Session 2 MOS Transistor for RF Circuits

Session 2 MOS Transistor for RF Circuits Session Speaker Chandramohan P.

Session Contents MOS transistor basics MOS equivalent circuit Single stage amplifiers Opamp design

Session objectives To understand the MOS device characteristics To derive MOS equivalent circuit and understand small signal model To design and analyze single stage amplifiers and Opamp

MOS I/V Characteristics Threshold voltage Derivation of I/V characteristics

MOS I/V Characteristics: Threshold Voltage () Threshold voltage (V TH ) - The value of V GS for at which a sufficient number of mobile electrons accumulate in the channel region to form a conducting channel is called the threshold voltage. V TH is given by: where Q dep V TH =Φ MS + 2Φ F +, Cox Φ MS - the difference between the work functions of the polysilicon gate and the silicon substrate, Φ F =(KT/q) ln(nsub/ni), q is electron charge, Also N sub is the doping concentration of the substrate, Q dep is the charge in the depletion region, C ox is the gate oxide capacitance per unit area. Q = 4ε Φ N : dep si F sub

MOS I/V Characteristics: Threshold Voltage (2) The PMOS device operates in the same manner as the NMOS device except that V GS, V DS and the threshold voltage V T are negative. As the gate-source voltage becomes sufficiently negative, an inversion layer consisting of holes is formed at the oxide-silicon interface, providing a conduction path between the source and the drain. VG -0. V p + p + n-substrate Holes Formation of inversion layer

Derivation of I/V Characteristics () S V G D S V G V D N+ N+ N+ N+ P-substrate a) equal source and drain voltages 0 X L P-substrate b) unequal source and drain voltages Channel charge density (charge per unit length) Q d = WC OX(VGS - V TH) Q (X) = WC (V - V(X)- V ) d OX GS TH

Derivation of I/V Characteristics (2) I D = -WC OX(VGS - V(X)- V TH)v The negative sign is inserted in the above formula as the charge carriers are negative. Also v denotes the velocity of the electrons in the channel I = WC (V - V(x)- V )μ D OX GS TH n dv(x) dx L V DS D OX n GS TH x=0 V=0 I dx = WC μ (V - V(x)- V )dv W I =μ C (V - V )V - V L 2 2 D n OX GS TH DS DS Here L is the effective channel length.

W I =μ C (V - V )V - V L 2 2 D n OX GS TH DS DS I D Triode Region Derivation of I/V Characteristics (3) Maximum drain current is given by: V GS2 V GS3 W I D,max = μnc OX (VGS - V TH) 2 L 2 V GS V GS -V TH V GS2 -V TH V GS3 -V TH V DS If VDS VGS - VTH the device operates in the triode region.

If Derivation of I/V Characteristics (4) V << 2(V - V ) DS GS TH W I μ C (V - V )V L D n OX GS TH DS then The drain current is a linear function of V DS. The linear relationship implies that the path from the source to the drain can be represented by a linear resistor equal to R = on W μnc OX (VGS - V TH) L If V DS << 2(VGS - V TH) the device operates in deep triode region.

Derivation of I/V Characteristics (5) S V G V DS S V G V DS2 > V DS N+ N+ N+ N+ 0 X P-substrate 0 X2 P-substrate pinch-off pinch-off V(x ) = V - V V(x 2) = VGS - VTH GS TH Pinch-off behavior

Derivation of I/V Characteristics (6) W I D = μnc OX (V ' GS - V TH) 2 L 2 I D V GS -V TH V GS2 -V TH V GS3 -V TH Saturation Region Fig: Saturation of drain current V GS3 V GS2 V GS V DS L With the approximation L ' a saturated MOSFET can be used as a current source connected between the drain and the source, an important component in analog design.

