Signal Circuit and Transistor Small-Signal Model Lecture notes: - PowerPoint PPT Presentation

Signal Circuit and Transistor Small-Signal Model Lecture notes: Sec. 5 Sedra & Smith (6 th Ed): Sec. 5.5 & 6.7 Sedra & Smith (5 th Ed): Sec. 4.6 & 5.6 F. Najmabadi, ECE65, Winter 2012

Transistor Amplifier Development Bias & Signal Bias Signal only = (Bias + Signal) - Bias + ? MOS : V , V , I , MOS : v , v , i , MOS : v , v , i , GS DS D gs ds d GS DS D = + ( v V v ,...) GS GS gs = + : , R V I R : v , i R : v V v D R R D r r D R R r = + ..... ..... i I i R R r ..... F. Najmabadi, ECE65, Winter 2012

Finding signal circuit elements -- Resistor Voltage Current iv Equation Resistor v R i R v R = R i R Bias + Signal: V R I R V R = R I R Bias: v r = v R − V R i r = i R − I R Signal: ?? = − = − = − v = v v V Ri RI R ( i I ) Ri r R R R R R R r r  A resistor remains as a resistor in the signal circuit. F. Najmabadi, ECE65, Winter 2012

Finding signal circuit elements -- Capacitor Voltage Current iv Equation Capacitor v C i C i C = C dv C /dt Bias + Signal: V C I C I C = C dV C /dt Bias: v c = v C − V C i c = i C − I C Signal: ?? − ( ) dv dV d v V = − = − = dv C C C C i i I C C C c = c i C c C C dt dt dt dt  A capacitor remains as a capacitor in the signal circuit. o Since V C = const., I C = 0 , i.e., A capacitor acts as an open circuit for bias circuit. F. Najmabadi, ECE65, Winter 2012

Finding signal circuit elements – IVS & ICS Independent Voltage Current iv Equation voltage source v IVS i IVS v IVS = V S = const Bias + Signal: V IVS I IVS V IVS = V S = const Bias: v ivs = v IVS − V IVS i ivs = i IVS − I IVS Signal: ?? = ≠ = − = − = v 0 , i 0 v v V V V 0 ivs ivs ivs IVS IVS S S  An independent voltage source becomes a short circuit! Similarly:  An independent current source becomes an open circuit! Exercise: Show that dependent sources remain as dependent sources F. Najmabadi, ECE65, Winter 2012

Summary of signal circuit elements  Resistors& capacitors: The Same o Capacitor act as open circuit in the bias circuit.  Independent voltage source (e.g., V DD ) : Effectively grounded  Independent current source: Effectively open circuit o Careful about current mirrors as they are NOT “ideal” current sources (early effect and/or channel width modulation was ignored!)  Dependent sources: The Same  Non-linear Elements: Different! o Diodes & transistors ? F. Najmabadi, ECE65, Winter 2012

Diode Signal Response v D   v   + = D Bias Signal : exp i I   D s i D   nV T V D   V   = D Bias : exp I I   I D D s   nV T v d     + V v V     ? = − = − D d D Signal : i i I I exp I exp     i d d D D s s     nV nV T T       V v     = × −   D d i I exp exp 1     d s     nV  nV  T T  A different iv equation!     v   = × −  iv equation is non-linear!   d i I exp 1   d D    nV   Related to bias value, I D ! T F. Najmabadi, ECE65, Winter 2012

Diode small-signal model: v d     ? v   = × − i d   d i I exp 1   d D   nV   T 2       v v 1 v       = + + + d d d Taylor Series Exapnsion : exp 1 ....             nV nV 2 ! nV T T T     v v v     << ≈ + d d d If 1 : exp 1         nV nV nV T T T       v I ≈ × +   − =    d  D i I 1 1 v     d D D     nV nV   T T nV = = T v i r i d d d d I D F. Najmabadi, ECE65, Winter 2012

Formal derivation of small signal model  Signal + Bias for element A ( i A , v A ) : i A = f ( v A )  Bias for element A ( I A , V A ) : I A = f ( V A )  Signal for element A ( i a , v a ) : i a = g ( v a ) = i f ( v ) A A ( 2 ) ( ) f ( V ) ( ) (Taylor Series = + ⋅ − + ⋅ − + 2 ( 1 ) A f ( V ) f ( V ) v V v V ... A A A A A A 2 ! Expansion) ( 2 ) f ( V ) = + ⋅ + ⋅ + 2 ( 1 ) A f ( V ) f ( V ) v v ... A A a a 2 ! ≈ + ⋅ ( 1 ) f ( V ) f ( V ) v A A a Small signal means: = + = + ⋅ ( 1 ) ( ) i i I I f V v A a A A A a ( 2 ) f ( V ) ⋅ >> ⋅ 2 ( 1 ) A f ( V ) v v A a a = = ⋅ ( 1 ) 2 ! i g ( v ) f ( V ) v a a A a ( 1 ) f ( V ) << ⋅ A v 2 a ( 2 ) f ( V ) A F. Najmabadi, ECE65, Winter 2012

