# High-Frequency Sections of Active Filters of Mixed-Signal

Authors: Sergei Krutchinsky; Nikolai Prokopenko
Journal: ISRN Electronics
 (2012)

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## Abstract

The sufficient conditions for the efficient use of active elements are formulated by analyzing the basic structure of second-order unit. The expediency of current amplifiers usage in HF and SHF filters is shown. The examples of a methodical nature are given and conclusions of application importance are formulated.


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## Body

## 1. Introduction

A modern mixed-signal systems on a chip (SoC) creating for communications and radio-technical systems are inextricably linked to the development of analog and analog-digital IP-blocks for a sufficiently accurate processing of HF and SHF range signals. Some of the basic components of these blocks are frequency filters, which are highly sensitive parametrically. As a rule, reduction of the parametric sensitivity causes the need to create fairly complex active elements, which consume much power. Exactly energy problems of precision filters reduce the attainable level of integration of the SoC and the area of practical usage. Typically, the frequency filter is designed for a specific task with given parametric and structural constraints that are defined by a specific technology. In HF and SHF ranges transimpedance amplifiers (voltage-current converters) are often used as the active elements for compensation of losses in the passive RC-circuits [1, 2]. However, the potentially achievable unity-gain frequencies of these elements give way to current amplifiers, thus ultimately limiting the product of the Q factor (Q) and the pole frequency (fp) with the existing restrictions on power consumption. That is why the potential properties of the filter based on current amplifiers have a practical perspective and require further theoretical research. At least, the limiting theoretical estimations are not yet available and practical recommendations are not formulated.


## 2. Problem Statement

As a rule, to increase the stability of the frequency characteristics of high-order filters, additional intersection connections are used to ensure the minimization of sensitivity to the instability of the passive RC elements in the range of operating frequencies [3, 4]. In HF and SHF bands, this schematic approach is almost never used because of the high and usually not accessible requirements to a parasitic phase shifts of used active elements. In this regard, the cascading of the second-order sections remains the main direction of the implementation of the necessary poles and zeros of the desired transfer function [5, 6]. That is why improving of their parameters stability and reduction (minimization) of appropriate parametric sensitivities is the main way of the general problem solution. In addition, achievable through the implementation in modern technologies the precision of resistors and capacitors in frequency-defining elements is not sufficient to ensure admissible errors of transfer function poles of the main frequency filter types.According to-above-mentioned things, solutions are necessary for the whole class of interrelated circuitry tasks which are aimed on the accuracy improving of the basic parameters realization in the second-order sections of HF and SHF ranges.Firstly, taking into account the dominating influence of the pole frequency implementation error(δfp) on the capacitive characteristics of high-Q sections, it is necessary to create structures with a minimum parametric sensitivity to the instability of active elements and components, and first of all, current amplifiers [7]. Secondly, from the point of view of frequency-band extension, similar structures with maximally effective usage of these amplifiers resources (component and energy costs for the product of Q-factor and pole frequency) are needed [8]. At the same time possible and acceptable trade-off is the equality of contributions of attenuation instability (δdp=-δQ) and pole frequency instability -δfpQ≈δQ. In terms of general problem solving it creates sufficient preconditions for energy consumption reduction or provides the structural degrees of freedom to increase the operating frequency range. Finally, and this is the most important from a technological point of view, the existing structural and hence the parametric degrees of freedom can be used for correction (adjustment) of the pole parameters (fp,Q) by mode-defining and therefore low-frequency circuits. Specified property of such circuit solutions permits not only to integrate IP-blocks and SoC, but also to implement during their exploitation a temperature and temporal correction of plural transformation tracts through programmable cores of the system and simple digital-to-analog converters.


