FUNCTIONAL AND DETERMINISTIC TUNING OF HYBRID INTEGRATED ACTIVE FILTERS

The two main methods of tuning hybrid integrated active filters, namely functional and deterministic tuning are described. Functional tuning implies the fine adjustment of an active network that is assembled and in operation so that its transmission characteristics (e.g. gain or phase response) can be monitored during the tuning process. One or several resistors are selected as tuning elements and these are adjusted (e.g. by laser, anodization, sandblasting, "postbox" selection of discrete resistors) until the desired response is obtained. Deterministic tuning involves the fine adjustment of individual resistors to values which are analytically predicted. The prediction is based on network equations which include parasitic effects and in which those component values that are not to be adjusted are obtained by measurements on the (non-operational) manufactured circuit. Manufacturing tolerances and parasitic effects are included in the comprehensive set of equations whose solutions provide the final adjustment values of a few ’tuning resistors’. Accuracy bounds and limitations of the two tuning methods are given and the pros and cons of the two methods are discussed. Finally it is suggested that a well balanced combination of the two methods may well provide the most efficient and least complicated solution to the tuning problem in hybrid integrated networks.


INTRODUCTION
As active hybrid-integrated filters are increasingly being developed for modern communication systems, 1-7 so the question of how to tune them to specifications most efficiently, and at minimum cost, is becoming ever more important.In practice two basically different tuning methods can be distin- guished, namely functional and deterministic tuning. 8unctional tuning implies tuning the critical parameters of a network while it is functional, i.e. in operation.Because the network is assembled as for operation in the final system, any parasitics built into the network are automatically taken into account and "tuned out" during the tuning process.
Functional tuning is generally iterative, particularly if the tuning steps are interactive (Figure 1).The number of iterations will increase with the degree of tuning accuracy required.The larger the number of iterative tuning steps, the more time consuming, and therefore the more costly, the tuning process will be.Functional tuning will generally be preferable for laboratory purposes and when production quantities are moderate or low.
Deterministic tuning implies tuning, or trimming to value, individual components of a network as predicted by a combination of comprehensive net- work equations (in which parasitic effects are taken 79 into account) and by component measurements (Figure 2).The solutions of equations (generally obtained by an on-line computation facility) provide the values of the components to be tuned.Tuning is carded out "to value", hence it makes no difference whether the network is operational or not.Since the components to be tuned are invariably resistors, this method consists of "resistor trimming", in contrast to the tuning of network characteristics (e.g.amplitude, phase, frequency) that occurs in functional tuning.The method is simple ("to-value" trimming of resistors is essential in any hybrid integrated circuit manufacturing plant) and rapid in execution (generally very few, if any, iterations are required).However, powerful computer programs will be required to solve the nonlinear network equations which must take first, and often second order parasitic effects into account.Deterministic tuning is the more efficient of the two methods, but the necessary expenditure of an on-line computation facility, and the initial computational effort required, can generally be justified only by very high pro- duction volumes.
In practice it will very often be found useful to combine functional with deterministic tuning.The initial adjustments will be carried out by deterministic tuning, where the values are obtained from either the idealized network equations or from

Runtuned Rtuned
Tuning Resistor R ing Tolerance

FIGURE
Functionally tuning to a parameter value No with the tuning resistor R. those containing at most first-order parasitic effects.
To overcome more subtle second-order parasitics, a fine-tuning step is then undertaken in which the circuit is operational (i.e.assembled and powered) and a functional adjustment of one or more of the critical parameters is carried out.In this way the computational complexity inherent in deterministic- only tuning can be considerably reduced.