Derivation of I/V Characteristics (7) A MOSFET when operated in the saturation region acts as a voltage-controlled current source i.e. changes in the gate-to-source voltage gives rise changes in the drain current I D. This is known as transconductance, g m which is expressed as: di W g = = μ C (V - V ) D m n OX GS TH dvgs L V =const DS g m can also be expressed as: W 2ID g m = 2μnCOX I D = L V - V GS TH

Derivation of I/V Characteristics (8) g m g m g m V GS -V TH I D V GS -V TH W/L Constant W/L Constant W/L Constant The above figures show MOS transconductance as a function of overdrive and drain current

Transistor Second Order Effect, Body Effect VD VB<0 VG VD p + n + n + VG VB<0 p-substrate Fig: NMOS device with negative bulk voltage Suppose V s =V D =0. If V B become more negative, more holes are attracted to the substrate.

Body Effect () VB=0 VG VD p + n + n + p-substrate Q D VB<0 VG VD p + n + n + p-substrate Q D

Body Effect (2) VDD I t t Ignoring body effect. As V in varies, V out closely follows the input because the drain current remains equal to I. w I= 2 μncox - - V TH, 2 L Now suppose the substrate is tied to ground and body effect is significant. As V in and hence V out become more positive, the potential difference between the source and the bulk increases, raising the value of V TH.

Channel-Length Modulation () The actual length of the inverted channel gradually decreases as the potential difference between the gate and the drain increases. In saturation: W 2 ID μncox VGS - VTH + λv DS, 2 L where λ is channel-length modulation coefficient, which represents the relative variation in length for a given increment in V DS. ID VGS2 VGS VDS

Channel-Length Modulation (2) With channel-length modulation, some of the expressions derived for g m must be modified. W g m =μncox VGS - VTH + λv DS. L 2μC ox D g m =. + λvds W/L I Since the dependence on V DS is much weaker, the drain-source voltage is not used to set the current.

Subthreshold Condition () In reality for VGS VTH, a weak inversion layer still exists and some current flows from D to S. Even for VGS<VTH, ID is finite, but it exhibits an exponential dependence on VGS called sub-threshold conduction. Tthis effect can be formulated for VDS greater than roughly 200 mv as V GS I D=I0exp, ζvt where ζ> is the nonideality factor and VT=KT/q.

decade M. S. Ramaiah School of Advanced Studies, Bangalore Subthreshold Condition (2) logid Exponenti al Square Law As the V GS falls below V TH, the drain current drops at a finite rate. With typical values of ζ, at room temperature V GS must decrease by approximately 80 mv for I D to decrease by one decade. The exponential dependence of I D upon V GS in subthreshold operation may suggest the use of MOS device in this regime so as to achieve a higher gain. 80mV VTH VGS MOS subthreshold characteristics

Voltage Limitations MOSFETs experience various breakdown effects if their terminal voltage differences exceed certain limits. At high gate-source voltages, the gate oxide breaks down damaging the transistor. In short-channel devices, an excessively large drain-source voltage widens the depletion region around the drain, creating a very large drain current (the effect is called punch through )

Small-Signal Model () Conductance Transconductance and body effect transconductance Small signal gain Capacitances of a transistor

Small-Signal Model (2) Derived by producing a small increment in a bias point and calculating the resulting increment in other bias parameters. Since the drain current is a function of the gate-source voltage, a voltagedependent current source equal to g m V GS is incorporated. The lowfrequency impedance between G and S is very high. G D V GS g m V GS S Fig: Small-signal model of an ideal MOSFET

Small-Signal Model (3) The drain current also varies with the drain-source voltage. This effect can also be modeled by a voltage-dependent current source. G D V GS g m V GS av DS S

Small-Signal Model (4) Tied between D and S, the resistor is given by: dvds r 0 = = = di di /dv W 2 μnc λi ox (VGS - V TH) λ 2 L D D DS D G D V GS g m V GS r 0 S

Small-Signal Model (5) The bulk potential influences the threshold voltage and hence the gate-source overdrive. The drain current is a function of the bulk voltage (the bulk behaves as a second gate). Modeling this dependence by a current source connected between D and S, the value can be written as g mb V BS G D V GS g m V GS r 0 g mb V BS S

