1 Small-Signal Modeling Dr. David W. Graham West Virginia University Lane Department of Computer Science and Electrical Engineering © 2010 David W. Graham

2 Small-Signal Modeling An approximation to the large-signal model around an operating point Assumes perturbations in the bias conditions are small Can use a linearized model for small changes

3 Small-Signal Modeling Assumptions –Transistor is biased at some current level –Saturated operation (typically) –Only small potential changes at the terminals (G, S, D, B) Consequences –DC bias levels (currents & voltages) can be ignored –Any voltage change at any terminal will modify the channel current

4 Different Small-Signal Models Bulk-referred model Source-referred model T model

5 Bulk-Referred Small-Signal Model (Low Frequency)

6 Bulk Referred Small-Signal Model Every potential is with respect to (wrt) the bulk Use VCCS to model changes in current from DC No gate current Increases in the gate & drain wrt the bulk increase the channel current (drain-to-source) Increases in the source wrt the bulk decrease the channel current Total Current (centered around I BIAS )

7 Transconductance, g m Changes in the gate voltage (ΔV g ) produce changes in the channel current Changes in V g change the current by g m  transconductance  creates a conductance at other terminals (not including the gate terminal)

8 Bulk-Referred Small-Signal Model Derive the small signal parameters for subthreshold operation –g m –g s –g d Bulk-referred model is typically used for subthreshold modeling (Source-referred model is typically used for above threshold modeling)

9 Source Conductance, g s Changes in the source voltage (ΔV s ) produce changes in the channel current Changes in V s change the current by g s  source conductance  creates a conductance at its own terminal

10 Drain Conductance, g d Changes in the drain voltage (ΔV d ) produce changes in the channel current Changes in V d change the current by g d  drain conductance Output Resistance

11 Total Small-Signal Current (Bulk-Referred Model)

12 Source-Referred Small-Signal Model (Low Frequency)

13 Source-Referred Small-Signal Model Use κ ≈ 1 Approximation All voltages referenced to the source instead of the bulk Traditional small-signal model for Above V T operation κ ≈ 1 Approximation

14 Source-Referred Small-Signal Model

15 κ ≈ 1 Sub V T Parameters

16 Simplification to the Small-Signal Models g 0 V ds is a current whose value linearly depends on the voltage across it  Resistor Typically use this simplification in both the source-referred and bulk referred models

17 Small Signal Models Bulk-Referred Source-Referred

18 Above V T Small-Signal Parameters, g m Typically use the source-referred model K is often referred to as the transconductance parameter

19 Above V T Small-Signal Parameters, g mb No V bs terms in the current expression However, V T depends on V bs λ = 1/V A

20 Above V T Small-Signal Parameters, r 0

21 Similarities Between Small-Signal Models Bulk-Referred ModelSource-Referred Model Comparison – KCL at the source for both cases Comparison

22 Consequences Therefore, you can use either model (bulk- referred or source-referred) for any analysis Use whichever model provides a simpler analysis Typically, we use |g s | for subthreshold and (g m +g mb ) for above threshold –Simply plug in the appropriate values at the end of the analysis

23 “Unified” Bulk-Referred Model Let g x = |g s | = g m + g mb (whichever is appropriate)

24 T Model Useful for specific situations For the derivation –Start from the source-referred model –Assume no body effect (can be added later) No current flows into the gate Two current sources in series do not add i g = 0 (KCL) vertical g m v gs is a “resistor”

25 Exploring the Small-Signal Parameters

26 Transconductance Efficiency Transconductance for a given bias current, g m /I D Sub VT has a constant, large transconductance efficiency This is related to the “inversion coefficient” Holds true for nFETs and pFETs of various processes and sizes

27 Maximum Intrinsic Gain Largest gain that can be achieved by a single transistor g m r 0 gmgm r0r0 gmr0gmr0 Sub V T Above V T Higher gain in sub V T Faster operation in above V T Example Configuration