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Chung-Yu Wu - Analog Circuit Design

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13-85

CHUNG-YU WU

[28]C. L. Portmann and T. H. Y. Meng, "Power-efficient metastability error reduction in CMOS flash A/D converters," IEEE J. Solid-State Circuits, vol. 31, no. 8, pp. 1132-1140, Aug. 1996.

[29]M. P. Flynn and D. J. Allstot, "CMOS folding converter with current-mode interpolation," IEEE J. Solid-State Circuits, vol. 31, no. 9, pp. 1248-1257, Sep. 1996.

[30]D. W. Cline and P. R. Gray, "A power optimized 13-b 5 Msamples/s pipelined analog-to- digital converter in 1.2/spl mu/m CMOS," IEEE J. Solid-State Circuits, vol. 31, no. 3, pp. 294-303, Mar. 1996.

[31]M. P. Flynn and B. Sheahan, "A 400-Msample/s, 6-b CMOS folding and interpolating ADC," IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 1932-1938, Dec. 1998.

[32]S. Tsukamoto, W. G. Schofield, and T. Endo, "A CMOS 6-b, 400-Msample/s ADC with error correction," IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 1939-1947, Dec. 1998.

[33]J. M. Ingino and B. A. Wooley, "A continuously calibrated 12-b, 10-Ms/s, 3.3-V A/D converter," IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 1920-1931, Dec. 1998.

[34]I. E. Opris, L. D. Lewicki, and B. C. Wong, "A single-ended 12-bit 20 Msample/s selfcalibrating pipeline A/D converter," IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 18981903, Dec. 1998.

[35]H. van der Ploeg and R. Remmers, "A 3.3-V, 10-b, 25-Msample/s two-step ADC in 0.35-um CMOS," IEEE J. Solid-State Circuits, vol. 34, no. 12, pp. 1803-1811, Dec. 1999.

[36]B. P. Brandt and J. Lutsky, "A 75-mW, 10-b, 20-MSPS CMOS subranging ADC with 9.5 effective bits at nyquist," IEEE J. Solid-State Circuits, vol. 34, no. 12, pp. 1788-1795, Dec. 1999.

[37]I. Mehr and D. Dalton, "A 500-Msample/s, 6-bit nyquist-rate ADC for disk-drive readchannel applications," IEEE J. Solid-State Circuits, vol. 34, no. 12, pp. 912-920, Dec. 1999.

14-1 CHUNG-YU WU

CH 14. MOS Switched-Capacitor Filter Design

§14--1 Preliminary Considerations

§14-1.1 Classification of systems and filters

1. Continuous-time, discrete-time, and sampled-data systems

Ampt.	Input x(t)	Ampt.	T	Input x(kt)	Ampt.	T	Input x(kt)
	Loss			Loss
		t			t		t
	Output y(t)			Output y(kt)			Output y(kt)
	τD			Output y(kt)		τD	Output y(kt)
	τD		τD	k: integer		τD

continuous-time system			sampled-data system	discrete-time system
e.q.: analog Filter			e.q.: SCF	e.q.: digital filter
differential equations			difference equations
2. Tine-invariant systems and causal systems
T.I.	:	x(kt)→ y(kt)	> x[(k-n)T]→ y[(k-n)T] for any x(kt) and n.
Causal	:	x(mt) => y(kT)=0 for k<m

3. Filter types:
(1) Low-Pass(LP)						H ( jω)
	Kωp			2		dB
H(s)= S 2 + (ω	Kωp			2
H(s)= S 2 + (ω		/ Q			)S +ω2			-N*20dB/decade
	p		p		p	Ideal		N: order
						Ideal
(biquad)						ωp	ωs	ω
Two complex poles (LHP)								ω
Two complex poles (LHP)

14-2 CHUNG-YU WU

(2) High-Pass(HP)

H(s)=	KS 2
	S 2 + (ωp / Qp )S +ωp2

Two complex poles(LHP)

