Wave Active Filters
Johan Borg
LiTH-ISY-EX-3401-2003 Linköping 2003-06-05
Wave Active Filters
Master’s Thesis
performed at Electronics Systems Linköpings Universitet byJohan Borg
LiTH-ISY-EX-3401-2003Supervisor: Emil Hjalmarson Linköpings Universitet Examiner: Professor Lars Wanhammar Linköpings Universitet
Spr ˚ak Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats ¨Ovrig rapport
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The first part of this thesis contains a literature study of current tuning tech-niques for continuous-time integrated filters. These tuning methods are charac-terised by which quantity they measure, their dependence on certain character-istics of the input signal, or matching of components on chip. The structure of the different tuning schemes are explained. The merits and drawbacks as well as achieved accuracies of previous works are summarised.
The second part is a study of wave active filters (WAFs), a less common structure for implementing active filters. In this structure the filter is realised by simulating the forward and reflected voltage waves present in the prototype filter. The main advantage of this is that the inherent low sensitivity of doubly terminated ladder-filters is better preserved than in many other structures. Two Mosfet-C realisations of Wave Active Filters have been suggested and high-level simulations have been used to compare them to the originally proposed implementation as well as a leapfrog implementation.
Electronics Systems,
Dept. of Electrical Engineering 581 83 Link¨oping 2003-06-05 — LITH-ISY-EX-3401-2003 — http://www.ep.liu.se/exjobb/isy/2003/3401/
Studies on Tuning of Integrated Wave Active Filters Studie av avst¨amning av integrerade aktiva v˚agfilter
Johan Borg
× ×
Abstract
The first part of this thesis contains a literature study of current tuning tech-niques for continuous-time integrated filters. These tuning methods are char-acterised by which quantity they measure, their dependence on certain characteristics of the input signal, or matching of components on chip. The structure of the different tuning schemes are explained. The merits and draw-backs as well as achieved accuracies of previous works are summarised. The second part is a study of wave active filters (WAFs), a less common structure for implementing active filters. In this structure the filter is realised by simulating the forward and reflected voltage waves present in the proto-type filter. The main advantage of this is that the inherent low sensitivity of doubly terminated ladder-filters is better preserved than in many other struc-tures. Two Mosfet-C realisations of Wave Active Filters have been suggested and high-level simulations have been used to compare them to the originally proposed implementation as well as a leapfrog implementation.
Table of Contents
1 Introduction 1
1.1 Background 1
1.2 Outline of this Thesis 2
1.3 Purpose of this Thesis 2
2 On-Line Tuning 3
2.1 Master-slave Frequency Control 3
2.1.1 Gm or R - only Tuning 3
2.1.2 Capacitor Charge Based Tuning 4
2.1.3 Integrator and First-Order Filter Based Tuning 7
2.1.4 Phase-Locked Filter 8
2.1.5 Phase-Locked Oscillators 10
2.2 Master-Slave Q-value Control 12
2.2.1 Phase-Locking an Integrator 12
2.2.2 Amplitude Locking Passband Gain 13
2.2.3 Envelope Based Q-value Tuning 16
2.3 True On-Line Tuning 18
2.3.1 The Correlated Tuning Loop 18
2.3.2 Orthogonal Reference Tuning 20
2.3.3 Tuning by Using Common Mode Signals 21
3 Off-Line Tuning 23
3.1 Frequency-Tuning 23
3.1.1 Step Response 23
3.1.2 Forced Oscillation 24
3.2 Combined Frequency and Q-value Tuning 24
3.2.1 Sweeping the Frequency Control Voltage 24
3.2.2 Two Reference Frequencies 24
3.2.5 Model Matching 26
4 Wave Active Filters 29
4.1 Introduction to Wave Active Filters 29
4.2 Sensitivity 31
4.2.1 Time Constant Errors 32
4.2.2 Gain Errors 34
5 Mosfet-C Implementation of WAFs 37
5.1 Background 37
5.2 Possible Structures 38
5.3 Sensitivity to Component Errors 40
5.4 Sensitivity to OP-Amp Bandwidth Variations 41
6 Mapping of S-parameter Errors to Passive Components 45
6.1 Analytical Mapping 45
6.2 Approximate Mapping by Optimization 46
7 Tuning Strategies for Wave Active Filters 51
8 Conclusions and Future Work 53
8.1 Tuning of Continuous-Time Integrated Filters 53
8.2 Wave Active Filters 53
1 Introduction
1.1 Background
Even though continuous-time integrated filters are usually replaced by switched capacitor filters where feasible, many important applications remain, such as anti-aliasing filters for high-speed data-converters and read-channel equalizers for hard-disk drives.
The main reason for using continuous-time filters is their speed. A compaira-ble switched capacitor filter for signals in the MHz-range or higher would require excessively high clock frequencies, with high power consumption and clock-feedthrough as a result. Furthermore, high-performance operational-amplifiers (OP-Amps) will be required to obtain settling-times sufficiently low for the switched capacitor circuits to reach steady state within half a clock period.
On the other hand, the main reason for using switched capacitor filters is their stability. Since all passive elements are realised using capacitors only, the fre-quency characteristics will only depend on the capacitor sizes and their rela-tive accuracy, which are typically less than 0.1% [1], and the clock frequency. For continuous-time filters this is not true, both capacitors, and either resis-tors or transconducresis-tors are used to realise the filter, the ratio of their sizes will determine the overall frequency characteristics. Unfortunately, chip to chip variations of RC or Gm/C can be in the order of 30% [2].
Because of this, it is usually necessary to implement some form of frequency control, “tuning”, to ensure that the filter meets the specification.
Integrated filter design is further complicated by the fact that high perform-ance filters are sensitive to component variations. Because of this, it is often necessary to introduce some type of control over other parameters in the fil-ter, in order to compensate for effects such as parasitic loads and device mis-match.
The sensitivity to component variations is also highly dependent on the struc-ture of the filter, for example lattice, filters are generally only suitable for crystal filters, as they are extremely dependent on element stability, while doubly terminated LC-ladder filters are relatively insensitive to small changes in component values.
1.2 Outline of this Thesis
• Chapter 2 - On-Line Tuning Chapter 3 - Off-Line Tuning
These two chapters contain the results from a literature study on the sub-ject of tuning of continuous-time filters.
• Chapter 4 - Wave Active Filters
A background on wave active filters as well as an initial study of their performance in respect of component variations.
• Chapter 5 - Mosfet-C Implementation of WAFs
Attempts at finding a Mosfet-C implementation and the performance of the resulting candidates.
• Chapter 6 - Mapping of S-parameter Errors to Passive Components
Further studies of the relation between filter defects in S and component domains.
• Chapter 7 - Tuning Strategies for Wave Active Filters
Some words on proposed tuning strategies for WAFs • Chapter 8 - Conclusions and Future Work
• Chapter 9 - References
1.3 Purpose of this Thesis
The purpose of this thesis:
• Perform a literature study of present works on tuning of continuous-time integrated filters.
• Study possible MOSFET-C implementations of wave active filters. • Investigate if it is possible map scattering-parameter errors of wave active
2 On-Line Tuning
Because the parameters of integrated active filters depend on temperature, supply-voltage and ageing, a tuning method that is active at all times (referred to as “on-line”), is usually required. The opposite is off-line tuning, where the filter is only tuned when inactive.
