A New Method for kTC Noise Analysis in Periodic
Passive Switched-Capacitor Networks
Assim Boukhayma
Christian Enz
CEA-LETI, Grenoble, France
EPFL, Neuchâtel, Switzerland
Email: assim.boukhayma@epfl.ch
EPFL
Neuchâtel, Switzerland
Email: christian.enz@epfl.ch
Abstract—This work presents a new method for kTC noise
calculation in periodic switched capacitor circuits. Two basic
applications are overviewed. Namely the switched capacitor lowpass filter and the N-path band-pass filter. Analytical noise
calculation using the new method is performed for both examples.
The obtained analytical results are confirmed with SpectreRF
Pnoise analysis and ELDO transient noise simulations.
(a) Timing diagram
I. I NTRODUCTION
Switched capacitors (SC) are widely used in analog signal
processing circuits, RF applications [1][2] and particularly
filters [3]. Conventional analog techniques rely on the quality
of elements like resistors or inductors. Such elements suffer
from various non-idealities like mismatch, non linearity at
certain frequencies and high power dissipation. Active SC
circuits offer a good alternative, however they rely on high
gain operational amplifiers. Such active circuits can suffer
from stability and sensitivity problems. Not to mention the
limitations related to the downscaling of technology.
The first implementation of modulated RC passive networks
using periodically operated switches has been introduced in the
early sixties by Franks and Sandberg [4] who gave it the name
of N-path filters. The first implementation of such filters using
integrated CMOS switched-capacitors has been introduced in
[5]. These filters offer many advantages. They allow design of
band-pass and band-reject filters without the use of inductors.
The bandwidth of the N-path filter is independent of the center
frequency. The center frequency of the N-path filter is only
determined by a clock frequency which makes it easily tunable
and robust against mismatch and process defects. Finally the
N-path filter design rely on capacitors, analog switches and
digital circuitry, thus, its performance does not get highly
affected by technology downscale. It has been demonstrated
in recent works [6] that N-path filters offer high selectivity
with a low layout footprint.
SC networks are mainly limited by analog switches nonidealities, charge injection, mismatch and thermal noise commonly referred to as kTC noise. The kTC noise calculation,
in such circuits, using classical methods is difficult as the
noise transfer functions change over time (LTV system). This
paper presents a new method of kTC noise calculation adapted
to passive SC networks. It is organized as follows: Section
II overviews tow basic applications of passive SC networks,
978-1-4799-8893-8/15/$31.00 c 2015 IEEE
(b) SC first order low pass filter
(c) Equivalent circuit
Fig. 1. A schematic and an equivalent circuit of the first-order switchedcapacitor filter.
namely, a low-pass filter and band-pass N-path filter. Section
III introduces the new noise calculation method and presents
a detailed noise calculation of the two practical examples
presented in Section II based on the new method. Section
IV presents noise simulation results confirming the analytical
calculation.
II. S WITCHED CAPACITOR PASSIVE FILTERS
A. Low-pass filter
The schematic of the switched capacitor first-order low-pass
filter and its timing diagram are depicted in Fig. 1. This filter
is based on the equivalence between a resistor and a switched
capacitor. The circuit of Fig. 1 (b) is equivalent to an RC
filter where R is given by TCS0 and TS is the period of the non
overlapping switching as shown in Fig. 1 (a). The derivation
of the transfer function is out of the scope of this work. The
transfer functions of the switched capacitor passive low pass
filters is obtained using SpectreRF PSS and PAC analysis for
a switching frequency fS = 400kHz and C0 = C1 = 10pF .
It is shown in Fig. 2.
400KHz
800KHz
Fig. 2. The transfer function of a first order SC low-pass filter for fS =
400kHz and C0 = C1 = 10pF obtained with SpectreRF PSS and PAC
analysis.
Fig. 5. Impact of
C1
C0
on the quality factor of the N-path filter.
analysis of SpectreRF simulator. The simulation results show
that the selectivity of the N-path filter depicted in Fig. 3 is
independent of the number of paths. In fact the calculated
quality factor of this filter is given by
C
= πr,
(1)
C0
where Ci = C for i = 1, ..., N . The filter quality factor is
only dependent on the ratio r. Fig. 5 shows the impact of
r on the filter selectivity based on SpectreRF PAC simulation
for Ci = C1 = 10pF and C0 = 100f F , 1pF and 10pF .