Small-Signal Model (6) In saturation region, g mb can be expressed as: di D W dvth g mb = =μnc ox (VGS - V TH) - dvbs L dvbs dvth dvth γ = - = - (2φ F + V SB ) dv dv 2 Also -/2 BS Thus mb m m SB γ g = g = ηg 2 2φ + V As expected, g mb is proportional to g. Last equation also suggests that incremental body effect becomes less pronounced as V SB increases. g m V GS and g mb V BS have the same polarity, i.e., raising the gate voltage has the same effect as raising the bulk potential. F SB

Capacitances of Transistor The complete small-signal model also includes the device capacitances. C GD G D C GS V GS g m V GS r 0 g mb V BS C GB V BS S C SB C DB B

Single Stage Amplifiers

Common Source Stage vdd M off RD V out =V in -V th V in M V out V in Vth V A out v VDD R V V out in R D D 2 n C n C ox ox W L W L V V 2 in th Vth gmrd

CS Stage with Diode Connected Load () Ix V GmV ro Vx Diode-connected NMOS and PMOS devices Small signal equivalent circuit

CS Stage with Diode Connected Load (2) vdd M Ix Vx Arrangement for measuring the equivalent resistance of a diode-connected MOSFET

CS stage with Diode Connected Load (3) I vdd vdd M2 M Vou t vout vdd M2 I t Cp Vdd VTH2 t A v G m G m G mb2 G G m m2

CS stage with Diode Connected Load (4) vout Vdd Vdd VTH2 M2 A VOUT M VTH A v G G m m2 M. S. Ramaiah School of Advanced Studies, Bangalore

CS Stage with Current-Source Load M2 vdd A v g m R D g m r0 r 02 Vb M2 M g r m 0 2 w l n C ox I d I D M. S. Ramaiah School of Advanced Studies, Bangalore

CS Stage with Triode Load vdd vdd M2 Vb Ron2 M max =VDD

CS Stage with Source Degeneration ID RD V M ID gm V RS RS G m g g m m R s A v G m R D gmr g R m D s

Small-signal Equivalent Circuit of a Degenerated CS Stage I D V in V G M V G MB V BS r o R S

Source Follower () Vdd M Rs VTH (b) (a) Source follower Its input-output characteristic Input-output characteristic can be expressed as: 2 n C ox W L (V in V TH V out ) 2 R S V out

Source Follower (2) Calculate the small-signal gain of the circuit by differentiating both sides with respect to V in Since V V TH in 2 Also note that Consequently, W ncox 2( VTH ) L V V out in V V out g in m, nc 2 ncox 2 A n v C ox W L ox (V W L in W L 2(V 2(V gmr (g g m S mb V V V in TH )R S in TH in V V V V V TH out TH V ) out in V out R out )R S )R S S V V ( ) out in

Source Follower (3) The result is more easily obtained with the aid of a small-signal equivalent circuit. V in -V =V out, V bs =-V out.,and g m V - g mb V out =V out /R s are obtained below: + Av + - V - gmv gmbvbs.0 +ή Rs VTH Small-signal equivalent circuit of source follower Source follower using an NMOS transistor as current source

Source Follower with NMOS transistor as current source () Vdd Vdd M M I Vb M2 (a) Source follower using an NMOS transistor (b) as current source To gain a better understanding of source followers, calculate the small-signal output resistance of the circuit. Using the equivalent circuit and V =-V x : I x g m V x g mb V x 0

Source Follower with NMOS transistor as Current Source (2) (b) Vdd + Vdd ac V gmv gmbvbs ac M - M Ix (a) Rout Vx + - Vx Ix + - (b) Fig: Calculation of the output impendence of a source follower R out g m g mb (c)

Source Follower with NMOS transistor as Current Source (3) The magnitude of the current source g mb V bs is linearly proportional to the voltage across it. Such behaviour is that of a simple resistor equal to /g mb,yielding the smallsignal model. M. S. Ramaiah School of Advanced Studies, Bangalore + V gmbvx gmv + V gmv - - Vx Ix + - Vx Ix + - (a) Fig: Source follower including body effect Without /g mb,the output resistance equals /g m,, concluding that R out g m g mb g m g mb (b)