The most common way to tune an integrated filter is by using a “master-slave” tuning scheme. One or more filters are used as reference, continuously tuned by a control circuit to meet some reference performance. The control signals from this process can then be used for tuning the filter(s) processing the actual input signal. One way of implementing this is shown in Fig. 1. Since the slave filter is never measured on, the accuracy of the tuning will be limited by the matching of the master and slave filters. Another problem with having the reference filter and the tuning-circuit operating continuously is the possibility of undesired signals from the tuning process leaking into the main signal path.
2.1 Master-slave Frequency Control
2.1.1 Gm or R - only Tuning
Where the required frequency accuracy is low, simply making sure that the transconductances (for Gm-C filters) or resistances (for (R-)MOSFET-C) are correct provides a simple solution. Since variations in capacitor values are not taken into account, the accuracy of the filter after tuning will be limited by the process variations of the capacitor values, which is usually about 10% [2].
Figure 1: The principle of master-slave tuning
Vref master filter Vc slave filter Vin Vout control 1 Vcn . . . . . . . .
For example, in Fig. 2, the voltage difference Vbwill make the transconduc-tor output a current, Iout=GmVb. At the same time, there is a voltage difference Vbover the off-chip resistor Rext, resulting in a current I=Vb/Rext. If these cur-rents are equal, no current is going into the integrator. An incorrect transcon-ductance Gm will cause a difference in currents, this difference will be integrated over time, until the control-voltage Vfhas changed enough to cor-rect the Gm value.
In [3] a 7th-order equiripple lowpass filter tuneable over 30-100MHz was built. Since it is designed for hard-disk read-channel equalizing, no data on cut-off frequency accuracy is available, as this is secondary to the group delay ripple.
Similarly, in [4] an elaborate scheme for tuning ratios of conductances and time constants are presented. Maintaining these ratios is in this case neces-sary to ensure that the filter meets the group delay ripple specification. On the other hand, the cut-off frequency control is mentioned only as “external”.
2.1.2 Capacitor Charge Based Tuning
A more accurate method is to replace the reference resistance with a switched capacitor equivalent, and thereby control a Gm/C or R/C ratio directly. In the-ory this would look like Fig. 3, but that approach is usually not realistic, due to the high clock frequency required if Gm and C have similar value to those used in the slave filter (which is preferable to achieve good matching). This can be solved by using the circuit in Fig. 4, which scales down the clock fre-quency a factor N, by using two currents of a ratio 1:N.
Figure 2: Gm - only tuning
Rext C -Vb Gm I Vf Iout I
A lowpass filter is usually required to reduce reference-signal leakage into the slave filter. While simple, these filters can become quite large, due to the large capacitances required for large time-constants. In some cases an off-chip capacitor has been used [5].
Another option is to lock the time-constant directly to a reference clock, like in Fig. 5, where the capacitor C is charged with a current determined by the transconductor, and the peak reached during 1/2 clock cycle is compared to the voltage Vb [6].
Figure 3: Gm/C tuning by using SC-circuit
Figure 4: Improved Gm/C tuning using SC-circuit
Figure 5: Gm/C tuning by locking the time constant to the period of a
reference clock C -Vb Vf C I m Gm LP filter 2 φ 2 φ 1 φ 1 φ C -Vb Vf 1 C I m Gm NIB IB LP filter φ 1 φ 2 φ 2 φ Vb Vf Gm Peak Hold Tclk Vb G 1 φ
For active R-C and similar filters, the circuit in Fig. 6 could be used, either with Vfdirectly controlling a bias to the mosfet-resistances in case of a Mos-fet-C filter, or through a comparator controlling a counter, in turn switching different R or C elements in or out [8].
Here the current Vb/R through the resistance will be balanced against the cur-rent -fVbCm transferred by the switched capacitor.
All methods discussed so far have the advantage of being very simple, and as opposed to most other methods, the reference-signal is not required to be a sine wave with low distortion.
Using a reference clock of a frequency considerably lower than the operation frequency of the filter, like in Fig. 4, will reduce the problem with reference-signal leaking into the main reference-signal path.
Accuracy will largely depend on offsets in the active components, but also on achieving good matching with the slave filter. This may be difficult since the structure of the master filter is fundamentally different from the slave filter. This may result in parasitics affecting the nodes differently, with a systematic error as result. Tracking of production spread and temperature variations are also likely to be relatively low when these methods are used.
In [7] a 4th-order 10.7MHz bandpass filter was tuned to a frequency accuracy of 1% by using a circuit similar to that in Fig. 4.
In [8] a 14th-order Chebyshev bandpass filter operating in the 165-505kHz range, was tuned to an accuracy of 1% by a circuit similar to that in Fig. 3. Here a reference-frequency well above the operating frequency of the filter was used.
In [9] three different 78kHz active-RC filters with 5 bit binary weigthed switchable capacitor arrays, controlled by a circuit similar to that of Fig. 6 operating in a dual-slope mode were implemented. Frequency accuracies of 5% were obtained.
Figure 6: R-C filter version of tuning using SC-circuit
Cm Vf R CI Vb 1 φ 1 φ 2 φ 2 φ
2.1.3 Integrator and First-Order Filter Based Tuning An ideal Gm-C integrator will have the transfer function
(2.1) Solving for |H(ω)|=1, we get:
(2.2) Which means that the unity gain frequency of the integrator will be
(2.3) As described by Fig. 7, this can be used to control the Gm/C ratio, comparing the peak level of the reference-signal before and after it has passed through a reference integrator. If the Gm/C ratio is correct, the output from the integra-tor should have the same amplitude as the input. Any difference in amplitudes will be integrated over time by the second integrator, and the control signal Vf changed to modify the value of Gmuntil the correct Gm/C ratio is obtained. The signal Vf is then used to control the transconductances in the slave filter.
For a non-ideal integrator it can be shown [10] that the frequency error will be below 0.1% if the DC-gain is larger than 40dB and the phase-error at unity gain is smaller than 1 degree.
Figure 7: Tuning using unity-gain frequency of the integrator
H s( ) Gm sC ---= ω Gm C ---= f 1 2π ---Gm C ---= Vref Vf LP filter C Peak Detect Peak Detect CI G Gm
When this method is used, the input offset of the transconductor must be low enough to keep it from saturating, since no DC loading or DC feedback exist. One way of avoiding this problem has been proposed in [11]. In Fig. 8, the new transconductor will simulate a resistance R=1/Gm, and the transfer func-tion becomes:
(2.4)
Here, instead of the unity gain, the -3dB frequency is used.
The choice of using a peak-detector or the square of the signal and low-pass filtering the result, for measuring a signal amplitude, seems arbitrary in most cases, but here the latter might have an advantage. This is because taking the square of a signal with a relative amplitude of -3dB will result in an output DC-level of half that of a 0dB input signal. A peak-detector is on the other hand designed to preserve a linear relationship between input amplitude and the output voltage, and will thereby produce an output of times that of a 0dB signal. In this case, when a ratio of the signals should be 3dB, imple-menting the attenuator after the squaring amplitude detector may improve accuracy, since it is usually easier to implement accurate integer ratios. This type of tuning has also been implemented in [12], [13] and [14], for tun-ing different circuits, but no useful experimental data is available on tuntun-ing performance.