Q=π
(a) timing diagram
III. N OISE CALCULATION IN SWITCHED CAPACITOR
PASSIVE FILTERS
A. Noise calculation method
(b) schematic
Fig. 3. Schematic of an N-paths filter.
B. Band-pass filter
The N-path filter consists of a network of N passive SC lowpass filters as depicted in Fig. 3. It can be shown analytically
that the transfer function of the N-path filter corresponds to
a periodic bandpass filter centered at the harmonics of N1TS
[7]. Fig. 4 shows the transfer function, for N-paths filters of
N = 4, N = 8 and N = 16, obtained using PSS and PAC
In a periodic switched capacitor network, each capacitor can
be connected in one of the following configurations depicted
in Fig. 6:
• Case (a): Connected through a switch to a voltage source.
• Case (b): Connected through a switch to another capacitor.
Due to thermal noise originating from the ”on” resistance of
switches, a kTC noise is injected in the switched capacitors.
The noise variance of the voltage across a capacitor C in case
(a) can be calculated using the equivalent circuit of Fig. 6 (a).
Z ∞
kT
4kT R
2
Vn,C
=
df =
.
(2)
2
1
+
(2πRCf
)
C
0
(a)
Fig. 4. The simulated transfer function of the N-path filter for N = 4, 8 and
16.
(b)
Fig. 6. Implementations of capacitors in SC networks.
(a) at t = nTS
(b) at t = (n + 12 )TS
(c) at t = (n + 1)TS
Fig. 7. The first order switched capacitor low pass filter equivalent ircuits
during one switching periode from nTS to (n + 1)TS .
In case (b), using the equivalent circuit of Fig. 6 (b). The kTC
noise voltage variance across capacitors C1 and C2 can be
calculated as
Z ∞
C2
kT
4kT R
2
df =
Vn,C
=
2
1
C
C
1
1 + C2
0
1
1+ C
+ (2πRC1 f )2
C2
(3)
and
2
C1
kT
C1
2
2
Vn,C
=
Vn,C
=
.
(4)
2
1
C2
C2 C1 + C2
Each capacitor in case (a) holds the injected kTC noise charge
during one switching period. That noise sample is erased once
the switch is closed at the beginning of the next period.
Each capacitor in case (b) holds a sampled noise from an nth
to an (n + 1)th switching period. Unlike the case (a), that
noise charge held in the capacitor at the end of a switching
period does not get erased in the next one. The variance of
noise sampled at the nth switching period can be considered
2
as a sequence denoted Vn,C
(nTS ), where TS is the switching
period. A study of noise charges injection and sharing in one
period TS starting from nTS determines a recursive relation2
2
ship between Vn,C
((n + 1)TS ) and Vn,C
(nTS ). Based on that
recursive relationship, the expression of the noise variance
of the voltage across the capacitor C can be determined as
a function of n. The noise variance after a high number of
switching periods can be obtained by calculating the limit of
2
Vn,C
(nTS ) when n tends to infinity.
B. Noise calculation for the passive first-order SC low-pass
filter
The first order switched capacitor passive low-pass filter
depicted in Fig. 1 is a direct application of the noise calculation
method presented above. In fact, the capacitor C0 corresponds
to case (a) and C1 corresponds to the case (b). At nTS
(see Fig. 7 (a)), a noise charge is frozen in capacitor C1
2
corresponding to a noise voltage variance of Vn,C
(nTS )
1
across C1 . When Φ1 is opened, capacitor C0 holds a kTC
noise charge corresponding to a voltage variance of kT
C0 based
on (2). At (n + 21 )TS (see Fig. 7 (b)), the switch Φ2 is closed
and the noise charges held in capacitors C0 and C1 at nTS
are shared. Since those noise charges are uncorrelated, the
variance of the total noise charge shared by C0 and C1 is
given by
1
kT
2
Q2n,shared ((n + )TS ) = C02
+ C12 Vn,C
(nTS ).
1
2
C0
(5)
This shared noise charge corresponds to a voltage variance,
Q2
across capacitors C0 and C1 , of (Cn,shared
2.
0 +C1 )
At (n + 1)TS , when the switch Φ2 is opened an additional
kTC noise charge is injected in capacitor C1 . The corresponding voltage variance is calculated using (4). It is given
C0
by kT
C1 C0 +C1 . This injected noise adds to the shared noise
calculated in (5) leading to the expression of the noise voltage
variance across capacitor C1 at the end of the (n + 1)th
switching period
2 2
C02 kT
C0 + C1 Vn,C1 (nTS )
kT
C0
.
