CMOS Integrated Microwave Filters

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Motivation

Filters are often required in communication systems to remove interfering channels, reduce noise levels, and prevent amplifier saturation. They are often a limiting factor in wireless system design (in terms of cost, performance, size, etc.)

Common implementations include:

Surface acoustic wave (SAW)
Ceramic and crystal resonator (very narrowband notch and bandpass filters)
Microstrip coupled line filters
Lumped element (very flexible)
Microstrip stub filters (very flexible)

We will examine methods of designing planar filters using lumped elements and distributed elements. Other methods, such as coupled line filters, will not be discussed here.

Resources

Networks of commonly used RF lumped element filters

A good summary of microwave filter design is at: Microwave Filter Design Theory

Filter Design by Insertion Loss Method

Filter design may be approached from several perspectives. We will use an approach that is general, and is commonly used - the insertion loss method. The procedure is as follows:

Decide on filter specifications, including cutoff frequency, passband attenuation, and rolloff
Design a low pass prototype (maximally flat, equal ripple, elliptic, linear phase)
Scale in frequency and impedance and translate to the appropriate response (highpass, bandpass, bandstop)
Implement with appropriate elements (LC, stubs, high-Z-low-Z)

The response of a filter is often defined by its insertion loss or power loss ratio, defined by

$P_{LR} = \frac{Power available from source}{Power delivered to load} = \frac{P_{inc}}{P_{load}} = \frac{1}{1-|\Gamma(\omega)|^2}$

We want

Low insertion loss in the passband
High insertion loss in the stopband
Fast rolloff at cutoff

We can't practically have all three so need to decide what is most important. There are several responses we may chose from; we will look at two of them - equal ripple (Chebeshev) and maximally flat (Butterworth) responses. A comparison of RF filter responses is at RFCafe.com which compares filter responses by magnitude response and group delay.

Comparison of responses of four filter types. Source: http://en.wikipedia.org/wiki/Image:Electronic_linear_filters.svg

Maximally Flat (Butterworth)

Response of orders 1-5 of a prototype Butterworth filter. Image source: Wikipedia, http://en.wikipedia.org/wiki/Image:Butterworth_Filter_Orders.svg

A butterworth filter has the flattest passband response, but relatively poor rolloff. The low pass insertion loss is given by

$P_{LR} = 1 + \left( \frac{\omega}{\omega_c} \right)^{2N}$

where $ω c$ is the cutoff frequency and $N$ is the order of the filter. At $ω = ω c$ , the insertion loss is 1/2, i.e. 3 dB down from the passband. The response rolls off as 20 dB/decade/order.

Equal Ripple (Chebyshev)

Chebyshev filters offer a sharper cutoff, but have a ripple in the passband; the amount of allowable ripple can be specified in the design. The insertion loss for a low pass design is given by

$P_{LR} = 1 + k^2 T_N \left( \frac{\omega}{\omega_c} \right)$

where $T N (x)$ is the Chebyshev polynomial. $T N (x)$ oscillates between $\pm 1$ for $|x| \le 1$ . For large $x$ , the insertion loss increases at a rate of 20N dB/decade.

Low pass prototype design

Any lowpass LC prototype can be converted to any other type (bandpass, highpass, bandstop) by the appropriate transformation. This will change the type and cutoff frequency, but leaves the rolloff, passband ripple, etc., the same. It may then be converted to another implementation (stubs, high-Z-low-Z, etc.)

Lowpass prototypes, of which there are two types, are used for initial filter design.

Low pass prototypes come in two forms, and we need to find $g 1$ , $g 2$ , etc., as shown on the right. Note that either prototype will work, but one may produce more practical component sizes than the other.

In the lowpass prototype, we assume that the cutoff frequency $ω c = 1$ , selected for ease of calculation. This is selected because the impedance of a series inductor and the admittance of a shunt capacitor are linearly related to frequency. If we scale the values of L and C in the lowpass prototype, the response will be the same, but shifted in frequency.

We can calculate $g 1$ , $g 2$ , etc., for the lowpass prototype using the equations for $P L R$ and the impedance of the LC network. For example, for a Butterworth filter the coefficients are given as (source: Wikipedia, Butterworth filter)

$g_k = 2 \sin \left [\frac {(2k-1)}{2n} \pi \right ]$

Similar equations exist for other topologies like Chebyshev. However, these coefficients have already been calculated and are in many texts and online sites on filter design. Tables for Chebeshev and Butterworth prototype filters are at RFCafe.com.