FUNCTIONAL TUNING
Functional tuning is based on a set of network equations in which the changes of the specified net- work characteristics Fj,j-- 1,2 m are related to incremental changes in the components xi, 1, 2,..., n, by the sensitivity matrix, hence: | x;m I where the sensitivities are defined by dx .
Hence, if a (i 1,... n) are the characteristic tuning parameters of a network function F(ai) and Rai are the corresponding tuning resistors (Figure 3), then tuning the parameters from a + i to a will adjust the network function from its initial form F(ai + i) to the desired form F(ai). tting dai/ai S'a dRaj/Raj we then obtain the sensitivity relations: (3) dai/a cla./a.
where [S] is the sensitivity matrix.
In order to allow for a noninteractive, and hence noniterative, tuning procedure the sensitivity matrix must be a diagonal matrix, meaning that all off- diagonal matrix elements must be zero (Figure 4a).
In practice this will rarely be the case.However, it may be possible to arrange the sensitivity matrix such that it is triangular with the upper triangular elements being zero (Figure 4b).For each network character- istic ak there is then a tuning element Rak that leaves all previously tuned parameters aj Q" < k) unaffected.The tuning sequence is critical; it results directly from the sensitivity matrix after the latter has been arranged in triangular form.The obtained sequence provides a single-pass (or one-shot), noninteractive tuning procedure requiring no itera- tions.If a triangular matrix in the form of Figure 4b cannot be obtained accurately, it must be approximated by arranging the matrix elements such that they decrease in value to the right of the diagonal (Figure 4c).In this way, the number of tuning iterations can be minimized.Consider, for example, a second-order network possessing two conjugate complex poles p x, : -o + j.The corresponding transfer function will have the form" N(s) T(s) (5) (s + p )(s + p: ) Assuming that the two poles p and p: are specified and must be tuned for, we now derive the matrix equation corresponding to (4) with respect to the n resistors in the network.(Capacitors cannot be adjusted in hybrid-integrated circuits and are therefore not considered as tuning components.)We obtain: "dPl Pl dp: P: Since Pl and pz are complex conjugate, the variations of copl, cop and qp,, qp will not be independent of one another and we need consider only the variation dcop/cop and dqp/qv.fora second order network.
Let us now consider the second-order active band- pass network shown in Figure 5.The voltage transfer function is s s T(s) K K (s + p l)(s + P: ) + qp (10) In practice, cop and qp rather than the pole pair Pl and p will be specified, since the former are accurately measurable quantities.In addition, the constant K will be specified, although in general with a much wider tolerance than cop and qp.Deriving the sensitivity matrix with respect to the five resistors of the network, we obtain" dK/K dcop/cop dqp/qp -1 0 0 -1 -0.25 -0.25 -0.5 0 0 0 .25 -1.75 1.5 1.5 -1.5 -dR /R - We must now rearrange the sensitivity matrix in Eq. ( 11) so as to provide a tuning sequence com- prising three resistors and requiring a minimum number of iterations.(Clearly neither a purely diagonal nor a triangular matrix can be obtained.) The best we can do is with the resistors R1, Ra and where RK =RI ,Rwp Ra andRqp =R4.The optimum tuning sequence is then R (K) R a (cop) -R4 (qp!kwhereby a slight error will be accrued in K since SRq p :fi: O.

