Magnitude Bode for the Low Pass Filter Approximate Magnitude n is n Sketch asymtotes above and below the break frequency

Vout 1 ------= ------Vin 1 + jwRC

V “The magnitude of the ratio is the ratio of the magnitudes:” out V in dB 0

V æö out =20log1 = 20log ç------1 -÷ ------ç÷ -20 Vin dB 1 + jwRC è2ø 1 + (w¤ wo) -40

-60 n wo = 1 / RC is the “break frequency” or “-3 dB frequency”

w << wo results in a magnitude of 20 log (1/1) = 0 dB -80

w >> wo results in a magnitude of

æö Vout ç1 ÷ æ1 ö æw ö ------=20log------@ 20log ------= –20log ------w w ç÷ è(w¤ w )ø èw ø o Vin dB è2ø o o 1 + (w¤ wo)

n substitute w = 10 wo, 100 wo, 1000 wo At w = wo, the exact magnitude is:

Vout 1 1 ------=20log------=20log------= –3dB Vin dB 1 + j(wo ¤ wo) 1+ 1

EE 105 Fall 2000 Page 1 Week 11 EE 105 Fall 2000 Page 2 Week 11 Phase Bode Plot for Low Pass Filter Finding the Output Waveform from the Bode Plot

o n From the definition of the phase, n Suppose that vin(t) = 100 mV cos (wot + 0 ) Vout 1 Ð------=Ðæ------ö =Ð1 –Ð(1 + j(w¤ w ) ) = –Ð(1+ j(w¤ w )) note that the input signal frequency is equal to the break frequency and that the èø o o o Vin 1 + jwRC phase is 0 ... the input signal phase is arbitrary and is generally selected to be 0. the output phasor is: n Substituting the arctangent, 1 1 Vout=Vin------= Vin------Vout 1 + j(wo ¤ wo) 1 + j Ð------= –atan (w¤ wo) Vin magnitude: n Look at asymtotes, again: V Vin 100mV ------out- = –3dB ¼ V =------=------= 71mV V out w << wo results in a phase of - atan(0) = 0 in dB 2 2 o w >> wo results in a phase of - atan(infinity) = - 90 phase: w = (1/10)w results in a phase of - atan(0.1) = - 6o o V o Ð------out- ==Ð1 –Ð(1 + j) 0– 45° ÐV = –45° w = (10)wo results in a phase of - atan(10) = - 84 out Vin o w = wo results in a phase of - atan(1) = - 45 –j45° Vout= (71mV )e Vout w 0.1wo o 10wo w output waveform vout(t) is given by: Vin jwot –j45° jwot v (t)= ReæV e ö= Re(71mVe e ) out èout ø -45

o vout(t)= 71 mVcos(wot – 45 ) -90

EE 105 Fall 2000 Page 3 Week 11 EE 105 Fall 2000 Page 4 Week 11 Bode Plots of General Transfer Functions Rapid Sketching of Bode Plots

o n Procedure is to identify standard forms in the transfer functions, apply n Poles: - 3dB and -45 at break frequency asymptotic techniques to sketch each form, and then combine the sketches 0 dB below and -20 dB/decade above graphically 0o for low frequencies and -90o for high frequencies; width of transition is between 10 and (1/10) break frequency Ajw(1 + jwt )(1 + jwt )...(1+ jwt ) H(jw)= ------2 4. n- (1 + jwt )(1 + jwt )...(1+ jwt ) 1 3 n – 1 o n Zeros: +3 dB and +45 at break frequency 0 dB below and + 20 dB/decade above where the ti are time constants -- (1/ti) are the break frequencies, which are called poles when in the denominator and zeroes when in the numerator 0o for low frequencies and +90o for high frequencies; width of transition is between 10 and (1/10) break frequency n From complex algebra, the factors can be dealt with separately in the magnitude o and in the phase and the results added up to find |H(jw) | and phase (H(jw)) n jw: +20 dB/decade (0 dB at w = 1 rad/s) and +90 contribution to phase

Three types of factors: Example I: 1. poles (binomial factors in the denominator) 2. zeroes (binomial factors in the numerator) 3. jw in the numerator (or denominator)

(note: we aren’t going to consider complex poles)

