Low-distortion balanced transconductor-C ladder filters without common-mode feedback

Antonio Carlos M. de Queiroz COPPE/EP – Electrical Engineering Program

Federal University of Rio de Janeiro Rio de Janeiro, Brazil

acmq@ufrj.br

Igor Oliveira Gameleiro EP – Department of Electronic and Computer Engineering

Federal University of Rio de Janeiro Rio de Janeiro, Brazil

igor.gameleiro@lps.ufrj.br

Abstract—This work shows how balanced transconductor-C low-pass ladder filters using a different arrangement of the transconductors to reduce distortion can also have stable com-mon-mode natural frequencies, not requiring common-mode stabilization circuits. Simulated distortion results are presented, interesting sensitivity properties of these filters are discussed, and some limitations of the technique are mentioned.

I. INTRODUCTION Transconductor-C filters are regarded as one of the best

approaches for the realization of high-frequency integrated active filters, due to the relatively simple structure of the tran-sconductors, that don’t usually require speed-limiting frequen-cy response compensation networks. The resulting filters are also tunable in frequency, if the transconductances of all the transconductors are changed simultaneously while keeping their proportions. This is an essential feature for a precise fil-ter, because in an integrated circuit fabrication process the capacitors and transconductors are fabricated in different un-correlated steps, and while the capacitors and transconductors result well matched among themselves, capacitances and tran-sconductances are not precisely matched. Tuning can be easily achieved, but a problem that results is that there is no simple technique to produce a transconductor with good linearity that is also adjustable, so some nonlinearity is always present. For better immunity to interferences and cancellation of even-order nonlinearities, balanced structures can be used. The ba-lancing introduces another problem, which is the need of common-mode feedback circuits to keep the filter stable. The stability problem is caused by the increase in the filter order due to the balancing, with the creation of new natural frequen-cies affecting only common-mode signals within the filter, and that must be in the left semiplane for stability. Reference [1] presented a method for the generation of balanced transcon-ductor-C filters simulating passive filters without common-mode feedback circuits, where the common-mode transcon-ductances of the transconductors are arranged in a way that turns the filter stable, developing an idea suggested in [2]. It was also discussed in [1] the application of a rearrangement of the transconductor inputs in a balanced filter that reduces their

input voltages, reducing also the generated distortion, as sug-gested in [3][4] for biquad filters. Since then, some develop-ments were published about the last subject, as [5] proposing a different way to make the input circuit, and [6] proposing a solution to the problem that arises in the application of the technique to the simulation of a doubly-terminated LC filter. In this paper, section I shows that it is possible to obtain a low-pass ladder simulation balanced filter where all the tran-sconductor inputs present reduced voltages and common-mode feedback is not necessary, a fact not observed in [1] and studied for the first time here. Section II shows simulations of an idealized example filter with distortion figures predicted, section III discusses interesting low-sensitivity properties of the obtained filters, and some limitations of the technique are described in the conclusions.

L2

C1

Vin Vo

+

−

+

−C2 C3 C5

L4

C4

1

1

a c e

C1: 1.41517 F C2: 1.08537 F L2: 0.586082 H

C3: 2.13067 F C4: 0.364398 F

L4: 0.881628 H C5: 1.84422 F

Figure 1. LC doubly terminated normalized 5th-order elliptic filter.

II. BALANCED FILTERS WITH REDUCED INPUT VOLTAGES

In order to obtain an active filter with low sensitivity to variations on its components, the usual approach is to simulate the operation of a passive LC doubly terminated prototype operating with maximum power transfer at the passband. In the examples that follow the normalized filter shown in Fig. 1 will be used, realizing a 5th-order elliptic filter with 1 dB pass-band ripple, 40 dB minimum stopband attenuation, and pass-band border at 1 rad/s. A regular single-ended transconductor-C version of the same filter is shown in Fig. 2. The termina-tions are implemented by inverting transconductors with input and output interconnected, the inductors by pairs of gyrators

This work was partially supported by the CNPq.

978-1-4673-2527-1/12/$31.00 ©2012 IEEE 97

connected to grounded capacitors, and a transconductor feeds the input with current. All the transconductances are unitary in the normalized filter.

C1 C2 C3

L 2

v a b eVin

+

−Vo

+

−

C5

c

C4

L 4

d

Figure 2. Single-ended transconductor-C filter.