ACCURACY CONSIDERATIONS FOR FUNCTIONAL TUNING
When tuning for the frequency response of a network, we have the choice (inthe case of minimum-phase networks) of tuning to a specified amplitude or to a specified phase.The question is which of these two physical quantities provide more convenient indicators for the tuning process.If we tune for an amplitude response as indicated in Figure 6a we are concerned with the variation of the amplitude a (e.g. in dB) at a particular frequency coi with respect to the variation of a corresponding tuning resistor Rai, thus: [da(coi) t,e irca(''i)] [ dRai where: dot dR/R (12b) Similarly, tuning for a phase response (Figure 6b), the corresponding relationship between the phase (i) and the tuning resistor Ri will be" At first sight it would seem that the sensitities Eq. (12b) and Eq.(13b) will determine whether the amplitude a or the phase is a preferable tuning indicator for a given minimum-phase network.However, it can be shown a that no matter what the net- work, the phase is always a more accurate indicator with which to tune the response of a network.In fact the tuning accuracy of the amplitude response of a second-order all-pole network is within +2A% or _+0.2ACdB, when trimmed by means of a phase meter with an accuracy of -+A .T his is independent of the pole Q.Thus, with the attainable phase accuracy given by A degrees, the resulting amplitude accuracy in dB will be: It follows that a phase error corresponds to a 0.2 dB error in amplitude.Phase meters with up to 0.1 phase accuracy are available at prices considerably lower than voltage meters with comparable accuracy (i.e.0.02 dB).It follows that wherever possible, functional tuning by phase is to be preferred over tuning by amplitude.The pole frequency error resulting from a phase error A is then: fP [%1 -Tr 100/Xq (15) fp 360qp qp where A is measured in degrees and qp is the pole Q.
Note that the higher q p the smaller the frequency error resulting from a given phase error.Thus, with 0.1 phase error, the frequency error will be -0.1% if qp and -0.01%if qp 10.The accuracy of qp associated with a phase error of A degrees can be shown to be independent of qp.
Consider, for example, the lowpass amplitude and phase characteristic shown in Figure 7. From the corresponding transfer function and the phase response: (co) arg T(d'Co) (18b) To tune for the specified cop and qp values, we tune for -90 at the frequency cop/2r using an appropriate resistor Rto and for -45 (or -135) at the P frequency co4 s/2r (or co, a s/270 using an appropriate resistor Rqp.With a phase meter accurate to within A degrees, the errors in amplitude, frequency and qp will be as given by Eq. ( 14), ( 15) and ( 16) respectively.K can also be tuned for.Remember that an error in K implies an error in the dc gain of the network. 4. DETERMINISTIC TUNING For a given network, the deterministic tuning procedure follows the flow-chart presented in Figure 8. Thus for the second-order lowpass network shown in Figure 9, whose transfer function was given by Eq. ( 17), we proceed as follows 8 1) Derive the three characteristic network parameters K, cop and qp as a function of the circuit components, i.e.K= fr(fl, R,,R2, Ca, C4) Cop fwp(R,R2,R4, Ca, C4) qp fqp(fJ, R,R2,R4, Ca, C4) 2) Measure the capacitors Ca, C4 and the closed- loop amplifier gain/3 3) Compute the following resistor values as a function of quantities that are either specified or measured, i.e.R fR, (K, cop, qp, Ca, C4, [3) R2 fR (K, cop, qp, C3, C4, fl) R4 fR, (K, COp, qp, Ca, C4, t3) 4) Trim resistors R t, R2 and R to the values computed under 3).
At this point some comments seem in order.
The equations derived in step 1) and 3) are tuning equations and not design equations.The derivation of the design equations precedes the tuning process and determines the nominal value of each component including the gain .S ince there are more components than there are design equations, various optimization criteria can be taken into account in the design, such as minimum sensitivity, minimum gain-sensitivity product, ease of tuning, minimum component  etc.In any event, the design, or nominal values of each component are known before the tuning process is begun.Thus, in the example of Figure 9, although the nominal value of is known, the actual value of obtained in.manufacture is measured in step 2) by measuring resistors R6 and R 7 (Note that at this point the amplifier is assumed to be ideal.) For the Sallen-Key type circuits shown in Figures 5 and 9,/3 is not made equal to unity, but is larger (e.g. between one and two) if tight tolerances are specified for qp.If necessary a final functional correction of qt can thereby be carried out without affecting the pole frequency p.Although this entails the inclusion of an additional resistor, namely R7 (R6 is required to balance DC offset), this is a small price to pay compared with the increase in yield afforded by the possibility of a final touch-up tuning step.The deterioration of gain stability incurred by this slight increase in gain is negligible o in hybrid- integrated filters, due to the very close resistor tracking obtainable with either thick or thin film resistors.The combination of deterministic and functional tuning implied here will be further discussed under section 6.
The measurement of the capacitors (i.e.C3 and C4 in step 2) above) should preferably be carried out after the (chip) capacitors have been assembled on the substrate.(In the case of thin film capacitors there is, of course, no other choice).This permits parasitic capacitance and assembly drift to be taken into account during the measurement.Although methods exist for the accurate measurement of individual capacitors when they are connected to additional circuitry8, the preferred way of overcoming this problem is to provide provisional capacitor contacts to the substrate edge.These permit the measurement R7=R R6= (8-   To attain a one-pass tuning procedure, parasitic effects due to nonideal circuit components must generally be taken into account in the network equations derived under step 3 above.This complicates the required computations considerably.The main parasitic effects that must be contended with are: i) Non-ideal characteristics of the active devices (e.g.frequency dependent gain of the operational amplifiers).Referring to Figure 10, this means that instead of using the constant gain/30 in our equations, we must use/(s), thus: ii) Losses and frequency dependence of capacitors.
Thin-film capacitors, for example, are both lossy d frequency dependent.If the loss of a capacitor Ci is tan i, then, instead of Ci, we must approximate Ci as follows:

Ci
Ci Ci Ci(1 _/6i ) Instead of frequency-independent capacitors, thin- film capacitors will be frequency-dependent according to the relationship: whereby the time constt rc characterizes the tecology used.Assuming that the value of C(w) is measured at two frequencies (see Figure 11), the value of C(p) can be obtained by extrapolation as follows: log C(wp) log C(,) + rc(p ) where log C(= )/C(w ) r iii) Parasitic capacitances on the circuit substrate and resistive losses along conductance paths.
These parasitics must be taken into account, to the extent that the response of the final assembled circuit is to be as accurate as if it had been tuned functionally.In doing so, the computations required under 3) above become rapidly more complex, the equations highly non-linear and of third, or even higher order.With increasing complexity, only numerical solutions by computer can be expected, whereas with ideal components, analytical solutions may be obtained.A more detailed discussion of the tuning of hybrid-integrated circuits with nonideal components is given elsewhere. 8

