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Analog Electronics Equation Sheet

– 23 –1 – 19
Constants: k = 1.38 × 10 JK ; q = 1.602 × 10 C ; V T = kT ⁄ q ≈ 26mV at 300 °K ;
– 12
ε 0 = 8.854 × 10 F/m ; k ox = 3.9 ; C ox = ( k ox ε 0 ) ⁄ t ox
NMOS: k n = µ n C ox ( W ⁄ L ) ; V tn > 0 ; v DS ≥ 0 ; v ov = v GS – V tn
2
(triode) v DS ≤ v ov (or v D < v G – V tn ) ; i D = k n ( ( v ov )v DS – ( v DS ⁄ 2 ) )
2k n I D ; r s = 1 ⁄ g m ; r o = L ⁄ ( λ′ I D )
2
(active) v DS ≥ v ov ; i D = 0.5k n v ov ( 1 + λv DS ) ; g m = k n V ov = 2I D ⁄ V ov =
PMOS: k p = µ p C ox ( W ⁄ L ) ; V tp < 0 ; v SD ≥ 0 ; v ov = v SG – V tp
2
(triode) v SD ≤ v ov (or ( v D > v G + V tp )) ; i D = k p ( ( v ov )v SD – ( v SD ⁄ 2 ) )
2k p I D ; r s = 1 ⁄ g m ; r o = L ⁄ ( λ′ I D )
2
(active) v DS ≥ v ov ; i D = 0.5k p v ov ( 1 + λ v SD ) ; g m = k p V ov = 2I D ⁄ V ov =
( v BE ⁄ VT )
BJT: (active) i C = I S e ( 1 + ( v CE ⁄ V A ) ) ; g m = α ⁄ r e = I C ⁄ V T ; r e = V T ⁄ I E ; r π = β ⁄ g m ; r o = V A ⁄ I C
i C = βi B E ; i = ( β + 1 )i B ; α = β ⁄ ( β + 1 ) ; i C = αi E ; R b = ( β + 1 ) ( r e + R E ) ; R e = ( R B + r π ) ⁄ ( β + 1 )
vo
R x ≈ ( 1 + g m R S )r o R x ≈ 1 ⁄ gm + R D ⁄ ( gm ro )
Cascode: i v v v o ⁄ v i ≈ g m ( r o || R D )
–1
i sc ≈ – ( 1 ⁄ g m + R S ) v i i R D v oc ≈ v i RD
RS
vi ( Approx due to g m r o » 1 )
Diff Pair: A d = g m R D ; A CM = – ( R D ⁄ ( 2R SS ) ) ( ( ∆R D ) ⁄ R D ) ; A CM = – ( R D ⁄ ( 2R SS ) ) ( ( ∆g m ) ⁄ g m )
V os = ∆V t ; V os = ( V ov ⁄ 2 ) ( ( ∆R D ) ⁄ R D ) ; V os = ( V ov ⁄ 2 ) ( ( ∆ ( W ⁄ L ) ) ⁄ ( W ⁄ L ) )
–t ⁄ τ AM f t ≈ A M ω 3dB when A M » 1
1st order: step response y(t) = Y ∞ – ( Y ∞ – Y 0+ )e unity gain freq for T ( s ) = ---------------------------
1 + s ⁄ ω 3dB
( 1 + s ⁄ z 1 ) ( 1 + s ⁄ z 2 )… ( 1 + s ⁄ z m )
Freq: for real axis poles/zeros T ( s ) = k dc ----------------------------------------------------------------------------------------
( 1 + s ⁄ ω 1 ) ( 1 + s ⁄ ω 2 )… ( 1 + s ⁄ ω n )
OTC estimate f H = 1 ⁄ ( 2π ∑ τ i ) ; dominant pole estimate f H = 1 ⁄ ( 2πτ max )
Miller: Z1 = Z ⁄ ( 1 – K ) ; Z2 = Z ⁄ ( 1 – 1 ⁄ K )

Mos caps: C gs = ( 2 ⁄ 3 )WLC ox + WL ov C ox ; C gd = WL ov C ox ; C db = C db0 ⁄ ( 1 + V db ⁄ V 0 )

2
f t = g m ⁄ ( 2π ( C gs + C gd ) ) assuming C gd « C gs f t = ( 3µV ov ) ⁄ ( 4πL )
Feedback: A = A ⁄ ( 1 + Aβ ) ; x = ( 1 ⁄ ( 1 + Aβ ) )x ; dA ⁄ A = ( 1 ⁄ ( 1 + Aβ ) )dA ⁄ A ; ω Hf = ω H ( 1 + Aβ ) ; ω Lf = ω L ⁄ ( 1 + Aβ )
f i s f f
Loop Gain L ≡ – s r ⁄ s t ; A f = A ∞ ( L ⁄ ( 1 + L ) ) + d ⁄ ( 1 + L ) ; Z port = Z 0 ( ( 1 + L S ) ⁄ ( 1 + L O ) )
P
PM = ∠L(jω 1) + 180 ; GM = – L(jω 180) dB
Pole Splitting ω p1 ′ ≅ 1 ⁄ ( g m R 2 C f R 1 ) ; ω p2 ′ ≅ ( g m C f ) ⁄ ( C 1 C 2 + C f ( C 1 + C 2 ) )
2 2
Pole Pair: s + ( ω o ⁄ Q )s + ω o = 0 ; Q ≤ 0.5 ⇒ real poles ; Q > 1 ⁄ 2 ⇒ freq resp peaking
ˆ ˆ ˆ 2 2
Power Amps: Class A: η = ( 1 ⁄ 4 ) ( V o ⁄ ( IR L ) ) ( V o ⁄ V CC ) Class B: η = ( π ⁄ 4 ) ( V o ⁄ V CC ) ; P DN_max = V CC ⁄ ( π R L )
2
Class AB: i n i p = I Q
2-stage cmos opamp: ω p1 ≈ ( 1 ⁄ ( R 1 G m2 R 2 C c ) ); ω p2 ≈ ( G m2 ⁄ C 2 ) ; ω z ≈ ( 1 ⁄ ( C c ( ( 1 ⁄ G m2 ) – R ) ) )
ˆ
SR = I ⁄ C c = ω t V ov1 ; will not SR limit if ω t V o < SR

MOS Transistor; CMOS basic parameters. Channel length = 0.18µm

C db0
µC ox C ox ----------
-
Vt λ′ t ox L ov W
( µ A ⁄ V ) ( µm/V )
(V) 2 2
( nm ) ( µm ) fF-
( fF ⁄ µm )  -------
 µm

NMOS 0.4 240 0.05 8.5 4 0.04 0.3

PMOS -0.4 60 -0.05 8.5 4 0.02 0.3

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