Source Follower with NMOS transistor as current source (4) + g m V g m V + V out V in + - - V out V in + - gmb g mb gm Fig: Representation of intrinsic source follower by a Thevenin equivalent A v gmb g g m mb g m gm g mb

Source Follower with NMOS transistor as Current Source (5) A v ro ro2 RL gmb ro ro2 RL g g mb m Vdd + V gmv M + - - Vb M2 RL gmb ro ro2 RL (a) Small-signal equivalent circuit Source follower driving load resistance (b)

Source Follower with NMOS M. S. Ramaiah School of Advanced Studies, Bangalore Transistor as Current Source (6) Vdd Vdd M2 Vb Vb n-well n-well Contacts M (a) GND (b) Fig: PMOS source follower with no body effect

Source Follower with NMOS transistor as Current Source (7) Vdd ID X M2 M Vb M3 Fig: Cascode of source follower & CS stage

Source Follower with NMOS Transistor as Current Source (8) The load can be driven by a source follower with an overall R voltage gain of L SF V in RL g m Vdd Vdd RL M I RL The load can be included as part (a) of a common source V stage proving a gain of out V in CS g m R L (b)

Common-Gate Stage () Vdd RD RD M Vb C M Vb + - + - I (a) Common-gate stage with CG stage with capacitive direct coupling at input coupling at input Assume that V in decreases from a large positive value. For V in V b -V TH,M is off and V out =V DD. For lower values of V in : W 2 ID ncox (Vb VTH) 2 L if M is in saturation, As V in decreases, so does V out eventually driving M into the triode region. (b)

Common-Gate Stage (2) DD n OX W L (V If M is saturated, the output voltage can be expressed as V V out 2 V C DD 2 n b C OX V in W L V (V b TH ) 2 R V in D V V b TH ) 2 V R TH D Vdd Fig: Common-gate input-output characteristic VTH Vb-VTH

Common-Gate Stage (3) Obtaining a small-signal gain of Since V V TH in V V V V out in TH SB V V out in n n C C OX OX W L W L (V (V b b V V in in, the following is obtained V V TH TH ) ) V V TH in R D R D gm( ) RD The gain is positive. Interestingly, body effect increases the equivalent transconductance of the stage. The circuit can be analyzed with the aid of its equivalent. Noting that the current flowing through R S is equal to V out /R D,this is obtained: V RS 0 R D

Common-Gate Stage (4) Vdd + RD V gmv ro gmbvbs RD Vb M ro - X Rs Rs + - + - (a) CG stage with finite output resistance Since the current through r O is equal to V out /R D -g m V -g mb V,it can be written ro gmv gmbv RS R D RD Upon substitution for V, R ro (gm gmb) R D R (b) Small-signal equivalent circuit. S D V in R RD S V in V out

It follows that Common-Gate Stage (5) V V out in r O (gm g (g g )r m mb mb O )r R O S R S R The gain of the common-gate stage is slightly higher due to body effect. D R D Vdd + IX RD M V gmv ro gmbvbs RD Vb ro - (a) Rin IX VX + - Input resistance of a CG stage (b) Small-signal equivalent circuit Since V =-V X and the current through r O is equal to I X +g m V +g mb V =I X -(g m +g mb )V x, the voltages can be added up across r O and R D as R I r I (g g )V V Thus, D X V I X X O X R (g D m m r g O mb )r mb O X (g m X RD g mb )r O g m g mb

First suppose R Common-Gate Stage (6) D 0. Then, V I X (g ro g X m mb)r O gm gmb ro The input impedance of a common-gate stage is relatively low only if the load impedance connected to the drain is small. Vdd I In order to calculate the output impedance of the common-gate stage, R out ={ [+(g m +g mb )r O ]R S +r O } R D, Vb M ro ++(3.3)>> A v =(g m +g mb )r O + IX + - Fig: Input resistance of CG stage with ideal current source load

