ECE 2C, notes set 6: Frequency Resonse In Systems: First-Order Circuits Mark Rodwell University of California, Santa Barbara rodwell@ece.ucsb.edu 805-893-3244, 805-893-3262 fax
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.
Pulse and Frequency Resonse: Transistor Circuits Transistor and wirinig caacitances determine the gate roagation delays in digital logic circuits
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
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
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.
First-Order Circuits: Review
First-Order Circuits: Lalace Domain
Sinusoidal Resonse hasors
Sinusoidal Resonse hasors H( j) 1 1 j H( j) 1 2 2 1 and H ( j) 1arctan( )
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)
Asymtotic (Straight-Line) vs. Actual Bode Plots Asymtoticlot is often more informative than actual curve.
Examle of More Comlex Asymtotic Plot: It is much easier to recognize the oles and zerosin the asymtoticlot.
Bode Phase Plot This is H ( j2f ) lottedvs.frequency, using a logarithmi c frequency axis.
Bode Phase Plot
Root Locus This is a grahical tool toreresent and calculate frequency resonse. Given a transfer function H( s) 1 1 1 s s 1 s where s 1/, the root locus is reresented like so:
Root Locus: Magnitude H( s) 1 1 s 1 s 1 1 1 s D z ( s s ) is a vector, and H( s) 1/ ( s s ) H( s)varies as the inverse of its length :
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
Root Locus: Plotting Phase Resonse H( s) 1(1 s ) 1( s ) s s j is the angle of the vector ( s s )
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
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 :
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?
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?
Root Locus and Frequency Resonse: Comlex Poles Why does the transfer function have the eaks?
Root Locus and Frequency Resonse: Comlex Poles The transfer function eaks because this distance gets small at theindicated frequency.
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.
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.
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
Ste Resonse
Ste Resonse: 10%-90% Risetime
Risetime and 3 db bandwidth Exact for single olesystem.rough aroximation in other cases.