ELIMINATING LIMIT CYCLES
A General Second Order Analog Filter to Eliminate Limit Cycles For Constant Input Signals Deepak Garg1 and Vijay Kumar Tayal2 1
Department of Electronics & Communication Engineering, ABES Engineering College, Ghaziabad 201009, UP, India 2 Department of Electrical & Electronics Engineering, ABES Engineering College, Ghaziabad 201009, UP, India
[email protected], vktayal71 @yahoo. com
H ( s) =
Abstract -- Dither refers to a signal with small amplitude and zero mean that is intentionally injected into a system to effectively remove nonlinear distortions of the signal processed. The technique can be incorporated only when the parameters of limit cycles i.e. amplitude & frequency are known. This paper examines the determination of amplitude and frequency of limit cycle of two-dimensional electronic filters. Analog Filters consist of components such as op-amp, resistors, capacitors that exhibit non-linear characteristics. These non-linearities of electronic components leads to limit cycles in the electronic system. The limit cycles correspond to oscillations of fixed amplitude & period. The limit cycles result in marginal stability and is tolerable only if its amplitude is within some specified limits. The constant oscillations associated with the limit cycles can cause deteriorating performance or even failure of the system hardware and therefore, the limit cycle problem is an important issue that ⋅(•srequired Ao Ao s22+1) is to be solved when dealing with analog filters. 22 ss ++ααss++11 Determination of existence of limit cycles is not an easy task as they depend both on type and amplitude of the input signal. The author has used graphical technique suitable for computer graphics to determine the amplitude and frequency of limit cycles. The technique is implemented through second order analog filters. The results are compared with the help of digital simulation technique using MATLAB 7.5.
II. LIMIT CYCLE PARAMETERS OF T W O DIMENSIONAL ELECTRONIC FILTER CIRCUITS Considering a system consisting of 2nd order electronic filters, characterized as two-dimensional multivariable systems. Transfer functions and frequency response are as mentioned below. A. Voltage Transfer Equations of 2nd order Filters Filters are typically specified by voltage transfer equation.
H ( s) =
Vo ( s) Vi ( s )
Filters are of different orders e.g. 1st / 2nd /3rd and other higher orders and types e.g. Low pass, high pass, band pass, band stop or all pass filter. For the 1st order filter the response is not ideal as the rate of decay is small therefore higher order filters are implemented to improve the response. The 2nd order filters are used in the system for which the normalized transfer functions are,
Keywords: Multivariable system, Phasor diagram, Limit Cycle, Nonlinear system.
2nd order LPF
H ( s) =
Ao s 2 +α s +1
H (s) =
− Ao⋅α ⋅s s 2 +α s +1
2nd order HPF I. INTRODUCTION ANALOG filters are composed of different active, passive and other components. Active filters are built from active devices such as operational amplifier, which exhibits non-linear characteristics [1]. The non-linearities like relay, saturation etc leads to limit cycles in electronic systems [2-10]. The limit cycles describe the oscillations of fixed amplitude and period. The determination and elimination of limit cycle is an important issue in any analog circuit or electronic system [11-16].
B. Analog Filters Frequency Response The frequency response of the filters mentioned are shown in Fig.1(a,b,c,d)
The graphical technique [18] uses mathematical modeling of the system represented as two-dimensional structure. A large number of electronic applications like filters, radar antenna pointing system etc [17] can fit structure of a general two dimensional multivariable system [13].
C. Representation of Analog Filter System The block diagram of a general 2x2, two-dimensional analog filter system is shown in Fig.2. Each subsystem (S1 & S2) consists of non-linearity (N1, N2) e.g. Saturation, relay etc. and a stable part such as analog filter connected in series. This is
2nd order BPF 2nd order BRF
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AKGEC INTERNATIONAL JOURNAL OF TECHNOLOGY, Vol. 3, No. 2
(a) Low Pass
(b) High Pass Figure 3. Characteristics of Saturation.
The stable part is replaced by transfer function of filter,the phasor diagram for whole system in autonomous state is shown inFig.4.
