• Abstract

# Capacitive Crosscoupling Biquad Polyphase Filter

In this paper, the design of biquad polyphase filters with capacitive crosscouplings is presented. Based on the elementary equation s → s − jωc, which describes the frequency shifting, system theoretic modeling is used for design. Some important observations simplifying the design process are denoted. The developed structures have been simulated in Cadence using operational amplifier models with adjustable gain and bandwidth. Results are compared to those of a Tow-Thomas polyphase filter with comparable sum capacitance but with an operational amplifier amount twice as high. By limiting the gain bandwidth product of the models to reasonable values it is shown that operational amplifier performance does not need to be improved and hence a reduced power consumption can be expected.

SECTION I

## INTRODUCTION

Todays mobile devices often provide a large number of functions which require interaction with the outer world. Appropriate interfaces have to be provided consuming a noticeable amount of power. Batteries capacity, on the other hand, limits the time of action of these devices. Hence reducing the power consumption is an important issue in circuit design. In the following sections, the development of biquad polyphase filter (PPF) structures is presented which are likely to reduce the power consumption due to a reduced operational amplifier (opamp) amount compared to Tow-Thomas or single pole PPF structures.

SECTION II

## MODEL DERIVATION

### A. Basic Theory

The basics of PPF design are covered well in literature. In [1] this topic is treated extensively. Often, a direct synthesis approach as presented in [2], for example, is applied in polyphase filter design but this approach does not cover the case of frequency dependant feedback signals. Hence, the design method presented below will be based on system theoretic modeling.

### B. Core Circuit

For model derivation the core circuit of Fig. 1 is focused first. It resembles the single pole structure but the tapping is different. For simplicity, the single ended version is examined here. In most applications, the differential structure will be preferred so that the inverters are not required.

By using the superpostion principle with respect to ϕi and ϕq as well as the voltage divider law in its admittance form, the following sets of equations can be derived: TeX Source \matrix{ &\phi'_{i-} &= {G_w \over sC_1+G_q+G_c+G_w}V_{in,i}\hfill\cr &\phi'_{q-} &= \left.{G_w \over sC_1+G_q+G_c+G_w}V_{in,q}\quad \right\vert\cdot j \cr \noalign{\hrule width 175pt\vskip2pt}\cr \sum&\underline{\phi'_{-}} &= {G_w \over sC_1+G_q+G_c+G_w}\underline{V_{in}}\qquad \hbox{(1)}\cr &\phi''_{i-} &= {-sC_1+G_q \over sC_1+G_q+G_c+G_w}V_{out,i}\hfill\cr &\phi''_{q-} &= \left.{-sC_1+G_q \over sC_1+G_q+G_c+G_w}V_{out,q}\quad \right\vert\cdot j \cr \noalign{\hrule width 175pt\vskip2pt}\cr \sum&\underline{\phi''_{-}} &= {-sC_1+G_q \over sC_1+G_q+G_c+G_w}\underline{V_{out}}\qquad \hbox{(2)}\cr &\phi'''_{i-} &= {-G_c \over sC_1+G_q+G_c+G_w}V_{out,q}\hfill\cr &\phi'''_{q-} &= \left.{G_c \over sC_1+G_q+G_c+G_w}V_{out,i}\quad \right\vert\cdot j \cr \noalign{\hrule width 175pt\vskip2pt}\cr \sum&\underline{\phi'''_{-}} &= {jG_c \over sC_1+G_q+G_c+G_w}\underline{V_{out}}\qquad \hbox{(3)}\cr &V_{out,i} &= \nu\phi_{i-}\hfill\cr &V_{out,q} &= \nu\phi_{q-}\quad \vert\cdot j\hfill\cr \noalign{\hrule width 175pt\vskip2pt}\cr \sum &\underline{V_{out}} &= \nu\underline{\phi_{-}}\hfill\qquad \hbox{(4)}. }In these equations, the dashes indicate complex quantities and Gx (x ∊ {i, q, w, c}) is the conductance according to the resistance Rx. The transfer functions themselves are all real in nature. The selection of signs in the numerators of equation set three conforms with the rule of complex multiplication. Since, a complex transfer characteristic is achieved1. Equation (1) represents the outer loop feed forward transfer function. The feedback transfer characteristic is given by (2) merged with (3). Equation (4) finally denotes the innerloop feedforward transfer function. They can be combined to the system theoretic model given in Fig. 2.

Fig. 1. Core circuit.
Fig. 2. System theoretic feedback structure.