Two zeros at S=0

	H ( jω)			ωp
	dB			ωp		-N*10dB/decade
	dB			Qp		N: order
3) Band-Pass(BP)
	-3dB
K(ωp / Qp )S		+N*10dB/decade
K(ωp / Qp )S		N: order
H(s)= S 2 + (ωp / Qp )s +ωp2		N: order
H(s)= S 2 + (ωp / Qp )s +ωp2
center freq. gain: k
center freq.	ωp	ωs1	ωp1	ωp	ωp2	ωs2	ω
Two complex poles (LHP)		ωs1	ωp1	ωp	ωp2	ωs2	ω
Two complex poles (LHP)
One zeros at S=0

4)Band-Reject(BR)

H(s)=	K(S 2 +ωz2 )
	S 2 + (ωz / Qp )s +ωp2

ωp=ωz

Two complex poles(LHP) Two imaginary zeros

H ( jω)
dB
ωz	ω


ωp>ωz	High-Pass Notch filter (HPN)
ωp<ωz	Low-Pass Notch filter (LPN)
(5)Low-Pass Notch (LPN)		(6)High-Pass Notch

H ( jϖ)			H ( jϖ)
dB			dB
0dB			0dB
ωp	ωz	ω	ωz	ωp	ω

14-3 CHUNG-YU WU

(7)ALL-Pass (Delay Equalizer)

H(S)= S 2 −(ωp / Qp )S +ωp2

S 2 + (ωp / Qp )S +ωp2

Two complex poles (LHP) Two complex zeros (RHP) mirror-imaged

0dB	Gain
	Gain
	-180o
	phase
ωp	ω

§14-1.2 Sampling Process

§ ideal impulse sampling:
∞
S(t)= Sδ (t)= ∑S(t −kτ)
k =−∞
τ: sampling period
		∞	∞
xd(t)=x(t)Sδ (t)=x(t) ∑δ(t −kτ )= ∑x(t)δ(t −kτ)
		K =−∞	k =−∞
remember: ∫−∞∞δ(t − kτ)dt =1			δ(t −kτ) =0 for t ≠ kτ
∞
> xd(t)= ∑x(kτ)δ(t −kτ)
k =−∞
					∞
Fourier transformation of Sδ (t): SF(t)= ∑ ck e jkωst
				k =−∞
	1	τ		1
Where Ck ≡	1	∫−2τ s(t) e jkωst dt=		1
Where Ck ≡	τ	∫−2τ s(t) e jkωst dt=		τ
2

	∞
>xd(t)=x(t)SF(t)= ∑Ck x(t)e jkωst
	k =−∞
∞	∞
F[xd(t)]=F[ ∑	Ck x(t)e jkωst ] = ∑Ck F[x(t)e jkωst ]
k =−∞	k =−∞

multiplier

x(t)

xd(t)

S(t)=Sδ(t) for ideal impulse sampling

ωs ≡ 2τπ

∞
= ∑Ck X(jω-jkωs)	where F[x(t)] ≡X(jω), k=±integer
k =−∞
	base-band spectrum

14-4 CHUNG-YU WU

X(jω)

				−ωc		0	ωc			ω
Aliasing:				−ωc		0	ωc			ωs<2ωc
Aliasing:							X(jω-jkωs)			ωs<2ωc
introduces an
ambiguity into
X(jω-jkωs)
and prevents				−ωs−ωc						ω
the eventual				−ωs−ωc		0	ωc ωs		2ωc	ω
recovery of							X(jω-jkωs) no
X(jω)							X(jω-jkωs) no			ωs>2ωc
							aliasing
−ω	s	−2ω		−ω	c	0	ω		2ω ω	ω
−ω		−2ω	c	−ω		0	ω	c	2ω ω	s
			c					c	c	s

Sampling Theorem:

A function x(t) that has a Fourier spectrum X(jω) such that X(jω)=0 for

ω ≥ωs 2 is uniquely described by a knowledge of its values at uniformly spaced

time instants,τ instants apart(τ = 2π / ws )

2ωc: Nyquist rate. Anti-aliasing filter is required.

