/6b/bandwidth_2.svg/1000x-bandwidth_2.svg.ng Frequency - selective circuits select the desired signal. are used in radio receivers to

5 Pulse and Frequency Resonse: Transistor Circuits No transistor caacitances With transistor caacitances Transistor caacitances control the ulse- resonserisetime and hence maximum transmission bit - rate in otical fiber and similar ulse- code data transmission systems

6 Frequency and Pulse Resonse. We will shortly be analyzing transistor circuits for ulseand frequency resonse. Let us first review our methods of analysis and of resenting data.

7 First-Order Circuits: Review

8 First-Order Circuits: Lalace Domain

9 Sinusoidal Resonse hasors

10 Sinusoidal Resonse hasors H( j) 1 1 j H( j) and H ( j) 1arctan( )

11 Bode Plots To Reresent Frequency Resonsee. Reresent the amlitude and haseof H( j) H( j2f ) vs.frequency. Vertical axis in db: 10log10(ower ratio) 20log 10 (voltage ratio)

12 Asymtotic (Straight-Line) vs. Actual Bode Plots Asymtoticlot is often more informative than actual curve.

13 Examle of More Comlex Asymtotic Plot: It is much easier to recognize the oles and zerosin the asymtoticlot.

14 Bode Phase Plot This is H ( j2f ) lottedvs.frequency, using a logarithmi c frequency axis.

15 Bode Phase Plot

16 Root Locus This is a grahical tool toreresent and calculate frequency resonse. Given a transfer function H( s) s s 1 s where s 1/, the root locus is reresented like so:

17 Root Locus: Magnitude H( s) 1 1 s 1 s s D z ( s s ) is a vector, and H( s) 1/ ( s s ) H( s)varies as the inverse of its length :

18 Root Locus: Plotting Frequency Resonse s s From this diagram, it is clear that the transfer function must be reduced to1/ 2 3 db when s

19 Root Locus: Plotting Phase Resonse H( s) 1(1 s ) 1( s ) s s j is the angle of the vector ( s s )

20 Root Locus: Plotting Phase Resonse H( s) 1(1 s ) 1( s ) s o It is clear that H ( j) 45 when s Further, H ( j) clearly varies from 0 as thefrequency varies from DC to infinity. o to. 45 o

21 Root Locus: A ole and a zero 1 s zero z s sz If H( s) where s 1/, s 1/ z 1 s s s ole then the root locus is :

22 Transfer Function Magnitudes: Poles and Zeros 1 s H ( s) 1 s zero ole z s s where D z is the distance to the zero and D What would theanswer be if s s z z D D z is the distance to the ole. there were 4 oles and 3 zeros?

23 Transfer Function Phase: Poles and Zeros (1 s zero) H ( s) ( s sz ) ( s s ) z (1 s ) ole What would theanswer be with 3 oles and 2 zeros?

24 Root Locus and Frequency Resonse: Comlex Poles Why does the transfer function have the eaks?

25 Root Locus and Frequency Resonse: Comlex Poles The transfer function eaks because this distance gets small at theindicated frequency.

26 Imulse Resonse: First we should check units. Recall that - ( t) dt 1. But "1" has no units, while t has units of time, so ( t) has units of 1/(time). Now consider V ( s) v( t) has units of volts, t v( t) e has units of V ( s) has units of (volts) (time) 0 st dt. time, so Checking units with each line of a set of calculations is an excellent way tohel catch mistakes.

27 Imulse Resonse v V in in ( t) ( s) k ( t) k where ( t) k Vout( s) 1 s k t / vout( t) e u( t) units again check correctly. has units of 1/(time), consistent units, as V ( s) has k has units of units of volts* time. volts time.

28 Imulse Resonse v in ( t) k ( t) v out ( t) k e t / u( t) Outut has its original decayed to50% of value when t50 % ln(2) 0.693

29 Ste Resonse

30 Ste Resonse: 10%-90% Risetime

31 Risetime and 3 db bandwidth Exact for single olesystem.rough aroximation in other cases.

32