2.1.4 Phase-Locked Filter
The main feature of this method is that good matching between master and slave is relatively easy to obtain, since both are filters and can be built using similar structures.
Figure 8: Tuning using a degenerated integrator.
Gm LP filter C Envelope Detector .5 Envelope Detector Gm Vf Vref H s( ) 1 1 sC Gm ---+ ---= 1 2⁄ ( )
In Fig. 9, a sine-wave reference-signal is used as input to the master filter. The phase of the output signal from the filter is compared with that of the ref-erence-signal. The phase comparison is carried out by multiplying the sig-nals, as the DC-component of the product of two signals with the same frequency will depend on the phase difference. If there is a 90 degree phase difference the output will be zero. The output from the multiplier is integrated over time, and used as the frequency control signal. This will effectively lock the phase-shift through the filter at 90-degrees, as a different phase shift will produce a DC output, which will be integrated until the control signal has changed enough to correct the phase shift.
A second order lowpass filter is usually used for the master filter, as it will have a 90 degree phase shift at its -3dB frequency. This is true even when the slave filter is of a different type or order, because locking to a 90 degree dif-ference usually simplifies design. Filters of higher order may also have more than one frequency where the phase difference is 90-degrees. Thus, there is a possibility that the tuning-circuit may converge to the wrong frequency (pro-vided the tuning range is sufficiently large).
Other types of filters may be used, however using a filter with a degree phase shift at the reference-frequency usually simplifies the design. If 0 or 180 degree phase shift is used, either the quadrature component of the refer-ence-signal or an additional 90 degree phase-shift will be required.
In some applications it might still be advantageous to use a notch-filter instead [15], especially if the location of a zero in the transfer function is important. When using a notch-filter as a reference, the output signal approaches zero as the frequency of the zero in the notch-filter approaches the frequency of the reference-signal. This will theoretically reduce the refer-ence-signal leakage to the main signal path and reduce the size of the LP-fil-ter in the frequency control loop.
Figure 9: Tuning using a phase-locked filter
Vref LP Vf filter master filter 90 ±
The phase-comparator can be a major error source, as a phase-error of 1 degree will cause a frequency-tuning-error of 0.5%, if the reference-filter is a 2nd order lowpass with a Q-values of 2. Using higher Q-values will reduce this error, but may result in reduced matching of the master and slave filters. If the initial tuning error is large enough to make the reference-frequency fall well inside the stop-band of the master filter, the amplitude of the input signal to the phase-detector will be low. If the phase-detector is based on direct mul-tiplication of the signals, the decreasing input signal amplitude will lead to a reduction of output signal amplitude. For large tuning errors, this effect will overtake the phase-detection and cause an overall decrease in output from the phase-detector. Depending on the feedback loop design, this may cause con-vergence problems. A solution for this problem is to decrease the Q-value of the master filter, as this will make the slope of the phase shallower, and this make the variations in amplitude less dramatic. Alternatively, it should be possible to avoid this problem by using a feedback loop that contains an inte-grator, as the sign of the phase signal will always be correct, even if the amplitude shows inconsistencies.
In [16] an integratorless feedback loop was used with this type of phase-detector. This resulted in a requirement of Q<2 to ensure convergence over a 30% range.
In [17] a 5th-order elliptic 1.92MHz lowpass MOSFET-C filter was tuned, no data on absolute frequency accuracy were presented, but the temperature coefficient of the cut-off frequency is said to be 100ppm/ C.
In [18] an 2nd order 78kHz lowpass active-RC filter using digitally program-mable current attenuators was tuned to an accuracy of 5%, of which the quan-tization error may account for 1-3%.
In [19] an unusually large ratio of master/slave cut-off frequencies was used, this resulted in relatively large temperature and supply voltage dependencies for the center frequency and Q-value.
2.1.5 Phase-Locked Oscillators
To eliminate the requirement of a low-distortion sine wave reference-signal and the absolute accuracy of the phase-detector, phase-locking of an oscilla-tor implemented with a structure similar to that of the slave filter, is often used. However, in order to make sure the circuit forms a stable oscillator with the active elements operating in their linear regions, new elements like non-linear negative resistances, modified transconductors or limiters are usually
required. These changes make a good matching to the slave filter harder to achieve compared to a phase-locked filter. Another approach is to try to keep a filter section oscillating by increasing the Q-value to infinity. This, however, also tend to cause a systematic frequency error.
In Fig. 10 an oscillator is formed by inserting a limiter in the feedback loop from the output to the input of a bandpass filter, which must have a passband gain larger than unity. The limiter will crop the peaks of the signal to some level. This ensures that the amplitude of the input signal is low enough for the filter to be sufficiently linear. Too high input signal amplitude will make non-linearities in the filter significant, with a change of oscillation frequency as a result.
When the tuning is complete, the oscillator is phase-locked to the reference-signal and any frequency error will make the phase error increase over time. This in turn will change the DC-output from the phase-detector and adjust the control signals for the filter. Because the phase error is the frequency error integrated over time, no stationary frequency error will remain.
Depending on the phase-detector used, locking range may be limited to only one octave, which is sufficiently wide to handle the tuning range of most fil-ters. However, in some cases when this method is used with very wide-band tuneable MOSFET-C filters, means for avoiding locking to harmonics may be required.
In [20] a 5th-order 3.4kHz elliptic lowpass filter was implemented, with a production spread after tuning of 0.5%, and a temperature dependence of 0.1% over the range 0-85 C.
In [21] a 4th-order 70MHz bandpass filter was implemented, with a system-atic frequency error of 1.5% and a production-spread of 1%.
In [22] a 1MHz 2nd order active-RC using programmable capacitor arrays, with frequency errors within 2% after tuning were implemented.
Figure 10: Tuning using a phase-locked voltage controlled oscillator
Vref V f LP filter master filter °
2.2 Master-Slave Q-value Control
For a pole Pk, the Q-value is defined as
(2.5)
For biquad filters, this is directly applicable to each biquad individually, as they implement one pair of complex conjugated poles each. When a filter of a higher order than two is implemented in a single structure, Q-values will be defined only for the realized poles, with no direct connection to the filter implementation as such.
In any case, making sure that the poles of a filter doesn’t move too far from their desired positions will be critical for ensuring that the shape of the pass-band remains acceptable.
The Q-values present in active filters are usually determined by a ratio between values of similar components, like Gm1/Gm2or C1/C2. Since the size ratio between components of the same type is relatively insensitive to process and temperature variations, Q-values should also be relatively insensitive and therefore not require any tuning. This is usually true for low frequencies and for low Q-values, where the component ratios are small and the active com-ponents are nearly ideal. At higher frequencies and larger component ratios, nonidealities, parasitics and process variations may cause considerable devia-tions from the desired Q-value.
The common methods of adjusting Q-values in a filter, are either adjusting the ratio of the component values that determines the Q-value, introducing a controllable (positive or negative) resistance element in the circuit, or in case of 2-stage active elements, adjusting a compensation circuit inside the ele-ment.
When the frequency and value tuning are not entirely independent, the Q-control loop is usually made an order of magnitude slower than the frequency control to make sure that the Q-value tuning is preformed at the correct fre-quency.
