6.012 Recitation 11: Small Signal Model of MOSFET/MOSFET in Digital Circuits

Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 Recitation 11: Small Signal Model of MOSFET/MOSFET in Digital Circuits Small Signal Models On Tuesday we talked about small signal models of MOSFETs. Why do we need small signal modeling? To linearize circuits. Linear circuits are much easier to work with: we can use Thevenin/Norton equivalent circuits, superposition, etc. How to obtain a linearized circuit? If we limit our signals to a relatively small amplitude, (this is in fact most often the case), the non-linear IV curves (as seen in 1 for example) can be considered piece-wise linear =⇒ small signal model. Figure 1: Non Linear IV Curves 1 Recitation 11 Small Signal Model of MOSFET A MOSFET is a 4-terminal device (NMOS as an example) V || V Figure 2: NMOS • cut-off id = 0 • triode/linear id = • Saturation id = w VDS μn Cox (VGS − − VT ) · VDS L 2 w μn Cox (VGS − VT )2 · (1 + λVDS ) 2L In small signal modeling it is very important to differentiate: iD ← total signal (iD = ID + id ) ID ← DC signal id ← small signal 2 6.012 Spring 2009 Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 What Happens at Low Frequency? To obtain a small signal equivalent circuit, we need to find an operation point first: Q(VDS , VGS , VBS ). (Q is a specific point on Fig. 1) thus ID is also known. Then, what is in between D & S in Figure ?? depends on: δid  δid  δid   · VGS +  · VDS +  · VBS δVGS Q δVDS Q δVBS Q = gm · VGS + go · VDS + gMB · VBS three conductances in parallel (∵ they add up) id = • gm : trans-conductance (unit S) • go : output conductance (unit S) • gmb : backgate transconductance (unit S) On Tuesday, we derived the expression for gm , go , gmb in saturation regime gm = =  go = γo = gmb =  δ[ w μn Cox (VGS − VT )2 (1 + λVDS )]  δid   = 2L  δVGS Q δVGS Q(VGS,VDS ,VBS ) w μn Cox (1 + λVDS )(VGS − VT ) · 2 2L   w 2w w μn Cox ID → gm ∝ ID μn Cox (VGS − VT ) = L L L δiD  w μn Cox (VGS − VT )2 · λ  λID  = δVDS Q 2L 1 1 L = ∝ go λID ID δiD  δVT  w μn Cox (1 + λVDS ) · (−2)(VGS − VT ) ·  =  δVBS Q 2L δVBS Q w δVT  − μn Cox (VGS − VT )   L   δVBS Q gm VT 1 = +gm · γ ·  2 −2φp − VBS   = VTo + γ( −2φp − VBS − −2φp ) 3 Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 Now what does the small signal circuit look like? Figure 3: Low Frequency Small Signal Circuit. Two of them are voltage controlled current sources, one is a resistor. Why? What Happens at High Frequency? There are intrinsic or parasitic capacitances related to the MOSFET structure, as we know 1 Zc = jwC . At low frequency, Zc is very large, can be approximated to open circuit, however at high frequency, Zc is small enough we have to consider. We have 4 terminals. Considering the possible combinations between them, we have: Figure 4: Between D & S, we have conduction channel, no capacitance between D & S 4 Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 What are these capacitances (under saturation)? Figure 5: Capacitances (under saturation) 1. Cgs , two contributions, first is the MOS capacitor capacitance under inversion was Cox before, but since under saturation regime, we have large VDS , the inversion layer is nonuniform in charge density, need to do integration of qG to consider this. We will skip the math here since it is derived in lecture already. The result is wLCox −→ 23 Cox ·wL. The other contribution is from the overlap between the source and gate Cov 2 ∴ CGS = wL · Cox + w · Cov Cov unit is F/cm 3 2. CGD = w · Cov L, Ldiff, w, see Fig. 5 above 3. CBS or CSB = pn junction capacitance underneath the source area and side wall, is: (s) = w · Ldiff Cj (s) Cj = + (2Ldiff + w) · Cjsw (s) qs Na 2(φB − VBS ) Cjsw (s) is usually given, should be 5 qs Na ·d 2(φB − VBS ) Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 4. Cbd or Cdb = pn junction capacitance underneath the drain area and side wall, is: (D) = w · Ldiff · Cj (D) Cj = (D) + (2Ldiff + w) · Cjsw qs Na 2(φB − VBD ) 5. Cgb is due to the presence of inversion layer (screening) under inversion, the capacitance of Cgb can be ignored (it only present at cut off). MOSFETS in Digital Circuits Now we have both the low frequency and high frequency version of the small signal equivalent circuit. What has it to do with our MOSFET digital circuit discussion? For digital circuits, there are two important aspects: 1. Noise Margin: larger noise margin −→ higher noise immunity, better 2. Speed: the concept of “propagation delay”. We want circuit response to be fast, =⇒ low propagation delay Now the question is, when we design a circuit, what parameters affect the noise margin and what affects the delay? Noise Margin Our inverter is shown below: Figure 6: n-MOS inverter 6 Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 NMH = VOH − VIH NML = VIL − VOL In order to have a high noise margin, we want high slope at Vm =⇒ high gain Av at Vm . Gain How to find gain at Vm ? Use a small signal circuit: 7 Recitation 11 Small Signal Model of MOSFET 6.012 Spring 2009 || V Av = Vout = −gm (ro ||R)  −gm R Vin The larger the R the bigger the noise margin. However, as we will see later, the larger R the slower the circuit. There is a tradeoff since capacitances in the circuit add delay. 8