Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 343
The Active Filter for Use in Measurement of the Fast Moving
Object
Martin Friedl, Lubomı́r Fröhlich, and Jiřı́ Sedláček
Department of Theoretical and Experimental Electrical Engineering
Faculty of Electrical Engineering and Communication
Brno University of Technology, Kolejnı́ 2906/4, 612 00 Brno, Czech Republic
Abstract— In the field of a measurement of the fast one shot processes there is necessary to
use special frequency filters. This article deals about the synthesis and optimization of the ARC
ladder filters with transmission zeros based on frequency dependent negative resistors (FDNR).
Frequency filters designed using approximation functions with transfer zeros (like Inverse Chebyshev or Cauer functions) exhibit in comparison to approximation functions with monotonic magnitude response in stop band essential higher steepness of magnitude response in area of transitive
band of filter. Active RC filters synthesized using modern active elements grown from passive
RLC filter prototypes with very small sensitivity on passive elements can be in comparison to
their RLC prototypes realized relative easily. These filters designed using active FDNR blocks
as LP (low pass) filters or using SI (Simulated inductors) active blocks in case of HP (high pass)
filters can be designed with minimum active and passive elements. During resulting optimization
filter design process must be an influence of real lossy active blocks (FNDR, SI) on resulting filter
response respected. In contribution here are presented some possibilities of filter optimization
from this point of view. There are also discussed and in some examples prescribed ways of filter
synthesis with account of influence of lossy active blocks on resulting filter magnitude response
in case of Inverse Chebyshev and Cauer filter of LP and HP filters higher (from 3rd to 7th) filter
orders.
1. INTRODUCTION
The presented paper describes ARC filter, which is used as LP filter in block diagram for measurement of the fast moving objects. This measurement is based on the eddy currents. Principle of
the measurement is draws in Fig. 1. The moving object affects the coil sensor and thus amplitude
of the oscillator. The peak detector monitors this change of the signal and next steps are filtering
and comparing. The last block is used control unit for signal processing.
Synthesis of the filters was worked out for filter with transmission zeros, where are used lossy
FDNR elements. The realization of filters with transmission zeros requires only a small increasing in
complexity. However it enables to obtained essential increasing of filter steepness in transition band
of filter characteristics and significantly suppress the specific frequency area (zero transmission) [1].
The zero transfer in the transmission function of the filter is obtained by insertion an additional
resonant circuit into the classical ladder filter structure.
There are advisedly used lossy FDNR elements, by reason that by useful optimization the value
of the filter element can be achieved almost the same filter transfer response as by filters with
lossless blocks, however using only half number of OAs. In this paper influence of the quality factor
Q (corresponding with lossy of the real elements) is analysed, depending on the filter parameters.
2. PRINCIPLE OF THE LOSSY FDNR BLOCKS
In area of passive filter design are well-known problems with realization of passive elements namely
inductors. Consequently, for low and medium frequencies, passive RLC filters are preferably replaced by active filters RC (ARC). For the following simulations was chosen ladder filter with
FDNR
COIL
uP
MOVING
OBJECT
OSCILATOR
DETECTOR
LOW PASS
FILTER
COMPARATOR
CONTROL UNIT
Figure 1: Block diagram of the measurement of the fast moving objects.
PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27–30, 2012
344
L2
L1
R1
L3
C1
R1
R2
R3
Bruton
L4
R4
transformation
L5
R5
R2
C2
Cp2
Cp1
Rp1
C1
Rp2
C2
D1
D2
Figure 2: Design principle of active LP filter of the 5th order with transfer zero (Inverse Chebyschev filter).
CB
R
Rz
C
D
CA
Cz
Bruton
transformation
_
FDNR
realization
+
Figure 3: The parallel equivalent lossy circuits and their realization using grounded lossy FDNR.
T-configuration, where can be used FDNR elements. By design of these active filters using Bruton’s transformation [2], the initial RLC structure is transformed into equivalent circuit of the CRD
structure (Fig. 2). This new structure does not contain an element of inductive character, but uses
the properties of the synthetic element named as FDNR (Frequency Depended Negative Rezistor).
The active circuits realized of the FDNR elements can be divided into different types according
to their basic circuit characteristics and parameters [3]. The lossy FDNR grounded circuit (Fig. 3)
can be realized by the use one active element (OA) [4]. In this circuit the losses are represented
by a parallel capacitor. These losses correspond to resistors Rp1 and Rp2 parallel connected to
capacitors in the original circuit of the RLC filter (Fig. 2). Thus it is very easy these lossy to
simulate and change of the quality factor Q of the resonant circuit (which represents transmission
zero) and investigate filter response due to these variations of lossy.
