Common Source Stage Miller Approximation ZVTC Analysis Backgate Effect Claudio Talarico Gonzaga University Sources: most of the figures were provided by B. Murmann CS with Resistive Load Vout VDD RD Bias Point (Q) vout Idi D = ID + id + VOUT v = V + v VoutOU=T OUT out Signal vin V in - Bias VIN vin Vin VIN + + gmvin 2I D 2I D vin ro RD vout Av = − RD || ro ≅ − RD - - VOV VOV gm often but not always !!! R EE 303 – Common Source Stage 2 Does ro matter ? § ro can be neglected in the gain calculation as long as the desired gain |Av| is much less than the intrinsic gain gmro 1 1 1 | AV |= gm (RD || ro ) = + | AV | gm RD gmro | A | g R = V ≅| A | m D | A | V 1− V gmro for | AV |<<gmro NOTE: IB The small signal 2ID 1 2 v VOoutUT gmro ≅ ⋅ = parameters (that is V V λI λV in vIN OV D OV gm and ro) are determined by the DC bias point Intrinsic Gain EE 303 – Common Source Stage 3 Upper Bound on Gain § In the basic common source stage RD performs two “conflicting” tasks – it translates the device’s drain current id into the output voltage vout. – it sets the drain bias voltage (VDS) of the MOSFET § This creates an upper bound on the achievable small signal voltage gain V 2ID RD | AV |≅ RD = 2 VOV VOV VRDmax VDD −VOV min VDD | AV max |=2 = 2 ≅ 2 VOUT VOV min VOV min VOV min VDD A NOTE: VIN – VT only a limited range B of the VTC is useful for amplification VDD ⋅ RON C For VIN large: VOUT ≅ (resistivedivider) RD + RON D VIN VT VDD cutoff saturation triode EE 303 – Common Source Stage 4 Voltage Gain and Drain Biasing Considerations (1) VIN ID VDS=VDD−RDID For a given ID if RD ( VDS ) the bias point Q moves toward triode region ID VIN5 Small RD Intercepts D VIN4 at VDD/RD VIN3 C Load line Large R D ID = (VDD-VOUT)/RD VIN2 B VIN1 A VDS = VOUT VDD EE 303 – Common Source Stage 5 Voltage Gain and Drain Biasing Considerations (2) Id For a given ID if RD the voltage gain Small RD Intercepts Q = (VDS,ID) at VDD/RD Large RD Id =gmvin Q Q1 2 ID Vout VDD vout EE 303 – Common Source Stage 6 Importance of choosing the “right” bias point § Deciding the location of the bias point Q affects: – gain (the values of gm and ro depend on the DC bias) – allowable signal swing at the output Vout Vout vout Q vout Q Vin Vin vin vin EE 303 – Common Source Stage 7 Can we overcome the upper bound on gain ? § The upper bound comes from the fact that both gain and bias point depend on RD V | A | 2 DD – Want large R for large gain V max ≅ D VOV min – Want small RD to prevent device from entering triode region § We can overcome the upper bound, if we can find a way to set VOUT independently of RD Small Id Large RD RD Q ID Vout VOUT EE 303 – Common Source Stage 8 CS stage with improved drain biasing scheme VB −Vout at the point V =V we have I =I I = I + out B d B d B regardless the value of R RD D Id IB RD VB vVOUT IidD out Q IB vin VIN Vout VB We wish to set IB=ID and VB=VOUT EE 303 – Common Source Stage 9 Plot for IB=ID Small Id Large RD RD I Q D gmid vout Vout VOUT § We can increase RD (and gain) without changing operating point EE 303 – Common Source Stage 10 Infinite gain? § It is tempting to think we can make RD nearly “infinitely” large and get close to “infinite” gain § This is not possible in practice for two reasons – Finite dId/dVds of the transistor I B 2I 1 2 V D vOUoutT A = g r ≅ ⋅ = V m o V λI λV Vin vIN OV D OV – Sensitivity to mismatch between ID and IB will render the circuit impractical EE 303 – Common Source Stage 11 Bias point shift due to mismatch in IB and ID § For large values of RD, it becomes harder to absorb differences between ID and IB and still maintain an operating point that is close to the desired value Id 1/RD IB ΔI ID Q ΔV= ΔI RD Vout VB VOUT EE 303 – Common Source Stage 12 Solutions to bias point shift issue § Limit RD to values such that expected mismatch in bias currents causes acceptable bias point variations § Feedback – Somehow sense VOUT and adjust IB (or ID) such that the outputs sits at a proper operating point regardless of mismatch between ID and IB – More later … § In a realistic circuit implementation, the auxiliary current source IB can be built, for example, using a pMOS that operates in saturation – More later … (CS stage with current source load) EE 303 – Common Source Stage 13 How fast can the CS stage go ? § There are two perspectives on “how fast” a circuit can go – Which one of the two matters more is dependent on the application § Time domain – Apply a transient at the input (e.g. a voltage step), measure how fast the output settles § Frequency domain – Apply a sinusoid at the input, measure the gain and phase of the circuit transfer function across frequency § Knowing the time domain response, we can estimate the frequency domain response, and vice versa EE 303 – Common Source Stage 14 Common Source Stage revisited VB IB R RD V “Transducer” o Ri vi VI Ri models finite resistance in the driving circuit EE 303 – Common Source Stage 15 Matlab Design Script % % C. Talarico % filename: csdesign.m % Design of CS amplifier using gm/Id methodology % clc; clear all; close all; addpath('~/gm_ID_starter_kit_2014'); load 180n.mat; % specs. av0 = 10; Id = 400e-6; Ri = 10e3; RD = 2e3; Vdd = 1.8; VB = Vdd/2; % design choice Ln = 0.18e-6 % calculations gm = av0/RD gm_id = gm/Id wT = lookup(nch, 'GM_CGG', 'GM_ID', gm_id, 'L', Ln); fT = wT/2/pi cgd_cgg = lookup(nch, 'CGD_CGG', 'GM_ID', gm_id, 'L', Ln); cdd_cgg = lookup(nch, 'CDD_CGG', 'GM_ID', gm_id, 'L', Ln); cgg = gm/wT; cgd = cgd_cgg*cgg cdd = cdd_cgg*cgg; cdb = cdd - cgd cgs = cgg - cgd gmro = lookup(nch, 'GM_GDS','GM_ID', gm_id, 'L', Ln) ro = gmro/gm % finding input bias VI = lookupVGS(nch, 'GM_ID', gm_id, 'L', Ln) % device sizing id_w = lookup(nch, 'ID_W', 'GM_ID', gm_id, 'L', Ln) W = Id/id_w % neglecting ro is not a very accurate assumption R = (1/RD + 1/ro)^-1 Av0 = gm*R Av0db = 20*log10(gm*R) % pole calculations (dominant pole assumption) b1 = Ri*(cgs + cgd*(1+Av0))+R*(cdb+cgd); b2 = Ri*R*(cgs*cdb + cgs*cgd + cdb*cgd); fp1 = 1/2/pi/b1 fp2 = 1/2/pi*b1/b2 % zero calculation fz = 1/2/pi/cgd*gm ! EE 303 – Common Source Stage 16 What is the speed of the CS ? VB VDD = 1.8 V IB R RD VB = VDD/2 = 0.9 V V R = 10K “Transducer” o i Ω RD = 2 KΩ Ri VI = 0.627 V vi IB = 400 µA VI W/L = 21.34 µm/0.18µm Cgd Ri + RD + Cgd≈10.28 fF g v vi vgs Cgs m gs ro R Cdb vo Cgs≈ 33.26 fF - - Cdb≈ 15.72 fF R ro≅ 7.63KΩ R = RD||ro ≅ 1.58KΩ EE 303 – Common Source Stage 17 CS Frequency Response AC Response 20 0 −20 |H(f)| [dB] −40 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 f [Hz] 200 150 100 50 phase[H(f)] [deg] 0 4 5 6 7 8 9 10 11 10 10 10 10 10 10 10 10 f [Hz] § There seems to be two poles. Let’s analyze the situation in detail. EE 303 – Common Source Stage 18 Exact Analytical Analysis 1 Cgd 2 + + Ri g v vi/Ri vgs Cgs m gs ro R Cdb vo - - Applying KCL at nodes 1 and 2, and solving for vo/vi yields ⎛ C ⎞ g R 1 s gd − m ⎜ − ⎟ vo( s ) ⎝ gm ⎠ = 2 vi ( s ) 1+ s[(Cdb + Cgd )R + (Cgs + Cgd )Ri + gm Ri RCgd ]+ s Ri R(CgsCdb + CgdCdb + CgsCgd ) High “entropy” expression. Difficult to get any insight !!! EE 303 – Common Source Stage 19 Issue § We could in principle use this expression to plot the frequency response of the circuit and compute the 3-dB bandwidth – The result would match the Spice simulation result exactly § There are two issues with going in this direction for hand analysis – The procedure is quite tedious… • Imagine how complex the equations would get for a multi- transistor circuit – The derived expression is useless for reasoning about the circuit from an intuitive design perspective • By looking at this equation we cannot easily tell what exactly limits the bandwidth, or how we can improve it EE 303 – Common Source Stage 20 Simulation Result § Want to have a method that § Non-dominant, high-frequency let’s us estimate the dominant poles and zeros may or may pole quickly using intuitive not be important methods – If they are, it may be OK to do a – Without running into high entropy little more work expressions that tell us things we are not interested in… AC Response 20 0 Dominant pole −20 −40 Non-dominant |H(f)| [dB] pole(s) and zero(s) −60 −80 4 5 6 7 8 9 10 11 12 10 10 10 10 10 10 10 10 10 f [Hz] EE 303 – Common Source Stage 21 The Culprit § The main reason for the high complexity in the derived expression is that Cgd “couples” nodes 1 and 2 § For Cgd=0, the circuit becomes 1 2 + + Ri g v vi/Ri vgs Cgs m gs R Cdb vo - - τ1 τ2 vo (s) vgs (s) vo (s) 1 −gm R 1 = ⋅ = ⋅ = −gm R vi (s) vi (s) vgs (s) (1+ sRiCgs ) (1+ sRCdb ) (1+ sτ1)(1+ sτ 2 ) EE 303 – Common Source Stage 22 A Promising Trick: Miller Theorem Z 1 2 + General + Linear V V1 Network 2 - - V K ≡ 2 V1 1 2 General + + Z Linear K V Z V2 1 1− K Network K −1 - - EE 303 – Common Source Stage 23 Finding K(s) for Our Circuit 1 Cgd 2 + + g v vgs m gs R Cdb vo - - Applying KCL at node 2, and solving for vo/vgs yields: " Cgd % $ 1− s ' v (s) g K(s) = o = −g R$ m ' v (s) m $1+ sR(C +C )' gs $ gd db ' # & EE 303 – Common Source Stage 24 Circuit After Applying Miller Theorem 1 2 + gmvgs + Ri vi/Ri vgs Cgs R Cdb vo - Cgd[1-K(s)] Cgd[1-1/K(s)] - g ⎛ C ⎞ z = + m ⎜ 1− s gd ⎟ C v ( s ) g gd K( s ) o g R⎜ m ⎟ = = − m ⎜ ⎟ vgs( s ) 1+ sR( Cgd + Cdb ) 1 ⎜ ⎟ p = − ⎝ ⎠ R(Cgd +Cdb ) Intuitively, the zero is caused because at high frequency Cgd shorts the gate and drain of the device together, providing a direct path from the amplifier’s input to output.