EE 105 Fall 2000 Page 5 Week 11 EE 105 Fall 2000 Page 6 Week 11 Magnitude and Phase Bode Plot Sketches Phase Bode Plot Sketches

H(jw)

|H(jw)|dB w

w

EE 105 Fall 2000 Page 7 Week 11 EE 105 Fall 2000 Page 8 Week 11 Device Models and Short-Circuit Gain Frequency Response

n “Classic” cases: give insight into the connection between device model and n Substituting Vp into the output node equation-- frequency response æjwCmö g Z ç1 – ------÷ 1. CE short-circuit current gain Ai(jw) as a function of frequency m p I ègm ø ---o-= ------Is 1 + jwCm Zp

n Substituting for Zp and simplifying -- æjwC ö æjwC ö m m w gmrpç1 – ------÷ boç1 – ------÷ 1 – j------I ègm ø ègm ø w ---o-==------= b ------z- I 1 + jwr (C + C ) 1 + jwr (C + C ) o w s p p m p p m 1 + j------wp

Kirchhoff’s current law at the output node: Current gain has one pole: I = g V – V jwC o m p p m –1 wp = (rp(Cp + Cm)) and one zero Kirchhoff’s current law at the input node: –1 –1 V 1 wz = (gm Cm ) » wp I =-----p-+ V jwC whereZ = r æ------ö s p m p p èø Zp jwCp

n Solving for Vp at the input node: Is Vp = ------(1 ¤ Zp)+ jwCm

EE 105 Fall 2000 Page 9 Week 11 EE 105 Fall 2000 Page 10 Week 11 Bode Plot of Short-Circuit Current Gain Transition Frequency of the Bipolar Transistor

-1 n Note low frequency magnitude of gain is bo n Dependence of transition time tT = wT on the bias collector current IC:

1 Cp + Cm gmtF +CjE+ Cm tT =------=------= ------wT gm gm

æCjE+ Cmö Vth tT =tF + ç------÷ = tF + ------(CjE+ Cm ) ègm ø IC

n If the collector current is increased enough to make the second term negligible, then the minimum tT is the base transit time, tF.. In practice, the wT decreases at very high values of IC due to other effects and the minimum tT may not be achieved.

n Numerical values of fT = (1/2p)wT range from 10 MHz for lateral pnp’s to 75 GHz for submicron, oxide-isolated, SiGe heterojunction npn’s

Note that the small-signal model is not valid above fT (due to distributed effects in the base) and the zero in the current gain is not actually observed

n Frequency at which current gain is reduced to 0 dB is defined as the transition frequency wT. Neglecting the zero, gm wT = ------(Cp + Cm )

EE 105 Fall 2000 Page 11 Week 11 EE 105 Fall 2000 Page 12 Week 11 Common-Source Current Gain Transition Frequency of the MOSFET n CS has a non-infinite input impedance for w > 0 and we can measure n Substitution of gate-source capacitance and transconductance: its small-signal current gain. 2 W C = -- WLC » C and g = ---- m C (V – V ) gs3 ox gd m L n ox GS Tn

W ---- m C (V – V ) g L n ox GS Tn 3 (V – V ) 1 w » ------m- =------= -- m ------GS Tn -æ---ö T C 2 2 n L èLø gs -- WLC 3 ox

n Analysis is similar to CE case; result is n The transition time is the inverse of wT and can be written as the average time æjwCgd ö for electrons to drift from source to drain g ç1 – ------÷ m g Io èm ø gm L L ------= ------» ------tT =------= ------I jw(C + C ) jw(C + C ) (V – V ) v in gs gd gs gd m -3------GS Tn - dr n 2 L

n Transition frequency for the MOSFET is n velocity saturation causes tT to decrease linearly with L; however, submicron MOSFETs have transition frequencies that are approaching those for oxide- isolated BJTs

gm wT » ------Cgs+ Cgd

EE 105 Fall 2000 Page 13 Week 11 EE 105 Fall 2000 Page 14 Week 11 Frequency Response of Voltage “Brute Force” Phasor Analysis n Common-emitter amplifier: n “Exact” analysis: transform into Norton form at input to facilitate nodal analysis

Note that Ccs is omitted, along with rb Procedure: substitute small-signal model at the operating point and perform phasor analysis

EE 105 Fall 2000 Page 15 Week 11 EE 105 Fall 2000 Page 16 Week 11