The system theoretic model can also be developed in terms of equivalent circuits which is presented in [3]. By evaluation, the transfer function of the core circuit can be determined: TeX Source $${\underline{V_{out}}\over \underline{V_{in}}} = {\nu {G_w\over sC_1+G_q+G_c+Gw}\over 1-\nu {-sC_1G_q+j\cdot G_c\over sC_1+G_q+G_c+G_w}}.\eqno{\hbox{(5)}}$$If gain of the operational amplifiers is high enough, (5) simplifies to: TeX Source $${\underline{V_{out}}\over \underline{V_{in}}} \approx {G_w\over sC_1-G_q-j\cdot G_c} = {{1\over R_wC_1} \over s - {1\over R_qC_1}- j\cdot {1\over R_cC_1}}\eqno{\hbox{(6)}}$$

### C. Normalized Biquad Transfer Function

To obtain a biquad PPF, a second loop is required. A generalized form of the structure is depicted in Fig. 3.

Fig. 3. Generalized feedback structure.

The core circuit discussed before is used within the loop to set the inner loop feedforward transfer function H2. Of course, this circuit has finite gain and its input can not be assumed to be virtual ground. Hence, loading of this node by the core circuit has to be taken into account. Again, the superposition principle can be applied in a similar manner as shown above, no matter which passive components are added. Thus, the outer loop feedforward and the feedback transfer function, H1 and H3, can again be determined by voltage divider ratios in their admittance form. In consequence, the transfer functions are given by the admittance present in the respective signal path divided by the total node admittance. As H1 and H3 refer to the same node, their denominators are equal. This equality can be used to expand the generalized model of Fig. 3, where Ny (y ∊ {1… 3}) represents the numerator of Hy and Dy its denominator: TeX Source \eqalignno{\underline{H} &= {\underline{H_1}\underline{H_2}\over 1- \underline{H_2}\underline{H_3}}{\underline{N_1}\underline{N_2}\over \underline{D_1}\underline{D_2}-{\underline{D_1}\underline{D_2}\over \underline{D_2}\underline{D_3}}\underline{N_2}\underline{N_3}}\cr&={\underline{N_1}\underline{N_2}\over \underline{D_1}\underline{D_2}-\underline{N_2}\underline{N_3}}&\hbox{(7)}}N2 and D2 are determined by the core circuit and hence given by (6). They can be inserted into (7) to yield some more insight: TeX Source $$\underline{H} = {\underline{N_1}{1\over R_w}\over \underline{D_1}\left(sC_1-{1\over R_q}-j\cdot {1\over R_c}\right) -{1\over R_w}\underline{N_3}}\eqno{\hbox{(8)}}$$If this representation is compared with the normalized biquad polyphase filter transfer function (9), some important observations can be made. TeX Source $$\underline{H_n} = {Aw^2_0\over s^2 + s{w_0\over Q_0} -js2w_c-\left(w^2_c - w^2_0\right)- j {w_cw_0\over Q_0}}\eqno{\hbox{(9)}}$$

In this equation is the bandwidth of an equivalent low pass, is the center frequency, Q0 is the filter quality factor and A is the filter gain. Comparing (8) with (9) it becomes apparent, that N1 needs to be a real valued function which is usually the case. A second order polynomial will be generated by the product D1 D2 as D2 contains the total node admittance of the node being input to the core circuit. D2 must not have a complex coefficient preceeding the first order polynomial because a complex coefficient preceeding the second order polynomial of the complete denominator of H would arise else. N3 on the other hand does not suffer from such a restriction. This justifies the placement of crosscoupling capacitors into overall feedback as will be done below2.

### D. Filter Structure

The remaining task is to find a circuit structure which allows for proper adjustment of N3. As explained above, N3 contains the admittances in the feedback path. If we want to adjust the transfer function arbitrarily, we have to insert a feedback resistor tapping the core circuit at the inphase output, another one tapping at the quadrature output, a feedback capacitor tapping at the inphase output and finally a feedback capacitor tapping at the quadrature output into the inphase path. Signs can be varied by interchanging taps of differential paths. For the quadrature path, signs of the crosscoupling element taps have to be inverted. This will yield the feedback structure given in Fig. 4.

Fig. 4. Outer loop feedback structure.

The denominators of H1 and H3 contain the conductance Gw due to the loading effect mentioned earlier. A circuit offering this feedback structure is shown in Fig. 5(a).

Fig. 5. Capacitive Crosscoupling PPFs—single ended equivalent circuits of the differential structures.