2.2.1 Phase-Locking an Integrator
It can be shown that a phase error in the active components of a filter will have considerable effect on the Q-value. When a single integrator is used as
Qk Pk Re P( )k ---– =
master for frequency control, see section 2.1.3, this phase-error will cause the phase difference over the integrator (after frequency-tuning) to differ from the ideal 90-degrees. This has been used in the tuning scheme presented in Fig. 11.
Here the reference signal and the output signals are converted to logic levels and used as inputs to an xor-gate. If the phase difference is not 90-degrees, the output from the xor-gate will not have a 50% duty-cycle. This will cause a non-zero average output current from the transconductor, charging or dis-charging the capacitor CI and thereby adjusting the control voltage VQ. The accuracy of this method will depend on the achievable phase accuracy of the phase-detector. It should be remembered that only phase errors caused by nonidealities in the transconductor are measured and corrected, errors origi-nating from inaccurate component ratios, due to process variations or parasit-ics, are not.
In [10] a 4MHz 6th-order elliptic lowpass filter was tuned by this method, they claim good theoretical accuracy for the phase-detector based on compa-rators and a xor-gate, but no experimental data on the performance of the Q-value tuning is presented.
2.2.2 Amplitude Locking Passband Gain
Fig. 12 shows the most common way of implementing Q-value tuning, sim-ply using that the passband gain of a 2nd order bandpass filter will be propor-tional to the value. If we assume that the mid-band gain is equal to the Q-value, a too low Q-value will produce an output lower than that of the ampli-fied reference-signal, this difference will be integrated over time, until the control-signal VQhas changed enough to correct the Q-value. This signal is also used to control the slave filters.
Figure 11: Q value tuning using phase difference
Vref Gm LP filter C VQ C m I
If a biquad filter is used as master in the phase-locked filter frequency-tuning loop, a bandpass-filtered signal is usually already available in the circuit, oth-erwise, a separate Q-value tuning master is used.
Q-value tuning is often used to compensate not only for nonidealities of the active elements, but also for component mismatches caused by parasitics and process variations. There have been implementations with one Q-master identical to each stage in a chain of biquads. In [23] four stages were used to make sure that all stages were compensated correctly, instead of trying to scale the compensation circuits.
If a frequency-tuning-error is present, the reference-frequency will not be exactly in the center of the passband. Because of this, the gain meassured when the reference-signal is feed through the filter will not be the passband gain of the filter. This will result in a Q-value tuning error, since the tuning-circuit will make the meassured gain equal to the desired passband-gain, and the actual passband gain will be forced to some different level. This error will be approximately proportional to the Q-value, as the passband width is the inversely proportional to the Q-value.
It has been suggested [24] that this error can be reduced (ideally eliminated) by using the circuit in Fig. 13.
Figure 12: Passband amplification based Q-tuning
Figure 13: Improved passband amplification based tuning
Vref LP filter master filter Peak Detect Peak Detect Q VQ Vref LP filter 1/Q masterfilter + + + + -- VQ
Here the change in VQis calculated as
(2.6) Where Vref and Vbpare the reference-signal before and after it has passed through the (bandpass) master filter.µ is the integrator gain.
When the tuning is complete, and no frequency error is present, both ampli-tude and phase will be equal. In case a frequency-tuning-error is present, there will also be a phase shiftφtrough the filter, which will make this circuit adjust VQ until the following condition is meet:
(2.7) This means that when the tuning is complete, the gain of the filter will be
(2.8) However, for a second order bandpass filter
(2.9)
the phase shift trough the filter will be
(2.10) Eq (2.9) and (2.10) gives
(2.11) and the magnitude response as a function of the phase shift will be
(2.12) Comparing this with Eq. (2.8), we now see that the filter will ideally be tuned to the correct Q-value, even if the reference-frequency is not in the exact center of the passband.
This method can actually be seen as an Least Mean Square (LMS) adaptation algorithm implementation, where the output from the master filter is used as an approximative gradient signal.
V˙Q = µ(Vref–Vbp)Vbp Vbp = Vrefcosφ H( )φ = Qcosφ H s( ) ω0s s2 ω0 Q ---s ω02 + + ---= φ tan Qω0 2 ω2 + ωω0 ---= H( )φ i Q φ+i tan
--- iQcosφ(cosφ+isinφ)
= =
In [24] a 10.7MHz single biquad bandpass filter with a Q-value of 20 was manufactured and a Q-value error of 0.75% was measured (after tuning). Dis-crete tests of a similar circuit with a Q-value of 10 indicated that a 3% fre-quency error would result in an 1.1% Q-value error. If a normal amplitude comparing Q-value tuning-circuit had been used, a 3% frequency-tuning-error at a Q-value of 10 would have resulted in a 16% Q-value frequency-tuning-error.
Fig. 14 shows another proposed method [25], which eliminates the require-ment of a separate Q-tuning master filter when using a phase-locked oscilla-tor for frequency control. According to [25] the method reduces the sensitivity to offsets in the tuning-circuit compared to the previous method.
A 2nd order 100MHz bandpass filter with a Q-value of 20 was manufactured, and a tuning accuracy in the order of 1% was measured.
2.2.3 Envelope Based Q-value Tuning
When a step is applied to a second order lowpass filter, the envelope of the oscillations will be equal to the step-response from a first order low-pass fil-ter, with a -3dB frequency of half the 2nd order filters bandwidth, as shown in Fig. 15.
In [26] it was proposed that this may be used for tuning the Q-value of a filter as shown in Fig. 16
Here Vrefis a square wave reference-signal, with a frequency low enough to allow the filters to reach steady state after each transition.
Figure 14: Combined frequency and Q-tuning scheme
LP filter 1/Q master filter + -+ Vref LP filter fctrl VQ
The output signals from the two filters pass trough the envelope detectors, which produce an output proportional to the square of their inputs. The out-puts from the envelope detectors are then compared and the difference, inte-grated over time, used to control the Q-value of the 2nd order filter.
By controlling the sample and hold (S&H) circuit to only sample when the filters have reached steady state, the signal leakage to the slave filters can be reduced.
In [26] a board level test circuit was built, and Q-tuning errors of 3-7% meas-sured.
Figure 15: 2nd order vs 1st order lowpass filter
Figure 16: Envelope based Q-tuning
0 2 4 6 8 −1 −0.5 0 0.5 1 t V(t) Vref S/H & LP-filter reference (1st order) filter master filter (2nd order) envelope detector envelope detector VQ
In [7] a 10.7MHz 4th-order biquadratic bandpass filter with Q=20 for both biquads was implemented, with a systematic Q-tuning error of 20% and chip to chip variations of 10%. This is attributed to offsets in the comparing transconductor, inaccuracies in the envelope-detection and frequency-sensi-tivity proportional to Q.
While not as accurate as the improved amplitude locking described in 2.2.2, it may well be comparable to the classic amplitude locking method and, if prop-erly implemented, provide an acceptable level of accuracy.
The low frequency of the reference-signal will help reduce reference-signal feedthrough to the slave filter, and possibly reduce the power consumption.
2.3 True On-Line Tuning
Ideally, one would want to measure the characteristics of the actual filter, like in off-line tuning, and at the same time be able to have both tuning and signal processing active at all times.