ACCURACY CONSIDERATIONS FOR DETERMINISTIC TUNING
Using the deterministic tuning procedure, the accuracy attained depends on the accuracy with which capacitors can be measured, and resistors can be measured and trimmed.Assuming a worst-case measuring error of AC/C for all capacitors and an equal worst-case trimming error of zLR/R for all resistors, we obtain a frequency error given by < + -- (24) O9 max max max With a 0.1% accurate capacitance bridge and a capability of trimming and measuring resistors to within 0.05%, the worst-case frequency error will be 0.15%.
The accuracy with which the capacitor losses are measured will also effect the frequency accuracy, adding a term to the two in Eq. ( 24).Assuming a loss- measurement error of Ai, we obtain the additional term: AW ( ) w 2qp 4qp This error is significantly smaller than the one represented by Eq. (24).Measuring the capacitor losses to within 10% accuracy (i.e.A6 0.1) and assuming a pole Q of only 2 (i.e.qp 2), the frequency error given by Eq. ( 25) is 0.02%.In general, any function F that is tuned to value using a resistor RF will be accurate to within F(1 +-AF/F), where ARF (26) and F/RF is the trimming and measuring accuracy of the resistor R F. Consider, for example, the case in wch qp is to be tuned deterministically by adjusting the closed-loop gain .T his, in turn, is determined by a resistor ratio say Ro/R, where R is the trimming resistor (e.g.R6 or R v in Figure 9).The qp accuracy follows as (27) For the network of Figure 9 Eq.( 27) becomes: qp q R where R R6.If Rv were used for R the expression on the right-hand side of Eq. (28) would be negative, a With a capability for measuring and trimming R to thin 0.05% accuracy, and th the typical values of fl 2, q 0.4 and qp 20, We obtain qp accurately to within 1%.Note that with currently available measuring equipment deterministic tuning can readily compete with functional tuning in terms of accuracy, profided that a sufficiently thorough computational effort and adequate computer facilities are invested in the process. 6. COMBINING DETERMINISTIC WITH FUNCTIONAL TUNING Because deterministic tuning entails only component measurement, computation, and resistor trimming, it is the preferred tuning method, provided the production quantities are large enough to justify the initial computational effort (deriving the network equations including parasitic effects) and the cost of on-line computation facilities.Very often, the initial computational effort can be significantly reduced by deriving the equations of the idealized network and, after initial deterministic tuning, correcting the resulting error by a small number of functional tuning SELECT n OF steps.This procedure, outlined in Figure 12, eliminates the numerous and time-consuming iterative tuning steps generally required by the purely functional tuning procedure.It accomplishes this by reserving a small number (typically one or two) appropriate resistors for a final touch-up, or vernier functional adjustment after the circuit has previously been "coarse adjusted" deterministically using simple, i.e.

SUMMARY
The two main tuning procedures used for hybrid- integrated active networks have been described.Their main characteristics are summarized in Table I.Functional tuning is conceptually simple and very effective in that all stray and parasitic effects of a practical circuit can be tuned out "in situ" with the circuit in operation.The main disadvantages of the  Suitable for lab-purposes and Suitable for high production low production quantities quantities method are the required time consuming iterations and the high-accuracy measurements of such parameters as phase, amplitude and frequency.Such measurements are often alien to the typical hybrid- circuit manufacturing facility which is equipped for high-accuracy resistance and capacitance measure- ments only.Functional tuning thus becomes particularly useful for prototype and lab purposes and for low-quantity production.Deterministic tuning is based on the more sophisticated concept of predicting all relevant parasitic effects analytically and, by a combination of component measurements and computations of resistor values, trimming resistors to value such that the required network response is obtained.It is thereby a "one-shot" process in that, ideally, each resistor need be trimmed only once.The disadvantages of the method are the complexity of initial computation and the luxury of on-line computing facilities.Given sufficiently high production quantities, however, such initial outlays in brain and computer power may readily be justified and subsequently amortized.Finally, for the most common "intermediate" situation where production quantities are neither so high nor so low as to obviate the question of the most suitable tuning method, a combination of functional and deterministic tuning seems to offer the best results.

2 TFIGURE 7
FIGURE 7 Amplitude and phase characteristic of a second- order lowpass network.

FIGURE 10 Frequency
FIGURE10 Frequency-dependent gain of an operational amplifier.