## 3. The Main Properties of the Basic Structure of the Second-Order Section

In general case the structure (Figure1) consists of three autonomous units—the active element K, the frequency-dependent γ(p), and frequency-independent β circuits.Figure 1
Basic structure of second-order section.That is why its transfer function(1)F(p)=K(p)[γ1(p)-β1]1-K(p)[γ2(p)-β2],(K(p)=K1+pτa)
allows to realize any type of filter, amplitude, or phase corrector. In general case for the second-order section
(2)γ1(p)=N(p)p2+pDpωpc+ωpc2,(3)γ2(p)=γ0a2p2+a1pD0ωpc+a0ωpc2p2+pDpωpc+ωpc2,
where D0, Dp are the attenuation of zero and the pole of frequency-dependent circuit; ωpc is the frequency of its pole; γn is the scale transfer coefficient; ai is binary (0, 1) coefficients; N(p) is the polynomial that defines the type of implemented second-order section.From (1)–(3) follows that for function in general form
(4)F(p)=A(p)p2+pdpωp+ωp2
frequency ωp and pole attenuation dp for idealized schema (τa=0)
(5)ω(p)=ωpc1+β2K-Kγ0a01+β2K-Kγ0a2,(6)dp=Dp(1+β2K)-Kγ0D0a1(1+Kβ2-Kγ0a2)(1+Kβ2-Kγ0a0).Ratios (5) and (6) define structure of parameters influence of passive frequency-dependent (D0, Dp, ωpc, γ0), frequency-independent (β2) circuits, and active (K) elements and allow to receive the following practical conclusions. First, for exception of the most unstable parameter influence of the scheme K on frequency of a pole ωp, it is necessary to use symmetric frequency-defining circuits (a2=a0) when ωp=ωpc and active element influence is directed on increase of Q-factor Q=1/dp. Secondly, in case of use of noninverting amplifier or the converter (K>0) value dp (6) is minimized at a0=a2=0, a1=1(7)dp=Dp-γ0D0K1+Kβ2,
and at application of an inverting active element (K<0) this minimization is possible when a0=a1=a2=1, β2=0:
(8)dp=D0Kγ01+Kγ0+Dp1+Kγ0.Thus, in the first case (K>0) in feedback loop of an active element (the amplifier and the converter) K it is necessary to use the frequency-defining circuits of band-pass type, and in the second case (K<0)—rejection type. In this case frequency of a pole of the scheme will be always defined by frequency of passive circuit pole, and action of active element will be directed on increase of realized Q-factor.For passive RC circuits of the second order [5] of band-pass type the condition D0γ0=Dp-D0 is characteristic, so at β2=0(9)dp=Dp(1-K)+D0K.Therefore, by the choice of passive elements ratios of these circuits (e.g., T-bridges) and structure of the amplifierK it is possible to provide realization of ωp, and dp with low parametrical sensitivity. Apparently from (8), realization of this condition for inverting amplifiers is connected with limiting (K→∞) increase of its gain.As it is seen from (1), (2), the execution of the formulated conditions demands from passive frequency-defining circuits the additional condition realization of reproduction of the desired frequency characteristics in a circuit of direct transfer (γ1(p)).If to consider the influence of time constant of noninverting active elements (τa) on parameters of transfer function poles of basic structure then under condition of their low admissible relative changes (δωp, δdp) we receive
(10)δωp≈δdp≈-12ωpBSDpK1+β2K,
where BS=K/τa is the gain area of active element K of basic structure.In this case parametrical sensitivities(11)SBSωp=SBSdp=12ωpBSDpK1+β2K
are defined by numerical values of BS and pole attenuation Dp of a passive chain and can not be minimized by frequency-independent feedback depth (0≤β≤1).From the specified positions the most important qualitative parameter of passive frequency-defining circuits is their attenuationDp which reduction favorably effects on a working frequency range and realized Q-factor. For RC circuits of the second order [5] Dpmin=3, thus D0=2, and under condition of D0 minimization (see (13)) Dp=2Q and, therefore, the influence of BS on poles parameters increases. For reduction of Dp it is possible to insert into RC circuit the additional repeater k (Figure 2) that allows, in particular, to realize in schemes the additional useful structural and parametrical degrees of freedom.Figure 2
The feature of formation of SHF section poles.