C1 C0 + C1
(6)
Equation (6) defines a recursive relationship for the sequence
2
Vn,C
(nTS ) as mentioned in Section III.A. The expression of
1
2
Vn,C
(nTS ) can be derived as
1
2n !
C
kT
1
2
Vn,C
1−
(nTS ) =
.
(7)
1
C1
C1 + C0
2
Vn,C
((n + 1)TS ) =
1
(C0 + C1 )2
+
The noise voltage variance across capacitor C1 after a
high number of switching periods is given by the limit of
2
(nTS ) when n tends to infinity
Vn,C
1
2
2
Vn,C
= lim Vn,C
(nTS ) =
1
1
n→∞
kT
.
C1
(8)
The output noise is simply equal to the one of the equivalent
RC circuit shown in Fig. 1 (c). Thus, the SC resistor acts
exactly the same as a resistor in terms of thermal noise power.
This result suggests also that even if a series of switched
capacitors are used instead of one switched capacitor resistor,
the output noise remains the same and is independent of the
value of the switched capacitors and their period of switching.
C. Noise calculation for the N-path filter
For the N-path filter, each path corresponds to a passive first
order SC filter as depicted in Fig. 3. The noise variance at each
capacitor Ci , for 1 = 1, ..., N , can be calculated the same way
as section III.B since the switch connecting each path to the
output has no impact on the noise sampled in capacitors Ci .
Ci
Thus, for C
= r, the output noise variance converges to
0
2
2
Vn,out
= Vn,C
=
i ,1≤i≤N
kT
,
rC0
(9)
Q
1
where r = C
C0 = π . Thus for a given C0 the kTC noise
contribution of the filter is inversely proportional to its Q
factor.
Fig. 8. Simulated and calculated (8) RMS noise of the SC low pass filter of
Fig. 1.
Fig. 10. 16-path filter simulated output noise with Pnoise analysis for Q
factors of 8π, 32π and 128π.
Fig. 9. Pnoise simulated output noise PSD of a passive SC low pass filter
with 1, 2 and 3 switched capacitors in series with a 10pF output capacitor.
Fig. 11. simulated and calculated (9) RMS noise of the N-path filter of Fig. 3.
V. CONCLUSION
IV. S IMULATION RESULTS
In order to confirm the analytical noise calculation presented
in previous section we performed noise simulations. The
passive SC low-pass filter and the N-path filter are simulated
usign ELDO transient noise simulation and SpectreRF Pnoise
analysis. For both circuits, the switches are modeled with a
resistance whose value determines the ”on” resistance (Ron )
and two ideal switches. All the capacitors and switches of
1
the circuits meet the condition: fS Ron
C fth,max ,
where fth,max is a simulation parameter corresponding to the
maximum frequency of thermal noise. The simulations are
performed with a switching frequency fS = 1M Hz and a
temperature of T = 302K. Fig. 8 shows the simulated output
noise of the passive first order SC low-pass filter, compared
1
to the calculated noise, as a function of the ratio r = C
C0
for C0 = 10pF . Both transient noise and Pnoise simulations
match well with the calculated noise. In order to confirm that a
series of switched capacitors acts exactly the same as a series
of resistors, a Pnoise simulation of a passive SC low pass
filter, using consecutively 1, 2 and 3 switched capacitors in
series with a 10pF output capacitor, is performed. The result
is shown in Fig. 9. The thermal noise PSD changes for the
different configurations but the integrated total noise power
remains the same and corresponds to 20.3µV which is the
kTC noise of a 10pF capacitor. Fig. 10 shows the SprectreRF
Pnoise simulation results for a 16-path filter for quality factors
Q = 4π, 16π and 64π and C0 = 1pF . Fig. 11 shows the
results obtained with ELDO and SpectreRF compared with
the calculated noise of the N-path filter as a function of Q
π.
View publication stats
A new method for kTC noise calculation in periodically
switched capacitor networks is presented. It allows the calculation of the noise voltage variance across each capacitor
of the network by only calculating the noise charge sharing
and injection during one switching period. An application of
this noise calculation method in two examples of passive SC
filters is presented. Both ELDO Transient noise simualtion and
SpectreRF Pnoise analysis show a perfect matching with the
results obtained with the new noise calculation method. Such a
noise calculation technique can be applied to other SC circuits
to evaluate their kTC noise.
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