IEEE Transactions On Circuits & Systems I
Design Of A Monolithic
Passive Notch Filter
Mr. Jun-Hee Lim
University of Southern California
Dr. John Choma*
Professor of Electrical Engineering &
Chair, Electrical Engineering-Electrophysics
*University of Southern California
Department of Electrical Engineering-Electrophysics
University Park: Mail Code: 0271
Los Angeles, California 90089–0271
213–740–4692 [USC Office]
213–740–7581 [USC Fax]
jchoma@usc.edu
ABSTRACT:
The fundamental role of a notch filter in communication systems is the attenuation, if not
outright rejection, of signals whose frequencies lie within a specified passband. If the
passband identified for rejection lies at very high frequencies, passive notch filters prove
more effective than their active counterparts, but their design realization is necessarily
less systematic. In this paper, open circuit impedance parameters comprise the
foundation for a new approach to designing a passive RLC notch filter. The third order
circuit model of the filter is thoroughly analyzed in the interest of achieving reliable
performance prediction and meaningful design optimization. The mathematical analyses
are complemented by HSPICE simulations, wherein requisite inductors are presumed to
derive as planar on chip spiral metallization. These manual and computer-based
investigations lead to a pole-zero cancellation method for controlling the symmetry
between the low and high frequency transfer characteristics of the notch filter.
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IEEE Transactions On Circuits & Systems - Part I
P. Wijetunga & J. Choma
I. INTRODUCTION
Notch filters, which are otherwise known as band reject filters, are ubiquitous in
modern wireless communication systems. Their fundamental purpose is to reduce, if not to
eliminate, the effects of unwanted signals whose Fourier spectrum is dominantly comprised of
frequencies lying in the immediate neighborhood of the frequency at which the notch filter is
designed for maximal signal attenuation. The unwanted frequency is not necessarily the
byproduct of covert operating environments. In a single conversion, superheterodyne receiver,
for example, the unwanted frequency motivating the use of an appropriate notch filter may be the
signal image frequency, which is displaced from the carrier frequency identified for detection by
an amount equal to twice the intermediate frequency (IF) of the channel[1].
Notch filters realized as either active or passive architectures abound in the technical
literature. In general, active filters exploiting only capacitive energy storage elements in
conjunction with operational amplifiers or operational transconductors offer the advantage of
integration ease, particularly in the sense of requisite surface area. They also afford convenient
and often independent electronic tuning of notch frequency and effective quality factor[2]-[5]. But
active notch filter realizations suffer from input/output (I/O) signal transmission limitations
incurred by omnipresent frequency and phase response shortfalls in their utilized active
elements[6]-[7]. They are also plagued by potentially significant electrical noise, nonzero standby
power dissipation, diminished dynamic range, and sensitivities to both active device processing
vagaries and perturbations in the operating temperatures of active devices[8]. While passive
notch filters obviate the shortcomings of their active counterparts, they do present challenges of
their own, in addition to their inability to emulate the attributes of active topologies. Foremost
among these challenges is anemic quality factor in monolithic realizations that incorporate on
chip spiral inductances[9]-[11]. This dilemma has motivated the use of active elements to
compensate for deficient quality factor in otherwise passive filter structures. The most common
compensation scheme features an active negative resistance subcircuit that effectively offsets the
non-zero skin effect and eddy loss resistances of a monolithic inductance. This active
compensation strategy enhances inductor quality factor and thus, the quality factor of the overall
filter[12]-[13]. In another passive notch filter realized as a tuned inductance-capacitance tank, a
transistor is incorporated to achieve quality factor tuning, independent of notch frequency[14].
Unfortunately, the latter innovations remain plagued by potentially serious electrical noise
problems.
Since the notch filter proposed in this paper is earmarked for operation at signal
frequencies extending through a few tens of gigahertz, it is necessarily a passive RLC structure
that can be fully integrated in convention multilayer silicon processes. The theoretical
foundation that supports the filter focuses on the development of an automatic self-locking, selftuning strategy for attenuating signals whose frequencies lie in several narrow bands within the
system passband of interest. As such, the two port characteristics of the filter developed herein
emulates those of tunable band reject filters commonly exploited within phase locked loops to
achieve image signal rejection in communication receivers. By achieving rejection of several
signals whose frequencies lie in proximately located frequency bands, the filter at hand also
behaves effectively as a comb filter. Like earlier notch filter realizations, notch frequency tuning
can be accomplished through adjustments in capacitances that can be implemented as simple
back biased PN junction diodes. But unlike previous notch filter disclosures, the terminating
load impedance is appropriately conditioned to offer a pole-zero cancellation scheme that
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IEEE Transactions On Circuits & Systems - Part I
P. Wijetunga & J. Choma
mitigates the degradation of filter quality factor manifested by capacitances implicit to the
inductor layout.
II. BASIC FILTER TOPOLOGY
The basic schematic diagram of the proposed notch filter appears in Figure (1a), where
resistance R accounts for the net effective resistance of the metallization spiral producing
inductance L. In addition to being designable circuit elements, capacitances C1 and C2
incorporate to first order the oxide and substrate capacitances associated with the inductance.
Resistance Rg comprises the vehicle for determining the notch frequency. It can be implemented
as a simple diode connected MOS transistor to achieve a voltage-controlled notch. It is
interesting to note that the pi subcircuit comprised of the elements, L, R, C1, and C2, is
reminiscent of both a lumped approximation of a lossy transmission line and the approximate
circuit model of a monolithic inductance.
Using the open circuit impedance parameters (z-parameters), zij(s), to model the
aforementioned pi subcircuit, the circuit in Figure (1a) can be represented by the structure
depicted in Figure (1b). It is a straightforward matter to show that the impedance parameters,
z11(s), z22(s), and z21(s), are given by
z11 (s) 
z22 (s) 
V1
I1