3. OPTIMIZATION OF THE QUALITY FACTOR FOR LOSSY FDNR BLOCK
The resonant circuits in the transverse branches of the ladder filter determine the frequency of
transmission zero of the filter. The resonant frequency and Q-factor of these circuit are given very
known formulas:
1
√
,
2·π· L·C
Q = ω · C · RZ .
f =
(1)
(2)
Losses caused by resistor RZ in Fig. 3, respectively capacitor CZ directly related to the quality
factor Q. It is evident that the resonant frequency is not affected by the lossy resistor RZ (Fig. 3)
connected to the circuit, but the whole filter cutoff frequency yes, as will be shown later. Inserted
losses represent lossy in the resonant circuit of original RLC filter prototype. In the active CRD
circuit realization these losses are modeled with lossy capacitors Cp1, Cp2 (see Fig. 2).
Losses caused by resistor RZ in Fig. 2, respectively capacitor CZ directly determined the quality
factor Q of resonant circuits, which realize transfer zero of filter. Investigation of influence the Qfactor on resulting filter parameters is very important and enables the optimization by filter design
using active lossy elements (FDNR).
Table 1: The parameter p for different quality factor.
Q
1
2
3
4
5
6
7
8
9
10
20
30
40
50
p 0.1592 0.3183 0.4775 0.6366 0.7958 0.9549 1.1141 1.2732 1.4324 1.5915 3.1831 4.7746 6.3662 7.9577
Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 345
5000
1000000
4500
900000
y = 3,1831x
Inverse Chebyschev
5th order, 10 kHz, lossy
4000
y = 2,3873x
800000
Q=20
3500
700000
3000
600000
y = 1,5915x
Rz [Ω]
Rz [Ω]
Q=15
2500
2000
1500
Q 20
Q 15
Q 10
Q5
Q=10
500000
400000
Q= 5
300000
1000
200000
500
100000
0
20
y = 0,7958x
Calculation Rz,
frequency independent
0
30
40
50
60
70
0
80
100000 200000 300000 400000 500000 600000 700000 800000 900000 100000
0
1 / (f.C)
Minimum suppression [dB]
Figure 4: The dependence of the minimal suppress
filter on lossy resistor for diferent quality factor Q.
Figure 5: The dependence of lossy rezistor Rz on
the coefficient k.
0
10
-0,2
9,8
-0,4
Drop in transmission area,
5th order, loss
Cutoff frequency shitf,
5th order, loss
9,6
suppress 30dB
-0,8
drop 30dB
drop 50dB
drop 70dB
9,4
suppress 50dB
-1
f [kHz]
A [dB]
-0,6
suppress 70dB
-1,2
9,2
9
-1,4
8,8
-1,6
8,6
-1,8
-2
1000
10000
100000
1000000
8,4
1000
Rz [Ω]
10000
100000
1000000
Rz [ Ω ]
Figure 6: The dependence of the drop in the transfer of the filter and cutoff frequency shift of the filter on
lossy resistor RZ (5th order, LP, Chebyshev).
The following chart (Fig. 4) express for example (in case of the 5th order filter LP, inverse
Chebyshev) the dependence of the minimum suppress of the resulting filter on value of lossy resistors
for different quality factor Q.
There is evident from the relation (1) that for a resonant frequency can be used a lot of combinations of L and C value parameters thus the chart (Fig. 4) is valid only for one of the ratios
of elements L and C in the resonant circuit. In case of the frequency change by constant ratio of
elements L and C, the chart will be frequency independent. However, there is dependence on the
frequency for different ratio of elements L and C. Therefore, the chart drawn in Fig. 5 has been
normalized for all frequencies using the coefficient k = 1/(C · f ) (3), where C is the capacitor of
the resonant circuit and f is the resonant frequency for the for quality factor 5, 10, 15 and 20:
1
,
C ·f
= p · k.
k =
RZ
(3)
(4)
Thus the chart in Fig. 5 can be easily used also for calculation of the required slope of the line
for other different quality factors. In next step it can be determined lossy resistor RZ according to
Equation (4), where p is the slope parameter of the line just for the individual quality factor and
k is the coefficient according to Equation (3). The following table shows the various parameters p
for quality factor Q = 1 ÷ 30.