The conventional way to obtain a balanced version is to duplicate the circuit, as shown in Fig. 3, where the upper half processes “positive” signals and the lower half their “nega-tive” versions. The single-ended transconductors of Fig. 2 are transformed into differential-input balanced-output transcon-ductors. All the transconductances in the normalized filter are of Gm = 0.5 S, because their input voltages are doubled.

C1

C1

C2

C2

C3

C3

L2

L2

v

'v

a

'a

b

'b

c

'cVin

+

−Vo

+

−

C4

C4

C5

C5

L4

L4

d e

'd 'e

Figure 3. Conventional balanced transconductor-C filter.

All the transconductors in this version receive as inputs the differences between signals at the two halves of the circuit. In most of the passband the input signal levels are similar, around the level of one of the input signals (because the prototype has a passband gain of 0.5), or somewhat larger. The technique described in [1] amounts to interchanging the negative inputs of transconductors feeding the same pairs of nodes, what con-nects both inputs of all the transconductors to signals at the same halve of the circuit, significantly reducing the input le-vels without modifying the filter. There is a difficulty with what to do at the nodes a-a’, because there are three transcon-ductors feeding that pair of nodes. The solutions proposed in [1] were to leave the transconductor corresponding to the in-put termination unchanged, or to eliminate it by using a sing-ly-terminated passive prototype. In [5] it was proposed a dif-ferent combination, to reduce the effect of input capacitances, but that keeps the same problem. In [6], an interesting solution was proposed, which amounts to the duplication of the input transconductor, leaving four transconductor feeding the nodes a-a’, and then the application of the same input interchanging technique to the two pairs in them. This solution is effective, and will be adopted in this work in the form shown in Fig. 4. It’s the same arrangement proposed in [1], with an additional input transconductor which has interchanged inputs with the input termination transconductor. The dotted lines show which inputs of the structure in Fig. 3 were interchanged. To guaran-tee common-mode stability, the technique proposed in [1] was to have the common-mode transconductances (with values ±Gc) of the balanced transconductors with the same polarities of the transconductances in the single-ended filter in Fig. 2. This results in the “common-mode filter”, obtained when the

signals at all the opposite pairs of nodes are identical, having the same structure of Fig. 2, resulting in “common-mode poles and zeros” proportional to the poles and zeros of the balanced filter. The common-mode signal present at the input is just passed to the output filtered by this low-pass filter, generating predictable common-mode signals at all internal node pairs in correspondence to what happens in the internal nodes of that filter. Common-mode signals generated inside the structure by imperfect balancing are also filtered with the same stable poles, leaving mostly small DC common-mode offsets.

No solution was presented in [1] for the modified filter with rearranged inputs. By observing Fig. 4, it can be noticed that since the opposite signals have opposite polarities, no common-mode currents are injected in the opposite node pairs if the common-mode transconductances of the transconductors feeding the same pairs of nodes are identical. This is an essen-tial feature for that structure, otherwise common-mode tran-sconductances interfere in the operation of the balanced filter. For common-mode signals, the terminations act as in the sin-gle-ended filter, and so the common-mode transconductances there must be negative. For the other common-mode transcon-ductances, the key point is that they shall be arranged so the corresponding elements in the common-mode filter form gyra-tors, resulting in the polarities, from left to right if Fig. 4, {– – + – + –}. For common-mode signals, the structure in Fig. 4 then reduces to a variation of the structure of the single-ended filter, shown in Fig. 5, with all the transconductances equal to ±Gc.

C1

C1

C2

C2

C3

C3L 2

v

'v

a b

'b

c

'cVin

+

−

Vo

+

−

L 2

'a

C4

C4

C5

C5L 4

d e

L 4

'd 'e

Figure 4. Modified balanced transconductor-C filter.

C1 C2 C3

L 2

v a b eVin

+

−Vo

+

−

C5

c

C4

L 4

d

Figure 5. Common-mode filter for the modified version.