Derivation of diode small signal model     v v v D 1     = ⋅ − = = − = × nV nV ( 1 ) nV i I e 1 f ( v ) f ( v ) I e 1 f ( v ) I e T T T     D S D S S nV     T   V V D D   = = ⋅ − = − nV nV I f ( V ) I e 1 I e I T T   D D S S S       v V D   ⋅ + ⋅     nV nV I e I e I I T T = × = × = × = × ( 1 ) S S D S   i f ( V ) v v v v     d D d d d d   nV nV nV     T T T     = v V v D D     + I I I i D = × ≈ × D S D     i v v d d d     nV nV T T Diode can be replaced with a resistor in the signal circuit! v nV v d i = r ≈ d T d d r I d D i d r d = nV T /I D F. Najmabadi, ECE65, Winter 2012

Small signal model vs iv characteristics Small signal model is equivalent to approximating the non-liner iv characteristics curve by a line tangent to the iv curve at the bias point = × ( 1 ) i f ( V ) v d D d 1 nV = ≈ T r d ( 1 ) ( ) f V I D D F. Najmabadi, ECE65, Winter 2012

Derivation of MOS small signal model (1) MOS iv equations: i D = f ( v GS , v DS ) i G = 0  Signal + Bias for MOS ( i D , v GS , v DS ) : i D = f ( v GS , v DS ), i G = 0  Bias for MOS ( I D , V GS , V DS ) : I D = f ( V GS , V DS ), I G = 0  Signal for MOS ( i d , v gs , v ds ) : i d = g ( v gs , v ds ), i g = 0 + = = (Taylor Series Expansion in 2 variables) I i i f ( v , v ) D d D GS DS ∂ ∂ f f = + ⋅ − + ⋅ − + f ( V , V ) ( v V ) ( v V ) ... ∂ ∂ GS DS GS GS DS DS v v GS DS V , V V , V GS DS GS DS ∂ ∂ f f ≈ + × + × I v v ∂ ∂ D gs ds v v GS DS V , V V , V GS DS GS DS ∂ ∂ f f ≈ × + × i v v ∂ ∂ d gs ds v v GS DS , , V V V V GS DS GS DS F. Najmabadi, ECE65, Winter 2012

Derivation of MOS small signal model (2) W = µ − + λ = 2 i 0 . 5 C ( v V ) ( 1 v ) f ( v , v ) D n ox GS t DS GS DS L ∂ ∂ f f = ⋅ + ⋅ i v v ∂ ∂ d gs ds v v GS DS V , V V , V GS DS GS DS ∂ f W = × µ − + λ 2 0 . 5 C ( v V )( 1 v ) ∂ n ox GS t DS V , V v L GS DS GS V , V GS DS W µ − + λ 2 0 . 5 C ( V V ) ( 1 V ) 2 I n ox GS t DS L = × = ≡ D 2 g − m ( ) V V V GS t OV ∂ f W = λ × µ − 2 0 . 5 C ( v V ) ∂ n ox GS t v L DS V , V V , V GS DS GS DS W µ − + λ 2 0 . 5 C ( V V ) ( 1 V ) λ n ox GS t DS I 1 L = λ × = ≈ λ ≡ D I + λ + λ D ( 1 V ) ( 1 V ) r DS DS o v = ⋅ + = ds i g v i 0 d m gs g r o F. Najmabadi, ECE65, Winter 2012

MOS small signal “circuit” model v = = ⋅ + ds i 0 and i g v g d m gs r o Statement of KCL Input open circuit Two elements in parallel ⋅ = 2 I 1 2 2 V ≈ λ = = >> D A g r g r 1 ⋅ λ m o m o V I V V OV D OV OV F. Najmabadi, ECE65, Winter 2012

PMOS “circuit” small signal model is identical to NMOS PMOS* NMOS =  PMOS small-signal circuit model is identical to NMOS We will use NMOS circuit model for both! o For both NMOS and PMOS, while i D ≥ 0 and I D ≥ 0 , signal quantities: i d , o v gs , and v ds , can be negative! Exercise: Derive PMOS small signal model (follow derivation of NMOS small-signal model) F. Najmabadi, ECE65, Winter 2012