跨导电容滤波器

• Integrators
Transconductor
i o = G mV i Io G V ⎛ω ⎞ = m i = ⎜ t i ⎟Vi sC 1 sC 1 ⎝ s ⎠ G ϖ ti = m C1 Vo =
Vi
+ Rin = ∞
+ Gm -
io Vo
C1
Rout = ∞
A single-ended Gm-C integrator
P. K. CHAN Gm-C Filters 4
Gm-C Filters
• Integrators
io + Vi + + - io
I G V Vo = o = m i sC1 sC1
+ Vo C1 Vo
For both circuits
Vo = Vo − Vo io = GmVi
+ −
io + Vi + + -
P. K. CHAN Gm-C Filters 5
Gm-C Filters
• Integrators
2C1
io
+ Vi + + -
Cp +
Vo+
io
+ Vi + + -
+
+ -
+ Vo -
io
{ C1
Cp
io
2C1
Vo =
GmVi Cp ⎞ ⎛ ⎟ s⎜ C1 + ⎜ 2 ⎟ ⎝ ⎠
ωti =
Gm C C1 + p 2
Vo =
2I o GV = m i s(2C1 ) sC1 Gm C1
ωti =
A fully differential integrator maintaining symmetry and showing parasitic capacitances.
P. K. CHAN
A Gm-C op-amp integrator (also called a Miller integrator)
P. K. CHAN Gm-C Filters 15
Gm-C Filters
• Automatic Tuning
Rext VB
Gm
CI VB
Gm
Fixed CI Rext VB V/I
Icntl
Vcntl
(For both circuits) (a) (b) Gm = 1/Rext Constant transconductance tuning. Possible tuning circuits if the transconductors are (a) voltage controlled (b) current controlled.
10
Gm-C Filters
• Second-Order Filter
-ωo -ωo/Q
c
Vin(s)
ko/ωo
1/S
ωo
Vout(s)
u
1/S
k1 + k2S A block diagram of a general second-order continuous filter
P. K. CHAN Gm-C Filters 11
P. K. CHAN
Gm-C Filters
7
Gm-C Filters
• First-Order Filter
Cx
+ Gm2 Vin(s)
+ Gm1 -
Vout(s) CA
A single-ended general first-order Gm-C filter.
P. K. CHAN
Gm-C Filters
Gm-C Filters
• Second-Order Filter
Gm1
2CA
Gm2
2CB 2CB
Gm3
Vout 2CA 2Cx
Gm4
Vin 2Cx
Gm5
A general second-order filter using fully-differential Gm-C technology
P. K. CHAN Gm-C Filters 12
P. K. CHAN Gm-C Filters 16
Gm-C Filters
• Automatic Tuning
Φ1 Φ2 -VB Cm Φ1 Φ2
Gm
CI R1 C1 Vcntl
Gm = fclkCm Vcntl
A frequency – tuning circuit where Gm is set by a switched capacitor circuit. The switched-capacitor branch has a clock frequency of fclk. The low pass filter, R1 C1 is set to a frequency low enough to reduce the high frequency ripple voltage.
Gm-C Filters
V (s) = H ( s ) = out V in ( s ) k 2 s 2 + k1 s + k 0 ⎛ω ⎞ s 2 + ⎜ o ⎟ s + ω o2 ⎜ Q ⎟ ⎝ ⎠ ⎞ ⎛ Gm5 ⎞ ⎛ G m 2G m 4 CX ⎟+⎜ ⎟s + s2⎜ ⎜C +C ⎟ ⎜C +C ⎟ C A (C X + C B ) V out ( s ) B ⎠ B ⎠ ⎝ X ⎝ X = H (s) = V in ( s ) ⎛ Gm5 ⎞ G m 1G m 2 s2 + ⎜ ⎜ C + C ⎟ s + C (C + C ) ⎟ B ⎠ A X B ⎝ X By comparing k2 = k1 = k0 = CX CX + CB Gm5 CX + CB G m 2G m 4 C A (C X + C B ) G m 1G m 2 C A (C X + C B ) Gm3 CX + CB the coefficien ts, we have Q = ⎛ G m 1G m 2 ⎜ ⎜ G3 m3 ⎝ ⎞⎛ C X + C B ⎟⎜ ⎟⎜ CA ⎠⎝ ⎞ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎠
Gm-C Filters
• • • • • Integrators First-Order Filter Second-Order Filter Automatic Tuning A Gm-C Filter Implementation
P. K. CHAN
Gm-C Filters
1
Gm-C Filters
8
Gm-C Filters
G m1Vin ( s ) + sC X [Vin ( s ) − Vout ( s ) ] − sC AVout ( s ) − G m 2Vout ( s ) = 0 Vout ( s ) sC X + G m1 = Vin ( s ) s (C A + C X ) + G m 2 ⎛ k1 ⎞ ∴CX = ⎜ ⎜ 1 − k ⎟C A ⎟ 1 ⎠ ⎝ G m1 = k 0 (C A + C X ) ⎡ ⎛ C X ⎞ ⎛ G m1 ⎢ s⎜ ⎟ ⎜ ⎜C +C ⎟+⎜C +C X ⎠ X ⎝ A ⎝ A =⎣ ⎡ ⎛ Gm 2 ⎞⎤ ⎢s + ⎜ ⎜ C + C ⎟⎥ ⎟ X ⎠⎦ ⎣ ⎝ A ⎞⎤ ⎟⎥ ⎟ ⎠⎦
2C1
Vo =
+
Io s (2C1 )
Vo = −
−
Io s (2C1 )
io
2C1
ω ti =
Gm C1 Vo = 2I o G V = m i s (2C1 ) sC1 Gm C1
ω ti =
Fully differential Gm-C integrators: (a) single capacitor, (b) two capacitors.