This filter has no clear relation with the prototype filter, because the inductors are not simulated correctly with the gy-rators with these polarities, but the network after the two input transconductors is composed of lossy elements, the termina-tions, and lossless elements, the capacitors and the gyrators. The filter is then guaranteed to be stable. A passive prototype corresponding to it is shown if Fig. 6. The inductors have ideal inverting 1:1 ideal transformers at one side. The capacitors are the same, and the terminations and inductors are scaled as shown. The filter has 5 poles in the left semiplane and two

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pairs of transmission zeros in the real axis. For the example filter, assuming common-mode transconductances Gc = ±0.2 S and differential transconductances Gm = 0.5 S, Fig. 7 shows the positions of the poles and zeros seen for differential and common-mode inputs. In the first case the “common-mode poles” appear cancelled by zeros. For common-mode input signals the regular poles appear cancelled, and the real zeros of the common-mode filter are seen. The locations of the common-mode poles and zeros are:

P1 = –0.036873 Z1,2 = ± 0.35286 P2,3 = –0.042598 ± 0.14309j Z3,4 = ± 0.25076 P4,5 = –0.031306 ± 0.29751j

C1Vo

+

−

+

−

C2

C3 C5

C4

a c e1:11:1

/1 Gc

/1 Gc

24 / GcL2

2/ GcL

inV2

Figure 6. Passive filter corresponding to the common-mode filter for the modified structure.

Figure 7. Poles and zeros of the modified filter for differential input (left) and common-mode input (right).

For the modified filter, the common-mode filter shown is the only stable configuration. For the regular structure of Fig. 3, this same arrangement of common-mode transconductances also works, resulting in the same equivalent common-mode filter of Fig. 6. For the example filter, this form is actually better, because it results in the common-mode poles having lower Q (maximum Q = 4.78) than in the version with the arrangement in Fig. 2, where the common-mode poles are scaled version of the balanced filter poles (maximum Q = 10.1). Two other possibilities also exist, corresponding to the removals of one of the ideal transformers in Fig. 6.

III. DISTORTION IN THE MODIFIED FILTER An evaluation of the input voltages for all the transconduc-

tors in the ideal structures of the filters in Figs. 3 and 4 for a range of frequencies is shown in fig. 8. The two input voltages were set to ±1 V, corresponding to 0 dB. The regular filter had its input transconductance doubled to generate the same levels of the modified filter. The modified filter shows reduced input signal levels in all the passband for all transconductors, with large differences at low frequency but little difference at the passband border. To evaluate the reduction of distortion that the modification produces, idealized balanced transconductors

with the structures shown in Fig. 9 (not necessarily practical, but a simple solution to have different polarities of the com-mon-mode transconductances) were used in a set of simula-tions with sinusoidal inputs, where the input signal level was varied and the fundamental and third harmonics computed. The transconductors were designed with normalized transis-tors, modeled with SPICE level 1 model, for 1 V of maximum differential input (saturating the differential pair), small-signal differential transconductances of 0.5 S, and negligible output conductance, except for the current sources of the differential pairs, which generate the common-mode transconductances. The filters were also kept normalized. The distortion is domi-nated by the nonlinearity of the differential pairs. Input fre-quencies of 0.972 rad/s (last ripple peak close to the passband edge) and 0.324 rad/s (1/3 of the last ripple peak) were used.

25

-450.1 3rad/s

dB

v-v'

a-a'

b-b'

c-c'

d-d'

e-e'

25

-450.1 3rad/s

dB

v-a

v-b

a-c

b-d

c-e

d-e

Figure 8. Input signal levels for the transconductors in the regular filter (above) and in the modified filter (below).

vb

vss

vdd

v−v+

i± 0 i0∓

Figure 9. Balanced transconductors for distortion simulation, with positive or negative (with the current inverters) common-mode transconductances.

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The obtained results are compared in Fig. 10. The scales are for voltages in dB, where 0 dB corresponds to peak values of 1 V at one of the inputs, and of difference between the two opposite outputs of 1 V too. It can be observed in fig. 8 that 1 V of input produces a worst-case input voltage of about 0.56 V (5 dB below) at 0.324 rad/s, while in the regular filter a lev-el of about 2.5 V (8 dB above) would be expected, if the filter were linear. Close to the passband edge, however, there is little difference in Fig. 8, and Fig. 10 shows that both filters work acceptably only for input signals smaller than about 56 mV (–25 dB), due to the high levels that appear over the capa-citors simulating the two inductors. The third harmonic is about 8 dB below and 40 dB below at the two test frequencies in the modified filter. For both filters there is significant com-pression in the fundamental at –10 dB. IIP3 lines extrapolated from the curves show improvements, in voltage, of about 25 dB at 0.324 rad/s and 4 dB at 0.972 rad/s.