## 4. Low-Sensitive SHF Sections of Band Type

The created above conclusions show that for active elements influence reduction on the frequency of high-Q pole it is expedient to use current repeaters (Ki) in which feedback loop the frequency-dependent circuit of band-pass type with the additional voltage buffer repeater k separating this circuit into elementary differentiating (integrating) circuits is included. In this case for increasing of BS and, therefore, according to ratios (10) and (11) (β2=0), the reduction of sensitivity and parameters realization errors, the circuitry degrees of freedom are created or, at least, saved, and k usage allows to reduce attenuation of a passive circuit pole Dp. The structure of such section is shown on Figure 3.Figure 3
The structure of low-sensitive section of the band type filter.Using a standard notation(12)F(p)=K0pdpωpp2+pdpωp+ωp2,
it is easy to determine that
(13)ωp=1τ1τ2;(τ1=R1C1,τ2=C2(R2+ri+ru)),(14)dp=τ1τ2+τ2τ1(1-KiR1R2+ri+ru),(15)K0=Qτ1τ2,
where ri, ru are input resistance of current amplifier and output resistance of the buffer voltage amplifier (k=1).Second-order circuit separation by means of the bufferk allows to get
(16)D0=τ1τ2,Dp=τ2τ1+τ1τ2,
therefore parametrical sensitivities of the scheme can be minimized.From (13) follows that always
(17)SR1ωp=SC1ωp=SC2ωp=-12,SR2ωp=-12R2R2+ri+ru,Sriωp=-12riR2+ri+ru,Sruωp=-12ruR2+ri+ru.In addition, despite the differential realization principle of low pole attenuation (14) the minimization of its parametrical sensitivity to the changing of passive elements of discussed structure is possible. So, at execution of parametrical conditions
(18)R1=R2+ri+ru,τ1=τ2(1-Ki)
realized attenuation
(19)dp=21-Ki
is accompanied by extremely low sensitivity
(20)SC1dp=SC2dp=SR1dp=SR2+ri+rudp=0.Thus, to ensure high qualitative indexes of schematics of band-pass sections it is necessary to develop special current amplifiers with transfer coefficient numerical values that following from (19). In this case, as it is evident from (15), the gain
(21)K0=12
is enough small and by multistage filter implementation with additional intersection connections can significantly affect the realizable dynamic range.By such filter implementation it is essential according to (14), (15) to increase Ki and to optimize the ratio of resistive elements of the scheme. In the general case
(22)K1R1R2+r1+ru=K0(K0-1)Q2+1.As seen from the given ratio the required values ofK0 and Q can be realized not only by the special structure of the current amplifier but also by selecting of the ratio of resistive elements R1 and R2. Particularly, this ratio can be used for designated parameters correction. In some cases the emerging degree of freedom can be used to optimize the sensitivity of the pole attenuation. Thus, to implement K0=Q it is needed to meet the conditions
(23)τ1=τ2,KiR1R2+ri+ru=2-dp,
in this case the parametric sensitivity
(24)SC1dp=SR1dp=-SC1dp=-SR2+ri+rudp=Q-12
is practically defined by realizable Q-factor.When designing the multisection filters with additional intersection connection the required value ofK0 is limited to values that are close to unity [3]. In this case, for sensitivity reduction it is advisable to use conditions
(25)τ1=τ2dp2,KiR1=R2+ri+ru,
that ensure the equality of this sensitivity to unity.


## 5. Examples of Second-Order Band-Pass Sections

The simplest current amplifier (Ki=α) and repeater (k) are the stages with common base and common collector that have the maximum Bs. Therefore, the simplest scheme of band-pass section is the scheme in Figure 4. Here
(26)Ki=α1,ri=h11.1,ru=h11.2,
where h11.i≈φT/Ii,αi  are input resistance and static gain of emitter current of transistor with i index.Figure 4
The simplest low-sensitivity section schematic of the band-pass filter.According to (13) in this scheme the pole frequency adjustment is possible by changing of bipolar transistors operation modes (by sources I1 and I2). However, due to the influence of these parameters (expressions (14), (15)) on the pole attenuation dp and gain K0, it is feasible only for small Q=1/dp values, when conditions (19) and (20) are satisfied. Otherwise,  as is evident from modeling results of this scheme (Table 1), this leads to significant changes of specified parameters.Table 1
The results of schematic modeling.
Parameter
Scheme
f
p
δ
f
p
Q
δ
Q
K
0
δ
K
0
I
0
Conditions
GHz
%
—
%
dB
%
mA
1
0.8
8.6
72
13
22
3
C
1 = 294 fF,
Figure4
C
2 = 800 fF,
R
1 = 360
R
2 = 150
1
0.6
5
3
13
1.4
6
C
1 = 500 fF,
Figure5
C
2 = 800 fF,
R
1 = 260
R
2 = 150
Note: bipolar heterojunction transistors (m=1, le=840*10−9, We=420*10−9, x=8, y=2) Supply voltage ±2.5 V.To eliminate this effect in the scheme it is necessary to use a more complex current amplifier that eliminates restrictions on the ratios of resistive elements of the scheme (expressions (22), (23), (25)) and allows to change gain Ki regardless of the resistors ri and ru. One of the possible solutions of this problem is shown in Figure 5.Figure 5
Schematic of band-pass section with adjustable pole parameters.Here(27)Ki=α1+α4h11.1h11.3+h11.4=α1+α4I04I1.Thus, by the changing of the mode currentI0 it is possible at given values of h11.1 and h11.2 (currents I1 and I2) to change the Ki and, therefore, by retaining of any parametric condition (18), (19) (22), (23) to implement the required section parameters. The results of comparative modeling of reviewed schemes in software environment Cadence Virtuoso on the base of technological process SGB25VD components (technological inaccuracy of resistors and capacitors is ±10%) are presented in Table 1.The increasing of the current amplifier circuit complexity always generates additional structural degrees of freedom, which can be used to implement certain impedance conditions. So in reviewed scheme the input signal source can be connected to the base of VT4, and the load to the emitter of VT1. In this case, at pole parameters retention the gain at the pole frequency is determined by the following relation:(28)K0=-Qα4h11.1h11.3+h11.4τ2τ1R1R2+h11.1+h11.2.This property of scheme is convenient to use by additional intersection connections implementation in high-order filters and by integration of band-pass section into different IP-blocks [3].