Reconstruction filter is also required to recover x(t).

spec

			H ( jω)		Transition				The smaller the ωs-2ωc (TB),
			H ( jω)

									the higher the filter order!
						band			the higher the filter order!
						band
				Pass				Stop
				Pass				band
				band

−ω +ω	c	−ωc			ωc	ω	-ω	c		ω
−ω +ω		−ωc			ωc	ω	-ω			ω
s			0			s
			0

ωs-2ωc

14-5 CHUNG-YU WU

§ Finite -Pulse Sampling (non-ideal sampling):

∞

Sp(t)= ∑

[u(t −kτ −

) −u(t

−kτ +

)]

a>0

k =−∞

Ck=

Sp(t) e− jkωst dt

∫−2τ

1 ∫−2a e− jkωst dt = a Sin(kωsa / 2

τ kωsa / 2

τ 2τ 3τ 4τ

Now, we have sinα /α envelope onto X (jω-jkωs)xd(t)

α ≡ kωsa / 2

Pulse Amplitude Modulation

(PAM)

τ	2τ
§14-1.3 Z-Transformation	3τ	4τ	t
§14-1.3 Z-Transformation			t

∞

xd(t)= ∑x(kτ)δ(t − kτ)

k =−∞

∞

Laplace Transformation >

Xd(s)=L[xd(t)]= ∑x(kτ)e−ksτ

k =−∞

Let z=esτ

∞

> X(z)= ∑X (kτ)z−k

two-sided

z-transform

S=jω

z=ejωτ

k =−∞

∞

one-sided

z-transform

X(z)= ∑X (kτ)z−k

k =0

∞

example 1:

x(t)=u(t) >x(kτ) =1 >X(z)= ∑Z −k =

− z

−1

K =0

-at

−akτ

∞

-akz -k

example 2:

x(t)=e

u(t) >x(kτ) = e

>X(z)= ∑e

1−e

−aτ

−1

k =0

for

>e-aτ

For single input/output, linear, time-invariant, sampled data (or discrete-time) system:

N	M
y(kτ) + ∑bn y[(k-n)τ] = ∑an x[(k-n)τ]
n=1	n=0

M.N: non-negative integers

14-6 CHUNG-YU WU

Two cases:(1) bn = 0 for all n > nonrecursive system

M+1 tap transversal filter

Finite-Duration Impulse Response(FIR)Filter

(2)

bn ≠ 0

for n ≥ 1

Nth-order recursive system

z-transform:

Infinite Impulse Response(IIR) Filter

Y(z)(1+ ∑bn Z

−n ) =X(z)

∑an z −n

n=1

n=0

H(z)=

Y (z)

∑an z −n

pulse transfer function

n=0

X (z)

1+ ∑bn z −n

n=1

z=αi: poles

= ao (1 − β1Z

−1

)(1 − β

2 Z

−1

)......(1 − βN Z

−1

)

z=βi: zeros

(1 −α1Z

−1

)(1 −α2 Z

−1

)........(1 −αN Z

−1

)

Mapping between Z-plane and S-plane:

sτ

στ

jωτ

2π

z=e

s=σ+jω

z=e e

τ=

For ω

>2ω , the base-band response X(jω) over the range- − ωs ≤ω ≤ω

/ 2 is

sufficient to determine X(jω) for all ω.