(c)Band Pass
(d) Band Stop
Figure 4. Phasor Diagram for 2x2 Analog Filter System.
For a fixed frequency ‘ω’ the angles qL1 and qL2 are fixed. The sides of DOBD2 correspond to subsystem-1 and that of DOBD1 represent for subsystem-2. For various values of R1- it is necessary to construct separate phasor diagrams to check the conditions for a fixed frequency. If all the quantities are normalized w.r.t the magnitude R1, a single phasor diagram (for a particular value of frequency) can be used for checking possibility of limit cycle. This diagram is called as normalized phasor diagram shown in Fig.5. Figure 2. General 2×2 Electronic Filter System.
the most general structural representation of two-dimensional system. Replacing the non-linearity by its describing function (example describing function of saturation is given by Eq.1). Describing function of the Saturation DF = N(X) = 2m[sin-1(δ/X)+(δ/X){1-(δ/X)2}1/2]
(1)
Figure 5. Normalized Phasor Diagram for Second Order 2x2 Analog Filter System.
where δ: amplitude and m=slope of the saturation Characteristics of Saturation shown in Fig.3
F. Mathematical Modeling of the 2x2 Analog Filter System The mathematical modeling of the system shown in Fig.2 is performed using Fig.5. From the Fig.5, we can derive:
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ELIMINATING LIMIT CYCLES
OA =
C2 = −1.0 R1
For different values of ‘ω’, the values of X1/X2 from Eq.6 & 7 and from Eq. 8 are calculated and a graph is plotted between ‘ω’ and both X1/X2 calculated above. E. Illustration of Graphical Method W e have consider the system shown in Fig.2 with G1 as the LPF and G2 as HPF. G1(s)=1.586/(s2+1.414s+1) and G2(s)=1.586s2/ (s2+1.414s+1) are the transfer function of the low pass and of high pass filter respectively. The two non-linear elements N1 & N2 are Saturation non-linearities with there parameters as δ1=1.5, m1=1 and δ2=2, m2=1.
OB =
O Di =
C1 C1 = , i = 1or 2 R1 C2
BDi =
X1 R1
For different values of ‘ω’, we have calculated the ratio X1/X2 from Eq.6 & 7 and also from Eq. 8. The calculated values are tabulated in Table-1.
ADi =
Table 1 Calculated X1/X2 from Eq.6 & 7 and Eq. 8
In Fig. 4, C is the enter of circle OBDi of radius r=OC. Selecting O as the origin the co-ordinates of the points Di can be determined. From triangle DiCO the co-ordinates of the point C are (0.5, -0.5/tanqL1). The radius of the circle OC=0.5/sinqL1. The equation for straight line ADi is obtained as: u = (vcotθL2) – 1
(2)
The co-ordinates of the intersection points, Di of the circle and straight line ADi are obtained as:
R X123cot θ L 2 + cot θ L1 + [(3cot θ L 2 + cot θ L1 )2 − 8cos ec2θ L2 ]1/ 2 vi = = 1.0 2 cos ec 2θ L 2 RR11
θL1
θL2
X1/X2 from Eq. 6 & 7
X1/X2 from Eq.8
0.4 0.45 0.5 0.55 0.6 0.65 0.7
-33.96 -38.45 -43.32 -47.67 -52.98 -57.97 -62.74
-33.96 -38.45 -43.32 -47.67 -52.98 -57.97 -62.74
1.0754 1.0596 1.0353 1.0223 0.9985 0.9852 0.9723
1.256 1.157 1.098 1.046 1.020 0.9675 0.9223
From the tabulated values we now plot a graph between ‘ω’ and X1/X2 calculated from Eq.6 & 7 and Eq. 8. The point of intersection of both the X 1 /X 2 will give the frequency of the limit cycle. The plot is shown in Fig.6 and from the plot we get the ‘ω’ . From the value of ‘ω’ , the amplitude C1 and C2 are calculated.