For a given response (maximally flat or equal ripple) and a given order, we can look up the appropriate values for L and C in the lowpass prototype.

More detailed analysis of these low-pass fiter prototypes is in Optimal Use of Some Classical Approximations in Filter Design, H. Dimopoulos, IEEE TCASII, September 2007.

Scaling and Translation

Once the lowpass prototype is designed, the impedances of filter elements are scaled to the terminating impedance of the filter which is usually the system impedance $Z o$ , and translated to either a lowpass response with the appropriate frequency, or to a highpass, bandpass, or bandstop type. The appropriate scaling and translation for a lowpass, highpass, bandpass, and bandstop filters are as follows:

Prototype Element	Lowpass	Highpass	Bandpass	Bandstop
Inductor $L k$	$L_k ' = \frac{L_k Z_o}{\omega_c}$	$L_k ' = \frac{Z_o}{C_k \omega_c}$	Series LC circuit composed of: $L_k ' = \frac{L_k Z_o}{\omega_c \Delta}$ $C_k ' = \frac{\Delta}{Z_o \omega_c L_k}$	Shunt LC circuit composed of: $L_k ' = \frac{L_k Z_o \Delta }{\omega_c}$ $C_k ' = \frac{}{Z_o \omega_c L_k \Delta}$
Capacitor $C k$	$C_k ' = \frac{C_k}{Z_o \omega_c}$	$C_k ' = \frac{1}{Z_o L_k \omega_c}$	Shunt LC circuit composed of: $L_k ' = \frac{\Delta Z_o}{\omega_c C_k}$ $C_k ' = \frac{C_k}{Z_o \omega_c \Delta}$	Series LC circuit composed of: $L_k ' = \frac{Z_o}{\omega_c \Delta C_k}$ $C_k ' = \frac{C_k \Delta}{Z_o \omega_c}$

For lowpass and highpass filters, $ω c$ represents the cutoff frequency of the filter. For bandpass and bandstop filters, $\Delta = \frac{\omega_2 - \omega_1}{\omega_c}$ is the percent bandwidth of the filter when $ω 2$ , $ω 1$ , and $ω c$ are the top, bottom, and centre of the band of interest, respectively.

Example: Bandpass filter design

Design a bandpass filter with 0.5 dB equal-ripple (Chebyshev) response, N=3, center frequency of 1 GHz, a 10% bandwidth, and a system impedance of 50 Ohms.

From tables, the lowpass prototype for a 0.5 dB equal-ripple response has coefficients given by

g 1 = 1.5963 = L 1

g 2 = 1.0967 = C 2

g 3 = 1.5963 = L 3

For a 10% bandwidth ( $Δ = 0.10$ ) and $ω o = 1$ GHz, we can use our expressions to transform a lowpass prototype to a bandpass design:

Transformation of a lowpass prototype to a bandpass filter.

$L_1' = \frac{L_1 Z_o}{\omega_o \Delta} = 127 nH$

$C_1' = \frac{\Delta}{\omega_o L_1 Z_o} = 0.199 pF$

$L_2' = \frac{\Delta Z_o}{\omega_o C_2} = 0.726 nH$

$C_2' = \frac{C_2}{\omega_o \Delta Z_o} = 34.91 pF$

$L_3' = \frac{L_3 Z_o}{\omega_o \Delta} = 127 nH$

$C_3' = \frac{\Delta}{\omega_o L_3 Z_o} = 0.199 pF$

Distributed Filter Transformation

Richard's Transformation can be used to convert inductors and capacitors in a filter prototype into distributed transmission line elements.

At microwave frequencies the parasitics of lumped elements may cause the filter's behaviour to significant deviate from expected. It is often desirable to convert a lumped element filter into a distributed filter. This can be done quite simply using Richard's Transformation, which converts lumped elements into a $λ / 8$ section of transmission line with $Z o$ equal to the reactance of the element.