* −

ωs ≤ω ≤ωs

/ 2 , ∞<σ<∞

=> all Z-plane Z = −π →π

−

3ωs

≤ω ≤ −

ωs

, ∞<σ<∞

=> overlap on Z-plane

Z = −3π → −π

ωs ≤ω ≤

3ωs

∞<σ<∞

=> overlap on Z-plane

Z =π →3π

* ω=ω1,

∞<σ<∞ => a straight line from z=0 to z= ∞ with angle ω1τ

* jω axis => σ=0 =>

=1 unit circle

*σ>0

>1 for all ω => RHP → outside the

circle

*σ<0

<1 for all ω => LHP → inside the

circle

14-7 CHUNG-YU WU

* S=0	z=1	; ω=0 ∞<σ<+∞ > Real Z axis (positive)
First-order transfer function:					jω
First-order transfer function:
	1
H(z)=	1−az−1 Z=a is the pole				ω1		τ2
h(kτ) k=0, 1, 2, 3,…….
z=a=eστejωτ					0		σ
(1)a>1, diverging			unstable		τ1
(1)a>1, diverging			unstable			stable region
(2)a=1	sequence of 1's		unstable			stable region
(3)a≥ 0	a<1		stable		Im Z
					Im Z
(4) 1<a≤ 0			stable	σ<0,	ωτ=±π	τ	2
					ωkτ=±kπ	τ	2
(5)a= 1			unstable		ωkτ=±kπ
(5)a= 1			unstable			ω1τ
						ω1τ
(6)a< 1			unstable		τ1		Ro Z
(6)a< 1			unstable		τ1	Z =1
						Z =1
G(ω)=20log[ H (Z) z=ejωτ ]				dB	magnitude
	φ(ω)=tan-1	Im H (Z)		rad	phase
	φ(ω)=tan-1	Re H (Z)	z=ejωτ	rad	phase

The magnitude and phase can be determined graphically in the same way as those determined from the s-plane poles & zeros.

§14-1.4 Sample and Hold Circuit

Zero-order hold or S/H function:

Ho(s)=	1−e−sτ

		sτ
Ho(jω)=e		-jωτ/2	sin(ωτ / 2)
			ωτ / 2

14-8 CHUNG-YU WU

					X d ( jω)		h0(t)
							τ
			X r ( jω)		H0 ( jω)		1/
			X r ( jω)		H0 ( jω)		Impulse response
							Impulse response
−2ωS	−ωS	−ωC	+ωC	ωS	2ωS	ω	ho(t)= τ1 u(t)- τ1 u(t-τ)
			HO ( jw)
							τ
			π
−2ωS	−ω			ωS	2ωS	ω
−2ωS	−ω
	S	−π					x(t)
							xr(t)
							τ
							x(k)
* may serves as a reconstruction circuit							xr(t)=x(t) h0(t)
	Xr(jω)=Xd(jω)Ho(jω)
* The different between Xr(jω) & Xd(jω) at ω ±ωc can be eliminated by setting
ωs/ωc>>1.

§14-2 Switched-Capacitor Network System

General Switched-Capacitor Network (SCN):

ideal capacitors, ideal voltage-controlled-voltage sources (VCVS's), ideal switches & sampled-data voltage inputs.

VCVS: freq. indep. gain amps or infinite gain OP amps.

*Typically, the sampled-data voltage input is only single, not multiple.

*The input may be a continuous one.

*The effects of non-ideal switches, non-ideal OP amps, & non-ideal cap. should be considered as & second order effects.

14-9 CHUNG-YU WU

Block diagram:

Continuous

Anti-aliasing

Filter

(S/H)i
φ	φ

Switched-

(S/H)o

Continuous

Capacitor

Reconstruction

Network

Filter

Switched-Capacitor Network (Two-phase clock) (can be multi-phase)

General symbols:

φe φo

e	o	e	o
		e	o
V1 (kT )	C V2 (kT ) ≡ V1 (kT )		C V2 (kT ) ≡ V1 (kT )	C V2 (kT )

φe

even clock

Tc<T

φo

to avoid overlapping

odd clock

of φ

and φ°

τ: sampling period

τ=2T

*Generally, SCN is time-variant since the network topology is different in the case of φe and φ°. However, if we separate the input/output sampled-data voltage into one even component and one odd component and separate the whole SCN into one even part and one odd part, then we have two time-invariant networks coupled together. Analysis thus can be performed.

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