The following charts in Fig. 6 draw the dependence of the resulting parameters for the filter of
5th order (Inverse Chebyshev, LP) on losses (loss resistor RZ from Fig. 3). The left chart shows
the dependence of the drop of filter transfer and the right chart shows the dependence of the cutoff
frequency shift of the filter on lossy resistors.
All the charts enable by the process of active filter optimization to get an idea about the influence
of the losses on the resulting filter parameters (Inverse Chebyshev, LP). The chart on Fig. 4 shows,
for the required minimum suppress of the filter, minimum loss resistor RZ which must be used for
required quality factor of the resonant circuit (realized zero of the transmission). The loss resistor
RZ in this chart represents the total losses in the whole ladder filter which for the 5th filter order
PIERS Proceedings, Kuala Lumpur, MALAYSIA, March 27–30, 2012
346
0
-10
Chebyshev, 5th order
-20
ARC+RLC WITH LOSSY
IDEAL WITHOUT LOSSY
A [dB ]
-30
-40
-50
-60
-70
-80
-90
1000
10000
100000
1000000
f [Hz]
Figure 7: The RLC filter with lossy and corresponding filter with lossy FDNR elements (5th, LP, Chebyshev)
and their corresponding transfer characteristic.
is the parallel sum of the resistors Rp1 and RP 2 in Fig. 2. The normalized chart from the Fig. 5
achieves more universal validity, because it is frequency independent for the all combinations of
the parallel connection of the capacitor C and loss resistor RZ from Fig. 3. With substituting the
calculated parameter p and after putting it into Equation (4) we obtain the value of loss resistor
for the required quality factor Q. This loss resistor correspond only to parallel combination of the
capacitance C and loss resistor RZ and therefore losses will be larger because there are two loss
resistors in filter of the 5th order (total losses can be calculated simply by the parallel sum of the
loss resistors Rp1 and Rp2 ). The influence of the calculated lossy resistor RZ on the resulting filter
parameters can be found from the charts on the Fig. 6.
The filter which is shown on the Fig. 7 was designed according above prescribed procedure.
On the left of the Fig. 7 is transfer characteristic of the ARC and RLC filters with lossy and for
comparison there is showed the ideal characteristic without lossy. There is suppressed influence of
the voltage divider — the termination resistors R1 and R2 . The both characteristic are shifted up
6 dB. The circuit diagram of the original RLC filter with lossy resistors and the correspond active
filter with lossy FDNR elements are drawn on the right. The filter was designed for frequency
10 kHz and quality factor Q = 10 for each resonant circuit (corresponding to two transmission
zeros).
According to charts on the Fig. 5 and according to Equations (3) and (4) loss resistors RZ1 =
3257 Ω and RZ2 = 2296 Ω correspond to quality factor Q = 10 for each resonant circuit. After
application of the Bruton transformation and calculation of the lossy FDNR elements with an
operational amplifiers we obtain according to Fig. 2 the final circuit, see in Fig. 7 on the right.
The difference of the filter parameters in comparison with designed filter it can be obtained from
the PC simulations. The drop in the transmission area of the filter is 2.77 dB and drop by cutoff
frequency of the filter is 8.5 dB. The lossy of the designed filter we can read for higher values of Q
from the curve on Fig. 6.
4. CONCLUSIONS
The contribution prescribed optimization process by filter design of active ARC filters with transfer
zeros, where active filters are based on RLC ladder prototypes. As active blocks of designed filters
here are used simple and economical lossy FDNR active elements. The prescribed process of filter
optimization has been grown from wide analysis and investigations of filter response by influence
of lossy in resonant circuits. In this analysis the quality factor Q was chosen as the basic attitude,
which determines the starting point in calculations of possible losses. The resulting influence of the
losses on the filter parameters can be determined from the derived charts.
Active RC filters with transfer zeros based on passive RLC filter prototypes exhibit higher
steepness of magnitude response in area of transitive band of filter by very small sensitivity on
passive elements and can be realized relative easily. These filters designed using active lossy FDNR
blocks can be realized with minimum active and passive elements, therefore can be in process of
active filter optimization successfully used.
Progress In Electromagnetics Research Symposium Proceedings, KL, MALAYSIA, March 27–30, 2012 347
ACKNOWLEDGMENT
The research described in the paper was financially supported by grant of Czech ministry of industry
and trade no. FR-TI1/001, GACR 102/09/0314 and project of the BUT Grant Agency FEKT-S11-5/1012.
REFERENCES
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