For example, an inductor can be transformed to an open circuited transmission line. Recall that the input impedance of an open circuited lossless transmission line is simply

Z i n = j Z o tan(β l)

where $Z o$ is the characteristic impedance of the line, $β$ is the phase constant, and $l$ is the line length. In the case where the line is $λ / 8$ long, $\beta l = \frac{2 \pi}{\lambda} \frac{\lambda}{8} = \frac{\pi}{4}$ , in which case the input impedance to the line is simply

$Z_{in} = j Z_o \tan (\beta l) = j Z_o \tan \left(\frac{\pi}{4} \right) = j Z_o$

As a result, we simply pick the characteristic impedance of the line to be equal to the desired reactance of the inductor:

j Z o = j ω L = j X L

We can determine the necessary physical length of the line quite simply:

$l = \frac{\lambda}{8} = \frac{c}{f \sqrt{\epsilon_{eff}}} \frac{1}{8}$

where $c$ is the speed of light, $ε e f f$ is the effective permittivity of the transmission line, and $f$ is the frequency of interest.

A similar analysis can show that an open circuited $λ / 8$ line looks like a capacitor. The susceptance of a capacitor $j B C = j ω C$ maps to an input admittance of $j C tanβ l$ .

Hence:

an inductor with impedance $j X L$ can be converted to a short-circuited $λ / 8$ transmission line with characteristic impedance $Z o = X L$
a capacitor with susceptance $j B C$ can be converted into an open-circuited $λ / 8$ transmission line with characteristic impedance $Z o = 1 / B C$

Kuroda's Identity

Kuroda's identity can be used to convert a series stub into a shunt stub.

When we transform lumped elements into distributed stubs using Richard's tranformation, the series elements transform to series stubs. However, it is often difficult to implement a series stub, e.g. it is very difficult when using microstrip lines. To get around this problem, we can make use of one of Kuroda's Identities to convert series stubs into a shunt stubs. This requires that a unit element ( $λ / 8$ , $Z o = 1$ ) of transmission line be available. Since a unit length of transmission line may always be added to either port, this is not a problem.

Stepped Impedance Filters

A simple low-pass filter can be designed using a microstrip line with alternating characteristic impedances.

A simple way to implement a distributed filter is to convert inductors and capacitors directly into sections of transmission line with high characteristic impedance ( $Z h$ ) and low characteristic impedance ( $Z l$ ), respectively. The necessary electrical lengths of the transmission lines are

$\beta l_k = \frac{g_k Z_o}{Z_h}$

for a lowpass prototype inductance $g k$ and

$\beta l_k = \frac{g_k Z_l}{Z_o}$

for a lowpass prototype capacitance $g k$ .

The derivation and more detail regarding the stepped-impedance filter is here. Precomputed values for these filters are in Tables of Element Values for the Distributed Low-Pass Prototype Filter, R. Levy, IEEE MTT, September 1965, pp. 514-536.

Implementation using commercially available standard processes

One of the most significant issues related to filter design in CMOS processes is the relatively low-Q available. The first paper below provides a good comparison, in Figure 1, between using ideal elements and using elements with a Q of 6.

A 0-dB IL 2140±30 MHz bandpass filter utilizing Q-enhanced spiral inductors in standard CMOS, Soorapanth, T. Wong, S.S., IEEE JSSC, May 2002, Vol. 37, Iss. 5, pp. 579-586.

Abstract: A 3-pole Chebyshev bandpass filter, that employs on-chip passive elements with Q-enhancement technique, achieves an insertion loss of 0 dB and a passband of 60 MHz around a center frequency of 2140 MHz. The Q-enhancement technique is based on coupled-inductor negative resistance generator. In contrast to conventional negative resistance generator, this technique compensates resonator loss without introducing distortion in the filter response in the passband. Fabricated in a 0.25-μm CMOS, the filter consumes 7 mA from a 2.5-V supply. The filter occupies an area of 1.3 mm×2.7 mm.

A 1.3-V 5-mW fully integrated tunable bandpass filter at 2.1 GHz in 0.35 um CMOS, Dulger, F. Sanchez-Sinencio, E. Silva-Martinez, J., IEEE JSSC, June 2003, Volume: 38, Issue: 6, pp. 918- 928

Abstract: A 2.1-GHz 1.3-V 5-mW fully integrated Q-enhancement LC bandpass biquad programmable in fo, Q, and peak gain is implemented in 0.35 um standard CMOS technology. The filter uses a resonator built with spiral inductors and inversion-mode pMOS capacitors that provide frequency tuning. The Q tuning is through an adjustable negative-conductance generator, whereas the peak gain is tuned through an input Gm stage. Noise and nonlinearity analyses presented demonstrate the design tradeoffs involved. Measured frequency tuning range around 2.1 GHz is 13%. Spiral inductors with Qo of 2 at 2.1 GHz limit the spurious-free dynamic range (SFDR) at 31-34 dB within the frequency tuning range. Measurements show that the peak gain can be tuned within a range of around two octaves. The filter sinks 4 mA from a 1.3-V supply providing a Q of 40 at 2.19 GHz with a 1-dB compression point dynamic range of 35 dB. The circuit operates with supply voltages ranging from 1.2 to 3 V. The silicon area is 0.1 mm2.