P. K. CHAN Gm-C FiltBaidu Nhomakorabears 3
Gm-C Filters
• Integrators
V1 + Gm1 i1
V2
+ Gm2 -
i2 Vo = C1
1 sC1
(Gm1V1 − Gm2V2 + Gm3V3 )
V3
+ Gm3 -
i3
A three-input, single-ended integrator/summer circuit
⎛ k2 CX = CB⎜ ⎜1− k 2 ⎝ G m1 = ω o C A
wher e 0 ≤ k 2 < 1
ω o2 = ωo
Q
P. K. CHAN
G m 2 = ω o (C X + C B ) Gm3 = Gm4 =
=
ω o (C X + C B )
Q k 0C A Gm5 =
Gm-C Filters
P. K. CHAN Gm-C Filters 17
Gm-C Filters
• Automatic Tuning
NIB Φ1
Gm
IB Cm Φ1
CI
Low-pass filter
Φ2 Gm = NfclkCm
Φ2 Vcntl
Vcntl
A switch-capacitor frequency tuning circuit that operates at a lower clock frequency.
Vcntl
Indirect frequency tuning. Note that, although the tuning signal is shown here as a voltage signal , it may also be realized using a set of current controlling signals.
= 2 .513 × 10 7 rad / s
P. K. CHAN
Gm-C Filters
14
Gm-C Filters
Transconductance –C filter
• Automatic Tuning
Vin Vout
Extra transconductor Plus tuning circuitry
ωo
k1 (C X + C B )
13
• Example
Gm-C Filters
Given 2nd order bandpass response, fo=20 MHz, Q=5, G=1, CA=CB=2pF, find the transconductance and capacitor values.
Gm-C Filters 6
Gm-C Filters
• First-Order Filter
− ωo Ko
+ 1/s
Vin(s)
Vout(s)
K1s
A block diagram of a general first-order continuous-time filter.
H (s) =
V out ( s ) k s + ko = 1 V in ( s ) s+ωo
P. K. CHAN
Gm-C Filters
2
Gm-C Filters
• Example What transconductance is needed for unity gain freq. of 20 MHz and 2 pF integrating capacitor ?
G m = 2π x 20 MHz x 2 pF = 0 .251 mA/V
V out ( s ) Q = ωo Vin ( s ) 2 2 s + s + ωo Q Given G = 1, Q = 5 and ω o = 2π × 20 MHz H (s) = ∴ k1 = G G
ωo
s
ωo
Q Since k 0 and k 2 are zero, we have C X = G m 4 = 0 and G m1 = ω o C A = 0 .2513 mA / V G m 2 = ω o (C X + C B ) = 0 .2513 mA / V Hence, G m 3 = G m 5 = k1C B = 50 .27 µ A / V
P. K. CHAN Gm-C Filters 18
Gm-C Filters
• Automatic Tuning
where
+ Vi -
+ + Gm1 - 2C A
+ + Gm2 - -
Vo
Gm 2 = ω o (C A + C x )
-
2C x
0 ≤ k1 < 1
A fully differential general first-order Gm-C filter
P. K. CHAN
Gm-C Filters
w here 0 ≤ k1 < 1
G m 2 = ω o (C A + C X )
P. K. CHAN
Gm-C Filters
9
Gm-C Filters
• First-Order Filter
2C x 2C A
+
Gm1 = ko (C A + C x ) ⎛ k ⎞ Cx = C A ⎜ 1 ⎟ ⎜1− k ⎟ 1⎠ ⎝