2.3.1 The Correlated Tuning Loop
In [27] a method for true on-line tuning of a filter is presented. It is similar to the methods described in 2.1.4 and 2.2.2 as it tunes the filter by observing the transfer function of the filter at a single reference-frequency. Instead of actively providing the filter with a known input signal, and measuring ampli-fication and phase shift at the output, these parameters are derived from the input signal. This tuning method assumes that the input signal has sufficient spectral contents at the reference-frequency, if this is not the case, conver-gence of the tuning loop can not be guaranteed.
For a linear system, the relation between the spectra at the input and output can be written
(2.13) where Gxy and Gxx are the cross power and input power spectral densities, respectively.
It can be shown [27] that signals Va,Vb, Vc and Vd calculated as
(2.14)
(2.15)
(2.16)
(2.17)
are orthogonal representations of the energy at the input (Vcand Vd) and out-put (Vaand Vb). The signals will be low frequency or DC, with a bandwidth determined by the lowpass filter hLP. It can also be shown that
(2.18) (2.19) (2.20) will be estimates of the average real and imaginary parts of Gxy and the Gxx, respectively. The center frequency can then be locked by using either Vx or Vy, depending on the filter tuned, as a measure of the error in phase shift trough the filter, and integrate this signal to create the frequency control sig-nal. Amplitude, and thus Q-value, can similarly be created from the signal not used for frequency-tuning, combined with Vref.
The proposed tuning system is shown in Fig. 17, where Vf(=Vx) is used for controlling center frequency of the filter, while VQ(=Vy-Vref) is used to force the gain to unity, in this case corresponding to a Q-value of 10.
Va y u( )sin(ω0u)hLP(t–u)du ∞ – t
∫
= Vb y u( )cos(ω0u)hLP(t–u)du ∞ – t∫
= Vc x u( )sin(ω0u)hLP(t–u)du ∞ – t∫
= Vd x u( )cos(ω0u)hLP(t–u)du ∞ – t∫
= Vx( )t = Vb( )t Vc( )t –Vd( )t Va( )t Vy( )t = Va( )t Vc( )t +Vb( )t Vd( )t Vref( )t = Vc2( )t +Vd2( )tHere the signals Foxand Foyare the reference-signals, with a phase shift of 0 and 90 respectively, x and y are the input and output signals of the system while Vf and VQ are frequency and Q control signals for the filter.
In [28] this tuning scheme was used for tuning an 2.5MHz 2nd order band-pass filter with a Q of 10, implemented in a 2µm CMOS process. With a full swing input signal a frequency-tuning-error of 0.2% and a gain error of 1.1dB was obtained.
2.3.2 Orthogonal Reference Tuning
If assumptions about the input signal, as in 2.3.1, can not be made, but the required signal to noise ratio is low, the tuning method proposed in [29], may be an option for true on-line tuning of the filter.
Here an approximative orthogonality is created between the reference and the input signal by phase modulating a reference-signal with a pseudo random sequence before adding it to the input signal, as shown in Fig. 18.
This will result in the reference-signal being spread out over a frequency band, with the width determined by the rate of the phase modulation signal. The output signal from the filter is then multiplied by the modulated refer-ence-signal and its quadrature components, producing estimates of the real and imaginary parts of the transfer function at the reference-frequency. The result is then used to tune the filter, as described in 2.3.1.
Figure 17: Tuning by using correlation of input and output signal
LP Filter LP Filter F (t)ox F (t)oy LP Filter LP Filter F (t)ox F (t)oy Vf VQ 2nd Order Bandpass Filter x(t) y(t) a V Vb c V Vd a V Vb c V Vd a V Vc c V Vd Vd Vb -Vf VQ ° °
A board level test circuit was built, tuning a 2nd order 10.7MHz bandpass fil-ter with Q-value of 100. To make the theoretical accuracy of the Q-tuning 10%, a carrier to reference (C/R) ratio of 20dB, using a control loop band-width of 10-4 times the modulation rate was required. For the test circuit a modulation rate of 10kHz and control loop bandwidths of 1.6Hz was selected.
A possible application for the proposed tuning scheme would be in receivers for wideband-FM and QPSK (of low dimensions) modulated signals. The test circuit was inserted in the signal path of a FM broadcast receiver, and the ref-erence-signal was virtually undetectable during listening tests when only the monaural part of the signal were used.
2.3.3 Tuning by Using Common Mode Signals
Integrated continuous-time filters are usually implemented as differential cir-cuits in order to improve linearity, by using differential transconductors or operational amplifiers. If the filter was instead designed as two identical sin-gle-ended structures, with input and output signals feed differentially
Figure 18: Orthogonal reference tuning
x(t) Tunable Bandpass Filter y(t) Control Loop Filter Lowpass Filter Reference Recovery Correlators Spreading Sequence Generator Vref Automatic Gain Control
between them, one could in theory have a tuning reference-signal present as a common-mode signal in the filter [30].
Due to mismatch of the two single-ended filters, some residual reference-sig-nal will be present in the output sigreference-sig-nal, and the input sigreference-sig-nal will have some influence on the tuning-circuit.
In [30] a 7th-order equiripple filter using three biquads and a first order low-pass filter was designed and simulated. The tuning-circuit only measure on the last biquad, but all three biquads are tuned based on this.
It is claimed that if the two single-ended filters are matched to 0.3%, the dynamic range would be 50dB if the levels of the input and reference-signals are equal, however, up to 80dB might be obtainable if the reference-signal level is reduced.
In [31] a single biquad 60MHz lowpass filter was tuned by this method by using phase-locking for tuning the cut-off frequency as described in 2.1.4 and amplitude locking for Q-tuning as described in 2.2.2. The reference-signal was added as common mode level at the input, and separated from the differ-ential output by adding the outputs. For recovering the differdiffer-ential output sig-nal a high CMR amplifier was used. For a 20mVp-preference-signal added at the input a residual level of 300uVp-pwas present at the output, this should be compared to the input signal range of 2Vp-p
3 Off-Line Tuning
As opposed to on-line tuning processes, like master-slave tuning, which are active while the filter is operational, off-line tuning is only performed while the filter is inactive, which may only be when the system is powered up, depending on the application.
The advantage of off-line tuning is that the main filter is characterised, instead of a reference circuit, thus, the accuracy of the tuning will no longer be dependent on the matching of these circuits
While the methods described herein are mostly suited for off-line tuning, they can in theory be used in a master/slave circuit. However, as the accuracy of master-slave tuning is limited by the matching of the master and the slave, using these methods are probably hard to justify, due to their larger power consumption and their area overhead.
3.1 Frequency-Tuning
3.1.1 Step Response
In [32] a frequency-tuning scheme based on the step-response of the filter was used, where the center frequency of a 16th-order 450kHz bandpass filter was tuned to an accuracy of 0.33%. A step was applied to the input of the fil-ter, and by using digital counters, the frequency of the resulting oscillations was measured and the control voltage adjusted accordingly. This process was carried out 3 times in a row, to reach the desired accuracy.
Since the chip was to be used in a time-division multiple-access (TDMA) environment, tuning could be repeated often enough to ensure that long-term parameter variations would not be a problem.
When implementing this method, one should remember that the oscillations resulting from a step on the input will have a frequency deviating slightly from the center frequency of the filter.
3.1.2 Forced Oscillation
Another, not very successful method, was proposed in [33]. A 250MHz 8th-order biquad R-C filter was forced to oscillate by changing the gain of one amplifier in each biquad. The oscillation-frequency was measured using dig-ital counters.