Millimeter-wave CMOS design, Doan, C.H. Emami, S. Niknejad, A.M. Brodersen, R.W., IEEE JSSC, Jan. 2005, Volume: 40, Issue: 1, pp. 144- 155

Abstract: This paper describes the design and modeling of CMOS transistors, integrated passives, and circuit blocks at millimeter-wave (mm-wave) frequencies. The effects of parasitics on the high-frequency performance of 130-nm CMOS transistors are investigated, and a peak f/sub max/ of 135 GHz has been achieved with optimal device layout. The inductive quality factor (Q/sub L/) is proposed as a more representative metric for transmission lines, and for a standard CMOS back-end process, coplanar waveguide (CPW) lines are determined to possess a higher Q/sub L/ than microstrip lines. Techniques for accurate modeling of active and passive components at mm-wave frequencies are presented. The proposed methodology was used to design two wideband mm-wave CMOS amplifiers operating at 40 GHz and 60 GHz. The 40-GHz amplifier achieves a peak |S/sub 21/| = 19 dB, output P/sub 1dB/ = -0.9 dBm, IIP3 = -7.4 dBm, and consumes 24 mA from a 1.5-V supply. The 60-GHz amplifier achieves a peak |S/sub 21/| = 12 dB, output P/sub 1dB/ = +2.0 dBm, NF = 8.8 dB, and consumes 36 mA from a 1.5-V supply. The amplifiers were fabricated in a standard 130-nm 6-metal layer bulk-CMOS process, demonstrating that complex mm-wave circuits are possible in today's mainstream CMOS technologies.

Implementation using customized processes

High-performance microwave coplanar bandpass and bandstop filters on Si substrates, Chan, K.T. Chin, A. Ming-Fu Li Dim-Lee Kwong McAlister, S.P. Duh, D.S. Lin, W.J. Chang, C.Y. , IEEE MTT, Sept. 2003, Volume: 51, Issue: 9, pp. 2036- 2040

Abstract: High-performance bandpass and bandstop microwave coplanar filters, which operate from 22 to 91 GHz, have been fabricated on Si substrates. This was achieved using an optimized proton implantation process that converts the standard low-resistivity (10 Ohm.cm) Si to a semi-insulating state. The bandpass filters consist of coupled lines to form a series resonator, while the bandstop filter was designed in a double-folded short-end stub structure. For the bandpass filters at 40 and 91 GHz, low insertion loss was measured, close to electromagnetic simulation values. We also fabricated excellent bandstop filters with very low transmission loss of 1 dB and deep band rejection at both 22 and 50 GHz. The good filter performance was confirmed by the higher substrate impedance to ground, which was extracted from the well-matched S-parameter equivalent-circuit data.

Low-loss microwave filters on CMOS-grade standard silicon substrate with low-k PCB dielectric, Lydia L. W. Leung, Kevin J. Chen, Xiao Huo, and Philip C. H. Chan, Wiley Microwave and Optical Technology Letters, Jan 5 2004, pp. 9-11.

Abstract: Benzocyclobutene (BCB), a low-k, photosensitive, easily patterned material is used as the interfacing layer that isolates the passive components from the lossy, low-resistivity silicon. Its performance is further improved by reducing the conductive loss with an electroplated and highly-conductive metal, namely, copper. Standard 50-Ohm microstrip lines and a microstrip-transmission-line-based microwave filter, exhibiting insertion losses of 0.8 dB and 1.1 dB at 10 GHz, respectively, are demonstrated.

Microwave coplanar filters on Si substrates, Chan, K.T. Chin, A. Kuo, J.T. Chang, C.Y. Duh, D.S. Lin, W.J. Chunxiang Zhu Li, M.F. Dim-Lee Kwong, IEEE Microwave Symposium Digest, June 2003, pp. 1909- 1912

Abstract: High performance band-pass and band-stop microwave coplanar filters operating from 22 to 94 GHz have been realized on Si substrates using a proton implantation process. Very good insertion loss and filter characteristics close to ideal EM simulation are measured that demonstrate excellent filter performance to 91 GHz.