Figs. 5(b) and (c) show two slightly different structures for which reasonable parameter values could be calculated. They differ from the structure of Fig. 5(a) by interchanged feedback taps with respect to the differential paths. This measure causes changing signs in the numerator of H3 or in the denominator of H2 respectively. The transfer functions are given as follows3: TeX Source \eqalignno{&{1 \over \underline{H}} = {s^2 + s \left[{1\over C_2 + C_3} \left({1\over R_i} + {1\over R_w} +{1\over R_{wo}} +{1\over R_{co}} \right) - {1\over C_1 R_q}- \right. \over {1\over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {\left.-{C_2\over C_1(C_2+C_3)} {1\over R_w}\right] -js \left[\pm {1\over R_c C_1} \pm {C_3 \over C_1 (C_2 + C_3)} {1\over R_w}\right]- \over {1 \over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {-{1\over C_1(C_2+C_3)} \left[{1\over R_q} \left({1\over R_i} + {1 \over R_w} + {1 \over R_{wo}} + {1\over R_{co}}\right) - {1\over R_w R_{wo}}\right]- \over {1 \over R_i R_w C_1 (C_2+C_3)}} \cr&\quad {-j{1\over C_1(C_2+C_3)} \left[\pm {1\over R_c} \left({1\over R_i} + {1\over R_w} + {1 \over R_{wo}} + {1 \over R_{co}}\right) \mp {1\over R_w R_{co}}\right] \over {1 \over R_i R_w C_1 (C_2+C_3)}}&\hbox{(10)}} (10) provides four selections of signs which determine the version realized. In order to obtain version 1, signs have to be chosen according to {+,+,+,−} in the order given. Version 2 will be obtained by selecting {−,+,−,+} while version 3 results from selecting {+,−,+,−}.

SECTION III

## SIMULATION RESULTS

In order to determine the parameter values, the coefficients of (10) have to be compared to those of (9): TeX Source \eqalignno{Aω_0^2 &= {1\over R_i R_w} {1\over C_1 (C_2 + C_3)}, &\hbox{(11)} \cr{ω_0 \over Q_0} &= {1 \over C_2 + C_3} \left({1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}}\right) - {1\over C_1 R_q} - \cr&\quad - {C_2 \over C_1 (C_2 + C_3)} {1 \over R_w}, &\hbox{(12)} \cr2ω_c &= \pm {1\over R_c C_1} +pm {C_3 \over C_1(C_2 + C_3)} {1\over R_w}, &\hbox{(13)} \crω_c^2 - ω_0^2 &= {1\over C_1(C_2 + C_3)} \cr&\quad \times \left[{1 \over R_q} {1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}} - {1\over R_w R_{wo}}\right], &\hbox{(14)} \cr{ω_0 ω_c \over Q_0} &= {1\over C_1(C_2 + C_3)} \cr&\quad \times \left[\pm {1 \over R_c} {1\over R_i\,\Vert\,R_w \,\Vert\,R_{wo}\,\Vert\,R_{co}} \mp {1\over R_{co} R_w}\right] &\hbox{(15)}}

As can be seen, there is a strong dependence between (12), (13) and (15). To eliminate this dependence, the product of the right hand sides of (12) and (13) has to be equated with two times the right hand side of (15). The auxiliary equation obtained this way can be solved for Rco for example. Using the result, a set of four equations remains. The values for filter gain A, equivalent low pass bandwidth f0, center frequency fc and filter quality Q0 can be substituted into these equations according to the specifications. Furthermore, values for the capacitance C1 and the input resistor Ri can be prescribed due to area or noise considerations, respectively. Values for the remaining filter coefficients can be calculated then. Specifying A = 10, ω0 = 2 π 1.25 MHz, ωc = 2 π 2 MHz and Q0 = 1.31, the values given in Table I are calculated.

TABLE I Filter Parmeters for a Filter Quality of Q0 = 1.31

Table II contains the parameter values for the same specifications but for Q0 modified to Q0 = 0.54.

TABLE II Filter Parameters for a Filter Quality of Q0 = 0.54

If two biquad polyphase filters with filter quality factors of Q0 = 1.31 and Q0 = 0.54 are combined, a Butterworth polyphase filter response is obtained. Component values according to Tables I and II have been selected to implement the three filter versions. In the Cadence simulations, models with finite gain bandwidth product (GBW) have been used to represent the opamps. A testbench according to Fig. 6 has been applied.

Fig. 6. Testbench used for simulations.

As the local oscillator inputs have been stimulated by the ports at a frequency of 1.57 GHz, the results taken at the output port are in the GHz range. The gain curve progressions of the filter versions have been examined under different conditions: First nearly ideal opamps (gain A0 = 10000 and bandwidth GHz) have been assumed. In order to model the opamps more realistic, gain has been reduced to A0 = 400 and the bandwidth has been limited to fp = 4 MHz. Due to these limitations, transfer characteristics are impaired by noticeable peaking. The latter can be avoided by appropriate filter tuning. This is shown in Fig. 7 for version 2 of the Capacitive Crosscoupling PPF (CCPPF), at which Rq and C1 have been tuned.