 1  sRC2  s 2 LC2 
1
,


2
s
C
C



1
2




I2 0
s
s
 
 
1 
Qn ωn
 ωn  

(1)
V2
I2




 1  sRC1  s 2 LC1 
1
,


2
s
C
C



1
2
 s  

I1  0
s
 
 
1 
Qn ωn
 ωn  

(2)
and
z21 (s) 
V2
V
 1
I1 I  0
I2
2






1
1
.


2
s
C
C



1
2




I1  0
s
s
 
 
1 
Qn ωn

 ωn  
(3)
In (1) -through- (3),
ωn 
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(4)
 C C 
L 1 2 
 C1  C2 
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IEEE Transactions On Circuits & Systems - Part I
P. Wijetunga & J. Choma
is the undamped natural frequency of the z-parameters, while the quality factor, Qn, of these
parameters derives from
 C C 
1
 R 1 2  .
Qn ωn
 C1  C2 
(5)
The combination of (4) and (5) leads to
Qn 
ωn L
,
R
(6)
which suggests that Qn is the quality factor of the inductor at frequency ωn.
An inspection of Figure (1b) reveals that the open circuit, or Thévenin, voltage transfer
function, HT(s), is
HT (s) 
z21 (s)  Rg
Vo
.

Vi I  0
z11 (s)  Rg
2
(7)
Since the filter transfer characteristic with a load terminating its output port is proportional to the
Thévenin transfer function, it is entirely proper to define the radial notch frequency, ωo, in
accordance with the implicit requirement,
z21  jωo   Rg  0 .
(8)
It should be clear that since the circuit in Figure (1a) is obviously physically realizable, which
ensures that the open circuit input impedance, z11(jω), is a positive real function for all values of
frequency ω, z11(jωo) + Rg cannot possibly be null. If (3) is inserted into (8), the notch frequency
is found to be identical to the undamped natural frequency of the z-parameters for the pi
subcircuit; specifically,
ωo  ωn 
1
 C C 
L 1 2 
 C1  C2 
.
(8)
Moreover, the combination of (3), (6), and (7) confirms that resistance Rg must be selected to
satisfy
Rg 
Qn
L

.
ωo  C1  C2 
R  C1  C2 
(9)
In addition to establishing a notch frequency at the designable value, ωo, the frequency
response of the open circuit transfer function displays symmetry about ωo. To this end, note
from (1) and (3) that z11(j0) = z21(j0) = ∞, while z11(j∞) = z21(j∞) = 0. Accordingly, (7) shows
that
HT (j0) 
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HT (j  )  1 .
(10)
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IEEE Transactions On Circuits & Systems - Part I
P. Wijetunga & J. Choma
REFERENCES
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Cambridge University Press, 2004, pp. 698-699.
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Transconductance Amplifiers: A Tutorial,” IEEE Circuits and Devices Mag., pp. 20-32, Mar.
1985.
[3]. M. Steyaert, J. Crols, and S. Gogaert, “Low-Voltage Analog CMOS Filter Design,” in LowVoltage/Low-Power Integrated Circuits and Systems, E. Sánchez-Sinencio and A. G. Andreou
(eds.). New York: IEEE Press, 1999, chap. 10.
[4]. J. Rogers and C. Plett, “A 5 GHz Radio Front-End With Automatically Q Tuned Notch Filter,”
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[6]. H. J. Orchard, “Gyrator Circuits,” in Active Filters: Lumped, Distributed, Integrated, Digital,
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3.
[7]. T. Bakken and J. Choma, Jr., “Gyrator-Based Synthesis Of Active On Chip Inductances,”
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[8]. Y. Chang, J. Choma, Jr., and J. Wills, “An Inductorless Active Notch Filter for RF Image
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[9]. S. S. Mohan, “Modeling, Design, and Optimization of On-Chip Inductors and Transformers,”
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[11]. C. P. Yue and S. S. Wong, “On-Chip Spiral Inductors With Patterned Ground Shields for SiBased RF IC’s,” IEEE J. Solid-State Circuits, vol. 33, pp. 743-752, May 1998.
[12]. H. Samavati, T. H. Rategh, and T. H. Lee, “A 5 GHz CMOS Wireless LAN Receiver Front
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[14]. J. Macedo, and M. A. Copeland, “A 1.9 GHz Silicon Receiver With Monolithic Image Reject
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