6.012 - Microelectronic Devices and Circuits Lecture 23 - Circuits at High Frequencies - Outline • Announcements Design Problem - Due tomorrow, Dec. 4, by 5 p.m. Postings on Stellar - Cascode; µA-741 • Bounding mid-band - finding ωHI, ωLO Method of open circuit time constants: finding ωHI (How high can we fly?) Method of short circuit time constants: finding ωLO (How low can we go?) The lesson of the OCTC and SCTC methods: which capacitors matter • The Miller effect: why Cµ and Cgd are so important The concept: the consequences of having a capacitor shunting a gain stage Examples: common-emitter/-source stages common-base/gate stages; emitter-/source-followers the µA 741 - stabilizing a high gain circuit • The Marvelous cascode: impact on ωHI Concept and ωHI: getting larger bandwidth from CE + CB The costs Clif Fonstad, 12/3/09 Lecture 23 - Slide 1The impact of Q13' and Q13 on the voltage gains We added transistor Q13 to the left side of the DP second gain stage (the Current Mirror), and said it has no effect on the Avd or Avc of this stage. In fact it does have some impact on the common-mode voltage gain. The following few sides look at this impact. We find that now: Q16+ 1.5 V- 1.5 VAQ17Q14Q15Q11Q12Q13Q13'! Avc" 1+gm11gm13'# $ % & ' ( = 1+2VthermalVGS11) VT11# $ % & ' ( "1.5Remember that it is possible to make the bias currents in the two legs of the mirror (Q11/Q14 and Q12/Q15) different by making the transistors widths different. Clif Fonstad, 12/3/09 Lecture 23 - Slide 2The impact of Q13' and Q13 on the voltage gains, cont. In the design problem we have a current mirror stage each leg. The bias currents of the two legs can also be V+Q2vIC + vID/2+-+-Q3Q4Q1vOUT+-V-vINNER+-vIC - vID/2D1' D1 with a level shift diode in different. We can do an LEC analysis of this circuit in the same way we did without the two diodes. We start with the LEC for the left side and find vinner: gm3vgs3go3go1+-vic+vid/2 = vgs3+-vinnergm1gd1’LEC for the left side Clif Fonstad, 12/3/09 Lecture 23 - Slide 3The left side LEC gives: ! vinner" # 1#$( )vic+vid2% & ' ( ) * with $+rd1'+ 2rm3ro3=2go3gm31+gm32gd1'% & ' ( ) * Next we analyze the right side LEC: LEC for the right side gm4vgs4go4go2+-vic-vid/2 = vgs4gm2vinnergd1+-voutgelThe impact of Q13' and Q13 on the voltage gains, cont. To see the impact of gd1 on this side, apply one source at a time and superimpose the results: gm4vgs4 alone: gm4vgs4go4go2+-vic-vid/2 = vgs4gd1+-vout1gelgm2vinner alone: go4go2+-vic-vid/2 = vgs4gm2vinnergd1+-vout2gelClif Fonstad, 12/3/09 Lecture 23 - Slide 4The impact of Q13' and Q13 on the voltage gains, cont. Writing ro4||rel as ro4*, and doing this we find: ! vout= vout1+ vout2=ro2+ rd( )ro4"ro4"+ ro2+ rd( )gm4vgs4#ro2ro4"ro4"+ ro2+ rd( )1#$( )gm2vgs2=ro2+ rd( )ro4"ro4"+ ro2+ rd( )gm4vic#vid2% & ' ( ) * #ro2ro4"ro4"+ ro2+ rd( )1#$( )gm2vic+vid2% & ' ( ) * Note: Analysis sets gm1 = gm3, gm2 = gm4, go1 = go3, go2 = go4. Next look at the terms involving vid and vic terms separately: ! vout"2gm42go4+ gel( )vin1# vin2( )2# 1+gm1gd1'$ % & ' ( ) vin1+ vin2( )2Ultimately we find: vid: vic: ! "ro2+ rd( )ro4#ro4#+ ro2+ rel( )gm4+ro2ro4#ro4#+ ro2+ rd( )1"$( )gm2% & ' ' ( ) * * vid2+ro2ro4#2"$( )ro4#+ ro2+ rd( )gm4vid2! "ro2+ rd( )ro4#ro4#+ ro2+ rd( )gm4"ro2ro4#ro4#+ ro2+ rd( )1"$( )gm2% & ' ' ( ) * * vic=rd+$ro2( )ro4#ro4#+ ro2+ rd( )gm4vic≈ unchanged by adding diodes ≈ 1.5, increased from ≈1 by adding diodes Clif Fonstad, 12/3/09 Lecture 23 - Slide 5Mid-band, cont: The mid-band range of frequencies In this range of frequencies the gain is a constant, and thephase shift between the input and output is also constant(either 0˚ or 180˚). log !log |Avd|!b!c!d!a!LO!LO*!4!5!2!1!3!HI*!HIMid-band RangeAll of the parasitic and intrinsic device capacitancesare effectively open circuits All of the biasing and coupling capacitors are effectively short circuits Clif Fonstad, 12/3/09 Lecture 23 - Slide 6gl+-vin = vgs+-voutvt+-rtgmvgsgods,bs,bgBounding mid-band: frequency range of constant gain and phase Common Biasing capacitors: typically in mF range (CO, CS, etc.) effectively shorts above ωLO Device capacitors: typically in pF range (Cgs, Cgd, etc.) effectively open until ωHI Source IBIASV-V+vin+-CECOvout+-+-vgs+-voutvt+-rtgmvgsgoCgsCgdds,bggob-vin+CSCOgslgelLEC for common source stage with all the capacitors Mid-band frequencies fall between: ωLO < ω < ωHI Common emitter LEC for in mid-band range Note: gl = gsl + gel What are ωLO and ωHI? Clif Fonstad, 12/3/09 Lecture 23 - Slide 7Estimating ωHI - Open Circuit Time Constants Method Open circuit time constants (OCTC) recipe: 1. Pick one Cgd, Cgs, Cµ, Cπ, etc. (call it C1) and assume all others are open circuits. 2. Find the resistance in parallel with C1 and call it R1. 3. Calculate 1/R1C1 and call it ω1. 4. Repeat this for each of the N different Cgd's, Cgs's, Cµ's, Cπ's, etc., in the circuit finding ω1, ω2, ω3, …, ωN. 5. Define ωHI* as the inverse of the sum of the inverses of the N ωi's: ωHI* = [Σ(ωi)-1]-1 = [ΣRiCi]-1 6. The true ωHI is similar to, but greater than, ωHI*. Observations: The OCTC method gives a conservative, low estimate for ωHI. The sum of inverses favors the smallest ωi, and thus the capacitor with the largest RC product dominates ωHI*. Clif Fonstad, 12/3/09 Lecture 23 - Slide 8Estimating ωLO - Short Circuit Time Constants Method Short circuit time constants (SCTC) recipe: 1. Pick one CO, CI, CE, etc. (call it C1) and assume all others are short circuits. 2. Find the resistance in parallel with C1 and call it R1. 3. Calculate 1/R1C1 and call it ω1. 4. Repeat this for each of the M different CI's, CO's, CE's, CS's, etc., in the circuit finding ω1, ω2, ω3, …, ωM. 5. Define ωLO* as the sum of the M ωj's: ωLO* = [Σ(ωj)] = [Σ(RjCj)-1] 6. The true ωLO is similar to, but less than, ωLO*. Observations: The SCTC method gives a conservative, high estimate for ωLO. The sum of inverses favors the largest ωj, and thus the capacitor with the smallest RC product dominates ωLO*. Clif Fonstad, 12/3/09 Lecture 23 - Slide 9Summary of OCTC and SCTC results • OCTC: an estimate for ωHI log !log |Avd|!b!c!d!a!LO!LO*!4!5!2!1!3!HI*!HIMid-band Range1. ωHI* is a weighted sum of ω 's associated with device capacitances: (add RC's and invert) 2. Smallest ω (largest RC) dominates ωHI * 3. Provides a lower bound on ωHI • SCTC: an estimate for ωLO 1. ωLO * is a
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