## 6. The Main Conclusions

In accordance with the stated task of the work and achieved results it is possible to formulate a number of theoretical conclusions. Firstly, the increasing of active elements use efficiency, in particular current amplifiers, is connected with the usage of symmetric frequency-defining chains in their feedback. This property ensures the independence of the pole frequency (5) from the active element gain and also maximal increasing of the Q-factor (Q=1/dp) (7), (8). Secondly, this property equalizes the impact of the active elements unity gain area Bs on the parameters of the poles (10), (11). In particular, the use of noninverting amplifiers (K>0) is associated with the creation of passive frequency-defining chains of band-pass type, at the same time according to (9) the required value of gain is reduced and sufficient structural degrees of freedom for gain area Bs maximization are implemented [9]. Finally, the degree of influence of Bs is directly determined by own attenuation of passive circuit pole Dp. From this point of view the expediency of separated RC circuits using is justified and an area of parametric trade-off between different components of the overall sensitivity of schematics is created. This statement requires some comment.As follows from the general relation (5) and its consequence (13), the pole frequency is determined mainly by the parameters of resistors and capacitors of passive circuit, hence the maximal deviation of amplitude-frequency characteristic and phase-frequency characteristic of sections is determined by the corresponding sensitivity and technological and temperature variations of resistors (ΘR) and capacitors (ΘC). That’s why this contribution is proportional to the realizable Q(29)[Δ|F(jω)||F(jω)|]max=Q(ΘR+ΘC),ΔargF(jω)=(ΘR+ΘC).On the other hand (expressions (9), (14)) the pole attenuation and Q-factor are determined by the accuracy of the implementation of these elements ratios (ΔΘR, ΔΘC), that are significantly less than ΘR and ΘC for all technologies. Therefore, at the practical problems solution, the realization of higher parametric sensitivity of this parameter may be accepted (almost up to equal the contributions of ΘR, ΔΘR and ΘC, ΔΘC). From (9) and the properties of symmetrical RC circuits of the second order (the functional dependence of Dp and D0) follows
(30)D0=2m,Dp=2m+1m,Smdp=3mQ,SKdp=-QK1m,
where m is the ratio of single-type parameters of passive elements (R or C) with accuracy of implementation that is determined by ΔΘR or ΔΘC.Thus, the condition of equality of the contributions is associated with the implementation of conditions(31)ΘR=ΔΘR3m2,ΘC=ΔΘC3m2,
that define acceptable maximal value of
(32)m≤2ΘR3ΔΘR,m≤2ΘC3ΔΘC.As can be seen from (30), (10), (11) mopt=1 is the sufficient condition for the nondominant nature of the influence of attenuation on the instability of the frequency characteristics of filters. In this case, Dp=Dpmin=3, D0=2 and, as follows from (9), K<3 for any Q. So
(33)Smdp=-SKdp=3Q;SBSωp=SBSdp<-92ωpBS,
and the implementation of the required quality factor is associated with the creation of qualitative current amplifier with 1≤Ki<3 [9].Similarly, for the separated symmetric chains (example in Figure4) can be obtained
(34)D0=m,Dp=m+1m
and shown that by mopt=1(35)Smdp=-SKdp=2Q,SBSωp=SBSdp<2ωpBS
and for any Q value it is required
(36)1≤Ki<2
the lower value of the gain at the wider range of operating frequencies.The last relation shows the benefits of schemes with separate RC circuits when the specified range of valuesKi is obtained by bipolar transistors geometry and provides greater stability of the gain.


## 7. Conclusion

The achieved results show that the limiting estimations of the effective usage of active elements for localization of transfer functions poles (ωp,dp) of electrical filters create additional opportunities to optimize the structure and parameters of current amplifiers. In this case, as it follows from (35), the optimal relation of parameters of frequency-defining passive elements not only does not increase the overall instability of the frequency characteristics of filters, but also permits to reduce the influence of gain area (unity gain frequency) on the attainable operating frequency range of scheme. That is what ultimately allows to implement the total decomposition of simple procedures in active filters [10]. As seen from reviewed example of band-pass section, by this approach the functional adjustment of their parameters is easily realized via the mode dependence of input impedance and current amplifier gain. The designing of such amplifiers is possible by the use of common microcircuitry principles of self- and mutual compensation of the parasitic parameters influence of both field effect and bipolar transistors.



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*Source: 319896-2012-09-27.xml*