-140

40

-40 20

3rd 0.324 rad/s

dB

dB

Output

Input

1st 0.324 rad/s

1st 0.972 rad/s

3rd 0.972 rad/s modified

regular

Figure 10. Simulated first and third harmonics for several input signal levels

at 0.324 and 0.972 rad/s for the two compared filters.

IV. SENSITIVITY PROPERTIES The modified filter has zero input voltages on all the tran-

sconductors at DC. This causes the DC gain to be independent of the differential transconductances, and the sensitivities to all the transconductances to be very low in the passband. In the regular filter, the sensitivities to the transconductances are a complex combination of the sensitivities of the prototype filter, and dominate the passband errors. The sensitivities to the capacitance values, the same in the regular balanced filter and in the modified filter, don’t copy perfectly the sensitivities in the passive prototype, because differences in the capacit-ances at the positive and negative sides of the filter don’t have a correspondent on the passive prototype. They are reasonably low, however. Fig. 11 shows a comparison of error limits cal-culated by sensitivity analysis for the regular and modified filters, considering 5% random errors in all the transconduc-tances and capacitances. The calculation was done by gain statistical deviation, eq. (1), where T(jω) is the transfer func-tion, ε = 0.05 (5%), and xi are all the differential transconduc-tances and capacitances. The common-mode transconduc-tances were not considered (they would increase a bit the low-frequency errors). It’s seen that the modified filter presents much lower passband errors, and even the stopband errors are smaller too. A similar analysis for common-mode input sig-nals (not shown) shows similar passband sensitivities but high stopband sensitivities for the modified filter. The common-

mode poles, however, have low sensitivities, while only the zeros have high sensitivities. The stability of the filter is then insensitive to component variations.

( )( ) dB Re)10Ln(

20)( 2∑ ωε±=ωΔi

jTxi

SjT (1)

0 3-60

10

-3

2

0 1.1

rad/s

rad/sdB

dB

regular

modified

Figure 11. Gain with error limits calculated by sensitivity analysis for the two compared filters. The peaks at the zeros are artifacts of the method.

V. CONCLUSIONS It was shown that the two techniques, of modifying the in-

put connections of the transconductors to reduce distortion and of arranging the polarities of the common-mode transconduc-tances to avoid the need of common-mode feedback, can be used simultaneously in a low-pass balanced transconductor-C filter. The modified filter has also interesting low-sensitivity properties. The technique has some limitations: In filters that are not low-pass, some node pairs are fed by just one tran-sconductor and there is no way to rearrange all the transcon-ductor inputs. Dynamic range scaling also can’t be done in the usual way, because the technique requires all the transconduc-tors feeding a node pair to be identical.

REFERENCES [1] Antonio Carlos M. de Queiroz, “Balanced transconductor-C ladder

filters with improved linearity,” 52nd IEEE International Midwest Sym-posium on Circuits and Systems, Cancún, Mexico. pp. 41-44, Aug. 2009.

[2] T. Sato, S. Takagi, S. Ao, and N. Fujii, “Realization of Fully Balanced Filter Using Low Power Active Inductor,” ICECS 2008, St. Julians, Malta, pp. 133-136, Sep. 2008.

[3] A. Tajalli and Y. Leblebici, “Linearity improvement in biquadratic transconductor-C filter,” Electronics Letters, Vol. 43, No. 24, pp. 1360-1362, Nov. 2007.

[4] A. Tajalli and Y. Leblebici, “Low-Power and Widely Tunable Linea-rized Biquadratic Low-Pass Transconductor-C Filter,” IEEE Transac-tions on Circuits and Systems II: Express Briefs, Vol. 58, No. 3, pp. 159 - 163, Mar. 2011.

[5] T. Choogorn and J. Mahattanakul, “Novel structure of low-distortion fully-differential doubly-terminated ladder Gm-C filters,” 2011 8th In-ternational Conference on Electrical Engineering/Electronics, Comput-er, Telecommunications and Information Technology (ECTI-CON, pp. 110-113, May 2011.

[6] Weinan Li, Yumei Huang, and Zhiliang Hong, “A 70–280 MHz fre-quency and Q tunable 53 dB SFDR Gm-C filter for ultra-wideband,” 2010 IEEE Asian Solid State Circuits Conference (A-SSCC), pp. 1-4, Nov. 2010.

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