Fig. 7. Gain curve progression of the CCPPF (version2) ① using nearly ideal opamp models, ② using models with A0 = 400 and fp = 4 MHz and ③ using the same models and filter tuning.

Owing to the use of models, the noise contribution of the opamps is neglected in the noise figure results of Fig. 8. These results have been obtained using nearly ideal models but do not differ significantly at reduced GBW. A Tow-Thomas PPF with approximately the same sum capacitance per signal branch (34 pF) has been implemented and used as reference. As the noise transfer functions from the opamp inputs to the filter outputs may differ, the comparison has been repeated with noise sources inserted in series to each differential amplifier input. A noise power spectral density of 6.25 · 10− 18 V2/Hz has been assumed for each source. Fig. 9 gives the results for opamp models with A0 = 400 and fp = 4 MHz as well as filter tuning.

Fig. 8. Noise figure results without noise contribution of the nearly ideal opamp models for ① the Tow-Thomas polyphase filter, ② version 1 of the CCPPF, ③ version 2 of the CCPPF and ④ version 3 of the CCPPF.
Fig. 9. Noise figure results with modeled opamp noise for ① the Tow-Thomas polyphase filter, ② version 1 of the CCPPF, ③ version 2 of the CCPPF and ④ version 3 of the CCPPF using opamp models with A0 = 400 and fp = 4 MHz as well as filter tuning.
SECTION IV

## CONCLUSION

In this paper, a design procedure to develop polyphase filter structures with capacitive crosscouplings has been presented. The component values calculated by this method have been used to implement the filter structures in Cadence. Finite GBW models have been used to represent the operational amplifiers in simulations. The influence of opamp noise has been examined using noise sources in series to the amplifier inputs. For comparison, a Tow-Thomas polyphase filter using approximately the same sum capacitance has been implemented, too. Results show, that the implemented polyphase filters suffer from a higher noise figure compared to the Tow-Thomas PPF structure if similar sum capacitances per signal branch are applied. However, version two of the CCPPF already shows good performance and it has to be pointed out, that the proposed structures allow for the reduction in both, area and power consumption due to a reduced opamp count. Thus, occupying the same chip area, the implemented filter versions might approach the noise performance of the Tow-Thomas structure closer while the advantage of reduced power consumption would be retained.

## Footnotes

Markus Robens, Ralf Wunderlich, and Stefan Heinen are with the RWTH Aachen University, Chair of Integrated Analog Circuits, D-52074 Aachen, Germany Email: mailbox@ias.rwth-aachen.de.

1. In math, j indicates the second entry in an ordered pair for which special rules of addition and multiplication are defined. Similarly it can be understood as path information here.

2. Adding a crosscoupling capacitor to the overall feedback will cause the same real, additive contribution to the first order polynomial of D1 and D3 but a complex contribution to the first order polynomial of N3.

3. Due to limited printing space, the inverse of the transfer functions are given.

## References

1. Complex signal processing is not complex

K. W. Martin

IEEE Trans. Circuits Syst. I, vol. 51, issue (9), p. 1823–1836, 2004-09

2. Low-IF topologies for high-performance analog front ends of fully integrated receivers

J. Crols, M. S. J. Steyaert

IEEE Trans. Circuits Syst. II, vol. 45, issue (3), p. 269–282, 1998-03

3. Sallen-key polyphase filter design

M. Robens, R. Wunderlich, S. Heinen

St. Julians, Malta
Proc. 15TH IEEE International Conference on Electronics, Circuits and Systems ICECS, 2008-09

## Cited By

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## Keywords

### IEEE Keywords

Mathematical model, Equations, Noise, Transfer functions, Bandwidth, Gain, Capacitance

### INSPEC: Controlled Indexing

phase shifters, biquadratic filters, operational amplifiers

### INSPEC: Non-Controlled Indexing

No Keywords Available

### Authors Keywords

No Keywords Available

### More Keywords

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No Corrections

## Media

No Content Available
This paper appears in:
International Symposium on Circuits and Systems
Issue Date:
2009
On page(s):
49 - 52
ISBN:
N/A
Print ISBN:
978-1-4244-3827-3
INSPEC Accession Number:
10760303
Digital Object Identifier:
10.1109/ISCAS.2009.5117682
Date of Current Version:
26 Jun, 2009