Common-Gate Stage (7) Vdd Vdd Vdd I I RD Vb M ro Vb M ro Vb M ro VX + - X X X Rs + - + - Rs (a) Figure 3.47 (b) Fig: Calculation of output resistance of a CG stage

Cascode Stage () M generates a small-signal drain current proportional to V in & M 2 simply rules the current to R D. Vdd Vdd RD RD gm M2 Vb Vb M2 + VGS2-VTH2 M X + VTH - X M - + VGS-VTH - Cascode stage Allowed voltages in cascode stage.

Cascode Stage (2) As V in exceeds V TH, M begins to draw current, and V out drops. Since I D2 increases, V GS2 must increase as well, causing V X to fall. Vdd + gmv V2 gm2v2 gmbvbs RD - Vb-VTH2 + VX + - V gmv - VTH Input-output characteristic of a cascode stage Small-signal equivalent circuit of cascode stage

Cascode Stage (3) An important property of the cascode structure is its high output impedance. To calculate R out,the circuit can be viewed as a common-source stage with a degeneration resistor equal to r o.thus, R out =[+(g m2 +g mb2 )r o2 ]r O +r O2 Rout Rout Rout M2 M2 Vb2 M3 ro Vb M2 M M Calculation of output resistance of cascode stage Triple cascode

Vdd Cascode Stage (4) If both M & M 2 operate in saturation, then G m g m & R out (g m2 +g mb2 )r O r O2, yielding A v =(g m2 +g mb2 )r o2 g mb r O I ID ID ID W L W 4L W L Vb M2 M (a) (b) Vb W L Cascode stage with current-source load Increasing output impedance by increasing the device length or cascoding. Now consider the output impedance achieved in each case. Since W gmro 2nCOX ID L I D (c)

Cascode Stage (5) If the gate bias voltages are chosen properly, the maximum output swing is equal to V dd -(V GS V TH )-(V GS2 V TH2 )- (V GS3 V TH3 ) - (V GS4 V TH4 ). Substituting G m g m & R out ={[+(g m2 +g mb2 )r o2 ]r O +r O2 } {[+(g m3 +g mb3 )r o3 ]r O4 +r o3 } A v g m R out is obtained. For typical values, the voltage gain is approximated as A v g m [(g m2 r o2 r O ) (g m3 r O3 r o4 )] Vb2 M2 + - Vx P Vb M As V x falls below V b2 -V TH2,M 2 requires a larger gate-source overdrive so as to sustain the current drawn by M. W 2 ID2 ncox [ 2(Vb 2 VP VTH2 )(VX VP ) (VX VP ) ] 2 L 2

Folded Cascode () Vdd Vdd Vdd RD RD I M M2 Vb M M2 Vb X M M2 Vb gm (a) I RD Simple folded cascode (b) Folded cascode with proper biasing (c) Folded cascode with NMOS input For V in >V dd - V TH,M is off & M 2 carries all of I (if I is excessively large,m 2 may enter deep triode region, possibly driving I into the triode region as well), yielding V out =V DD -I R D.For V in <V DD - V TH,M turns on in saturation, giving W 2 ID2 I pcox (VDD VTH ) 2 L

Folded Cascode (2) As V in drops, I D2 decreases further falling to zero if I D =I. For this to occur: Thus, V 2 p C V OX W L (V C DD V in 2I W L V TH V in DD TH p OX 2 ) I Vdd I ID ID2 Vdd -RDID Vdd -RDID Vdd - VTH Large-signal characteristics of folded cascode Vdd - VTH

Session summary Small signal model of MOS transistor consists of voltage controlled current source, drain to source resistance, bulk controlled voltage source Capacitances of MOS introduce delay CS is used for voltage amplification, CD is used as voltage follower and CG is used as impedance matching network In design of Opamp, phase margin should be 60 degrees for good stability