This resulted in a frequency-dependent systematic error of 5-10%. This is a larger shift than can be accounted for by the change of Q-value when the filter was forced to oscillate. One possible explanation might be the nonlinear effects encountered when the oscillation is limited by the linear range of the filter.
3.2 Combined Frequency and Q-value Tuning
3.2.1 Sweeping the Frequency Control Voltage
In [34] tuning by applying a (slow) triangular wave at the frequency control input of the filter was proposed. A constant frequency reference-signal is used as the input, and the resulting amplitude-variations of the output signal are observed and used to tune the filter. This method is only applicable for high-Q filters, with a well-defined peak in the amplitude-response.
This method is implemented by sampling the control voltage when the filter amplification passes a level slightly below the peak level, once for rising con-trol voltage and once for falling concon-trol voltage, and using the average of these voltages for controlling the filter. It is also possible to use the peak out-put amplitude to tune the filter Q-value by the method described in 2.2.2. In [35] this method was tested in an off-line-configuration for a single biquad bandpass filter tuneable over 105-120MHz, with frequency-tuning-errors below 0.3% for Q-values ranging from 34 to 83.
3.2.2 Two Reference Frequencies
In [36] a tuning scheme based on using a phase-locked VCO with a fre-quency-divider controlled by the tuning-circuit, producing 2 frequencies N+1 and N-1 times the reference-frequency was proposed. N/2 is approximately equal to the desired Q-value, and N times the reference-frequency is the desired center frequency of the filter to be tuned. A second order filter with bandpass and lowpass outputs are assumed.
As seen in Fig. 19, the 3dB frequencies of the bandpass filter will correspond to phase shifts of -45 and -135 degrees in the lowpass filter. These phase-shifts are measured, and the frequency-tuning loop is designed to converge to: (3.1) and the Q-tuning loop should converge to:
(3.2) It should be remembered that the statements that Q is equal to N/2, and that f0 equal to N times the reference-frequency are only approximately true, for N<10 (Q<5) the errors will be larger than 0.5%.
It has also been suggested [37] that this method may be used to tune the indi-vidual circuits in a filter built from a chain of biquads.
3.2.3 Three Reference Frequencies
In [38] a tuning scheme similar to 3.2.2 was proposed, but in this case three frequencies (N-1,N,N+1)ωrefare used, with the signal attenuated by a factor of two when Νωref is being generated. The center frequency of the filter is tuned to make the output amplitude from(Ν−1)ωrefand(Ν+1)ωrefequal, and the Q-value is tuned to make the amplitude fromΝωrefequal to that of one of the other reference-signals, locking the -6dB bandwidth to 2ωref.
Figure 19: Phase-frequency relation of a 2nd order bandpass filter
ω ω |H ( )|BP ω arg(H ( ))LP ω 3dB 0 -135 -90 -45 -180 φ((N+1)ωr) φ+ ((N–1)ωr) = –180° φ((N+1)ωr) φ– ((N–1)ωr) = –90°
A second order 200MHz bandpass filter with a desired Q-value of 28.6 was manufactured, and a frequency-tuning-error of 0.25% and a Q tuning error of 3% was measured.
3.2.4 Isolation of Sub-Circuits
If the shape of the passband is important, tuning center frequency and Q-value of the filter may not be enough. One approach is to isolate sub-circuits in the filter and tune them individually.
In [39] tuning of a leapfrog filter by isolating resonant loops in the filters, and separately measuring their resonance frequencies was proposed. The parts of the filter that are not part of a resonant loop may either be reconnected to form one, or they may be tuned by the methods described in 2.1.3. Another approach is to isolate the filter completely into first order sections, and apply the method from 2.1.3 to each part individually.
In [40] a 6th-order narrow band Chebyshev filter was tuned one resonator at a time, by shunting the others to ground. A frequency-tuning-error of 3% was measured, which is suggested to be caused by nonideal characteristics of the phase-detector used.
In [41] a 4th-order, 21.4MHz butterworth filter was tuned by isolating one resonator at a time, and employing the tuning schemes described in 2.1.4 and 2.2.2 for frequency and Q-tuning, respectively. They obtained a center fre-quency accuracy of 0.014%. Here a mixed-signal implementation of the tun-ing-circuit was used, where a D-type flip-flop replaced the multiplier as phase-detector, and a successive approximation scheme controlling current DACs replaced the integrator.
3.2.5 Model Matching
In [42] the use of a model-matching algorithm for tuning continuous-time integrated filters is proposed. Model-matching algorithms in general are orig-inate from control theory, where they are used to create a model of a simula-tion model of a physical system, by observing input and output signals only. Ideally this type of method can tune the position of all the poles and zeros in the filters.
In this case the least mean square (LMS) algorithm is used, where the coeffi-cients of the filter are updated by
(3.3) where e(t) is the difference between actual and ideal output from the filter, and is the gradient signal. The gradient signal is defined as the deriva-tive of the output signal with respect to parameter bn, thus, if both e(t) and are positive at a given instant, the output signal at this instant is too low, but if parameter bnhad been larger, the output would have been higher, so increasing bn, as indicated by will reduce the error. Normally the product of the gradient and the error signal is used, but in this
implemen-tation is used in order to simplify the multiplication circuit.
Linear time-invariant systems, such as ideal filters, can be described by a state-space representation
(3.4)
where u(t) is the input signal, y(t) the output, and xi(t) the internal states of the filter.
Filters for generating gradient signals (gradient filters) can then be derived as
(3.5)
The gradient for Aijcan be found from the state xi(t) in a gradient filter with the state xj(t) in the main filter as input, similarly, the gradient for biis found as the state xiin a gradient filter with u(t) as input signal. The gradients for c and d are the states of the main filter and the input signal, respectively. Depending on the filter structure used, the tunable parameters of the filter can be found from more or less simple relations to the parameters of the state-space description. In the article an orthonormal ladder filter was used, for which the coefficients of the filters are found directly in A and b.
If only one parameter is being tuned at a time, only one gradient filter is required, which takes its input from different points in the main filter, depend-ing on which gradient signal is bedepend-ing generated.
b˙ tn( ) 2µe t( )φb n( )t = φb n( )t φb n( )t e t( )φb n( )t e t( )sign(φb n( ) )t sX s( )=AX s( )+bU s( ) Y s( )=cTX s( )+dU s( ) Agrad=AT bgrad=cT cgrad=bT dgrad=d
The proposed tuning-circuit is shown in Fig. 20 (a), the predetermined input is generated by a pseudo random number generator followed by a digital to analog converter (DAC). The reference-signal is generated by another DAC, from a table of precalculated values.
This table of precalculated values is generated as shown by Fig. 20 (b). Here the desired continuous-time filter is simulated, and a digital filter tuned to minimise the difference between the continuous and discrete filter outputs. When the tuning is complete, the output from the digital filter can be saved and used for tuning the real continuous-time filter.
A working discrete tuning-circuit, tuning an integrated filter, was constructed, but no performance measures other than time to tune the filer, are given. Use of a dithered linear search algorithm for tuning filters has recently been proposed [43], eliminating the need for large gradient filters.
Adaptive tuning techniques can in theory also be used for tuning a filter while it is processing signals. This is done by implementing one more identical fil-ter, which is first tuned by an other method. This second filter is then feed the same input as the main filter, and the adaptive algorithm is used to tune the main filter until the output signals are identical. If the input signal has suffi-cient spectral contents, it would in theory be possible to tune the filter per-fectly using this method, as it does not depend on the matching of the filters.
Figure 20: Tuning by LMS model matching
Ideal Reference Filter Tunable Filter Adaptive Tuning Algorithm u(t) y (t) e(t) (t) δ Reference Signal Generator Tunable Filter Adaptive Tuning Algorithm u(t) y(t) e(t) Predetermined Input + (Continuous) (Digital) Predetermied Input n + (t) δn (a) (b)
4 Wave Active Filters
4.1 Introduction to Wave Active Filters
Wave active filters (WAF) was first proposed in [44], in an attempt to find an active filter structure with the same insensitivity to coefficient errors as wave digital filters (WDF). Instead of simulating passive filter components, as in gyrator-C filters, or node voltages as in leapfrog filters, the filter is described by the forward and reflected voltage waves.
Starting from the generic two-port N in Fig. 21 (with port resistances Ri, i=1..2), incident waves A and reflected voltage waves B are defined as
(4.1)
Although different port resistances are possible, they will be assumed to be equal in all cases discussed here.
The relationship between A and B is described by the scattering matrix S as:
(4.2)
The basic element when building wave active filters is the wave equivalent of a series inductor L, which can be shown to have the scattering matrix S:
(4.3)
with L=2Rτ,where R is the common port resistance. Τhis can be interpreted as a lowpass filter from input to reflected wave signal, and a complementary high pass filter for the transmitted wave signal. This functionality may be implemented using the circuit in Fig. 22.
Figure 21: Generic two-port
N R1 R2 V2 V1 A1 1 B I1 I2 A2 B2 Ai = Vi+RiIi Bi = Vi–RiIi B1 B2 S A1 A2 = S 1 1+sτ --- sτ 1 1 sτ =
As shown in Fig. 23, series and parallel connected inductors and capacitors can be created from this element by swapping outputs or inverting signals. More complex elements like parallel series resonators and series connected parallel resonators can also be realised from these blocks.
Figure 22: A simple implementation of the wave two-ports used in WAFs
Figure 23: Wave two-port equivalents of passive components
A1 1 B A2 B2 1 1 τ τ τ -1 τ C= /2Rτ C=2 /Rτ L=R /2τ -1 -1 -1 τ L=2Rτ C= /2Rτ L=2Rτ1 2 L=R /2τ C=2 /Rτ 1 2 τ2 1 τ τ2 1 -1 -1 A2 B1 A2 B1 B2 A2 B2 A2 A2 B2 A2 B2 2 B A1 A1 2 B 1 B A1 1 B A1 1 B A1 1 B A1
Terminating one port of the two-port adaptor with the resistance R is equiva-lent to having no reflected signal from B to A at that port. The output voltage would be directly available on port B. Similarly, connecting one port to a source with impedance R, simply means feeding the signal directly into A. For example, the Chebyshev and Cauer filters, both of the 5th order, shown in Fig. 24,can be realized as active wave filters according to Fig. 25
4.2 Sensitivity
The performance of wave active filter implementations presented so far has been worse than expected [45], and it was suspected that this might have been due to lack of reciprocity in the wave two-ports, caused by unavoidable com-ponent variations.
Reciprocity basically means that a two-port has the same transfer-characteris-tics in both directions, when the (possibly different) port impedances has been accounted for.
Figure 24: 5th order Chebyshev (a) and Cauer (b) lp-filters.
Figure 25: WAF realisation of 5th order Chebyshev (a) and Cauer (b) filters
R R L1 L3 L5 C2 C4 VIN U VOUT R R L1 L3 L5 C2 VIN U VOUT L2 C4 L4 (b) (a) τ -1 L V'OUT VIN OUT IN V' V 1 τL3 τL5 τ -1 -1 -1 -1 1 τ2 τ3 τ4 τ5 VOUT V' OUT VIN V'IN τL2 τC2 -1 τL4 τC4 -1 -1 (a) (b)
For a wave two-port where both ports have the same port resistance, reciproc-ity simply means that the transfer function from port A1to port B2is identical to the transfer function from port A2to port B1, or expressed from the scatter-ing parameters: s12=s21 [46].
In order to investigate this, the two 5th order wave active filters shown in Fig. 25 (based on Chebyshev and Cauer lowpass filters), were simulated with different types of errors introduced.
4.2.1 Time Constant Errors
The time constant errors were created by replacing the scattering matrix S from Eq (4.3), describing the ideal wave two-port for an inductor, with:
(4.4)
where e1..4 are error parameters, with en=1 when no error is present.
Randomly distributed errors in the range 0.99 to 1.01 were used, 10000 sets of parameters were tested, and the largest and smallest absolute values of the amplitude responses were plotted for 1000 frequencies in the range 0 to 2. In the graphs presented the curve in the middle represents the ideal frequency response. These and all simulations in the following chapters were performed using MatLabTM.
Fig. 26 shows the result when e1..4 are allowed to vary independently.
In Fig. 27 relations e3=e2 and e4=e1 between the errors in one two-port are maintained, as this ensures that reciprocity[46] is preserved for all the com-ponents derived as shown in Fig. 23.
In Fig. 28, the all errors in a two-port are equal.
S sτe1 1+sτe1 --- 1 1+sτe2 ---1 1+sτe3 --- sτe4 1+sτe4 ---=
Figure 26: Frequency response for the Chebyshev and Cauer filters with independent time-constant errors
Figure 27: Frequency response for the Chebyshev and Cauer filters with reciprocity preserving time-constant errors.
0 1 2 0 0.2 0.4 0.6 0.8 1 Chebyshev w |H(w)| 0 1 2 0 0.2 0.4 0.6 0.8 1 Cauer w |H(w)| 0 1 2 0 0.2 0.4 0.6 0.8 1 Chebyshev w |H(w)| 0 1 2 0 0.2 0.4 0.6 0.8 1 Cauer w |H(w)|
While there is a marginal positive effect of maintaining reciprocity, this change is highly marginal and hardly visible from the plots alone. From the very marginal result of reducing the number of error-sources by a factor of two, one could suspect that there are other ratios that are more important to maintain. Some attempts at finding such ratios were made, with limited suc-cess, as the results seemed to depend on which type of component, which fil-ter structure and where in the filfil-ter the component was.
If all the added errors are kept equal in each wave two-port, any errors can be mapped directly to component errors in the LC-filters, with the expected low sensitivity as a result.
4.2.2 Gain Errors
In order to evaluate the effect of gain errors in the wave two-ports, similar tests as in 4.2.1 were conducted, this time the scattering matrix
(4.5)
was used, where e1..4 are the error terms, en=1 when no error is present.
Figure 28: Frequency response for the Chebyshev and Cauer filters with equal time-constant errors
0 1 2 0 0.2 0.4 0.6 0.8 1 Chebyshev w |H(w)| 0 1 2 0 0.2 0.4 0.6 0.8 1 Cauer w |H(w)| S sτe1 1+sτ --- e2 1+sτ ---e3 1+sτ --- sτe4 1+sτ ---=
The same number of parameters was tested (10000) and error range (0.99 to 1.01) was used, with the largest and smallest absolute value plotted for 1000 frequencies.
Fig. 29 shows the result when e1..4 are allowed to vary independently.
In Fig. 30 relations e3=e2 and e4=e1 between the errors in one two-port are maintained, in order to ensure that reciprocity is preserved.
Figure 29: Frequency response for the Chebyshev and Cauer filters with independent gain errors
Figure 30: Frequency response for the Chebyshev and Cauer filters with reciprocity preserving gain errors.
0 1 2 0 0.5 1 1.5 Chebyshev w |H(w)| 0 1 2 0 0.5 1 1.5 Cauer w |H(w)| 0 1 2 0 0.5 1 1.5 Chebyshev w |H(w)| 0 1 2 0 0.5 1 1.5 Cauer w |H(w)|
The conclusions from 4.2.1 applies here too, only a very marginal improve-ment was seen from maintaining reciprocity, and the observed component dependency of which, if any, relations should be maintained to reduce the effect of errors, seemed to be roughly the same.
However, one should remember that the even distributions of random errors used in this and previous section, combined with plots of min and max, only give an idea about the worst case effects of errors, which may be highly pessi-mistic if some errors interact strongly. However, it does give a rough idea about the relative importance of the different errors.
5 Mosfet-C Implementation of WAFs
5.1 Background
The chip to chip absolute variations of transconductances, capacitances and resistances are considerable. This results in a cut-off frequency uncertainty in the order of 10-30%, if no correction is applied.
Fortunately, it is usually rather straight-forward to measure these variations and adjust the filter accordingly, using the techniques described in chapter 2 and 3, improving the final accuracy an order of magnitude, or more.
However, this does require that it is somehow possible to control either resist-ances, conductances or capacitresist-ances, depending on the implementation. In the case of Gm-C filters this is straight-forward, as the time-constants in the filter will be determined by capacitances and the output transconductance of the active elements. The transconductances are in turn determined by a bias voltage which can easily be changed.
For active-RC filters the time-constant control is usually implemented by realizing resistances with mosfet transistors working in the triode region. Unfortunately these resistances are not linear. This problem is be reduced by only implementing part of each resistance as mosfet transistor, and the rest as a passive resistance in series. By connecting the active part closest to the OP-Amp input, the voltage amplitudes over it will be low, with improved overall linearity as a result.
Wave active filters implemented as described in the previous chapter lack any such low-voltage node, making this type of control less useful. An alternative control-strategy is to have a bank of discrete valued components switched in or out to obtain the desired time-constant.
Based on this, it would be interesting to create a R-mosfet-C implementation of the wave two-port, where all resistors are connected to a virtual ground node at the input of an OP-Amps. In the study only structures suitable for dif-ferential implementations were considered.
5.2 Possible Structures
In an attempt to find a suitable structure, the signal flow graph of a wave two-port was transformed in various ways, the resulting graphs are shown in Fig. 31.
These were mapped to differential mosfet-C structures, the resulting circuits for Fig. 31 C, A and F are shown in Fig. 32, 33 and 34, respectively. All the other structures will in fact result in a circuit very similar to one of these. The values of all resistances and capacitances connected to the inputs of one OP-Amp are equal. The time-constant of the resulting wave two-port is τ=RC.
Figure 31: Signal flow graphs for the basic wave two-port
1/(1+st) A2 B1 A1 B2 st/(1+st) A2 B1 A1 B2 A 2 B1 A 1 B2 1/st 1/st st/(1+st) st/(1+st) A 2 B A 1 2 1/st 1/st st/(1+st) st/(1+st) B1 A 2 B 1 A 1 B 2 1/(st+1) st/(1+st) A 2 B 1 A 1 B 2 1/(st+1) st/(st+1) A 2 B 1 A 1 B 2 1/(st+1) 1/(1+st) A 2 B 1 A 1 B 2 1/(st+1) 1/(st+1) 1/st A2 B1 A1 1/st B2 1/st A2 B1 A1 1/st B2 1/st A1 B1 A2 1/st B2 1/st A1 B1 A2 1/st B2 (A) (B) (C) (D) (E) (F) (G) (H) (I) (J) (K) (L)
Figure 32: Two OP-Amp realisation of the wave two-port (C)
Figure 33: Tree OP-Amp realisation of the wave two-port (A)
Figure 34: Four OP-Amp realisation of the wave two-port (F)
+A1 -A1 +A2 -A2 +B1 -B1 +A1 -A1 +A2 -A2 +B2 -B2 +A1 -A1 +A1 +A2 -A2 -A1 +A2 -A2 +B2 -B2 +B1 -B1 +B2 -B2 +B1 -B1 -A1 +A1 +A2 -A2
5.3 Sensitivity to Component Errors
In order to evaluate the sensitivity to component value errors for the struc-tures in section 5.2, monte-carlo analysis based on circuit level simulations was performed on an 5th order Cauer filter. The filter used is the pi-type equivalent of the t-mode Cauer filter used earlier. Four different active-RC implementations were examined, three wave active and one leapfrog imple-mentation (for comparison).
The first wave-active implementations used the circuit in Fig. 22, with the unity gain buffers implemented as single-ended OP-Amps with the output connected to the inverting input. The second WAF used the two OP-Amp active-RC implementation of Fig. 32. The last WAF was implemented using the three OP-Amp implementation of Fig. 33, where only a single time-con-stant is used.
Finding realistic figures of component variations within a chip proved diffi-cult; according to [1] a matching of 0.01% can be achieved for untrimmed capacitors on the same chip. However, this is the matching between identical components. No useful figures on accuracy of non-integer ratioed compo-nents were found, for the simulations a matching error with a standard devia-tion of 0.1% was used, as this seemed to be a reasonable value.
In all the analysed filter structures it is possible to make all resistances equal, at the expense of capacitor ratios, but according to [1], this seems to be pref-erable, as the achievable matching of equal sized resistors was claimed to be in the order of 0.1%.
The results are shown in Fig. 35. Dashed lines represents the 5th and 95th percentiles when only resistor errors are present, dotted lines corresponds to capacitor errors, and solid lines the combined errors.
One should note that the buffer-based wave active filter is clearly superior in this respect. The 3 OP-Amp/two-port (single time-constant) implementation have similar sensitivity to capacitor variations, which makes it relatively well suited for MOSFET-C implementation, if adequate tuning-circuitry is imple-mented to correct the resistance values.
5.4 Sensitivity to OP-Amp Bandwidth Variations
In the previous chapter ideal OP-Amps have been assumed, however, in an actual implementation the presence and location of poles and zeros in the transfer function of the OP-Amp will influence the transfer function of the fil-ter.
In this chapter an OP-Amp based on example 5.7 in [47] is used, which has a transfer function of
(5.1)
where ωz=120MHz, ωp1=4.2kHz,ωp2=143MHz and A0=80dB. This results in an unity gain frequency of about 100MHz. These values are later scaled to
Figure 35: Component error sensitivity of 5th order Cauer filters
0 1 2
0 0.5 1 1.5
WAF, using buffers
w |H(w)| 0 1 2 0 0.5 1 1.5 WAF, 2 OP−Amps/two−port w |H(w)| 0 1 2 0 0.5 1 1.5 WAF, 3 OP−Amps/two−port w |H(w)| 0 1 2 0 0.5 1 1.5 leapfrog w |H(w)| H s( ) A0 1 s ωz ---+ 1 s ωp1 ---+ 1 s ωp2 ---+ ---=