(3) ui = (vi/tanθL2)-1
‘ω’
(4)
ODi = (ui 2 + vi 2 )0.5 OA= -1.0 2
2
0.5
BDi = [(1 − ui ) + vi )]
;
ADi = [(1 + ui ) 2 + vi 2 )]0.5
(5)
N1(X1) = Y1/X1= C1/ (G1X1) =ODi/ (G1BDi)
(6)
N2(X2)=Y2/X2=C2/(G2X2) =OA/G2ADi)
(7)
X1/X2=BDi/ADi
(8)
Figure 6. Comparative Plot X1/X2 and ω.
From the plot, the frequency of limit cycle, ω = 0.62. The amplitude of limit cycle calculated at this frequency are: C1 = 0.5835, C2 = 0.3185.
Corresponding to the value of N 1 and N 2 mentioned as describing function in Eq.1, the ratio of X1 and X2 can be found from Eq (6) and (7). Similarly, the ratio of X1/X2 is again obtained from Eq (8). This process is repeated for different values of ‘ω’ .
The Graphical method is also applied for other types of electronic filters with various other non-linearities and the results are tabulated in Table-2.
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AKGEC INTERNATIONAL JOURNAL OF TECHNOLOGY, Vol. 3, No. 2 Table 2 Graphical Method Results Type of Filter
Type of Non-Linearity&Parameters
G1
G2
N1
LPF BPF BRF BRF
HPF LPF LPF HPF
Saturationδ=1.5, Saturationδ=1.5, Saturationδ=1.5, Saturationδ=1.5,
m=1 m=1 m=1 m=1
Graphical Method Results
N2
ω
C1
C2
Saturationδ=2.0, m=1 RelayD=1.0 RelayD=1.0 Saturationδ=2.0, m=1
0.62 0.63 0.625 0.63
0.5835 1.0567 0.4666 0.4685
0.3185 0.5575 0.1839 0.2660
F. MATLAB Implementation The system shown in Fig. 2, illustrated by an example mentioned in (E) has been simulated using the MATLAB7.5. The SIMULINK block representation is shown in Fig 7. Transfer Function represents the LPF and Transfer Function1 represents HPF.The non linearity N1 is replaced by Saturation and N2 as Saturation-1. The simulation is run to obtain the time history plots as shown in Fig.8(a & b). From the time history plot the frequency and amplitude of the limit cycles for the analog filter system mentioned above are computed. Similarly, for other analog filters and nonlinearities the SIMULINK model is represented and time history plots are obtained from which the frequency and amplitude of limit cycles are computed as shown in Fig.9 – 12.
(b) Output C2 Figure 8 (a) Output C1 of LPF with Saturation NonLinearity. (b) Output C2 of HPF with Saturation NonLinearity.
For other electronic filters with various nonlinearities the time history plots are as shown in Fig 9-12 using SIMULINK .
Figure 7. MATLAB / SIMULINK Model of 2-Dimensional. Electronic Filter System with Saturation Non-linearity (a) Output C3
(b) Output C4
(a) Output C1
Figure 9(a) Output C3 of LPF with Dead Zone NonLinearity (b) Output C4 of HPF with Saturation NonLinearity.
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ELIMINATING LIMIT CYCLES
(a) Output C5
(a) Output C9
(b) Output C6
(b) Output C10
Figure 10 (a) Output C5 of LPF with Relay Non-Linearity (b) Output C6 of HPF with Saturation Non-Linearity.
Figure 12 (a) Output C9 of BRF with Saturation Non-Linearity. (b) Output C10 of LPF with Saturation Non-Linearity.
The Table-3 shows the SIMULINK results for various filters implemented with different nonlinearities from Fig. 8-12. III. COMPA R ATIVE RESULTS The Table-4 is amalgamation of Table-2 & Table-3 and it gives a comparison between the results obtained from Graphical Method and SIMULINK simulation. From the results we can see a slight discrepancy between the graphical and simulated results, which is a consequence only of the approximation in Describing Function Theory (i.e. the waveforms in this region are highly non-sinusoidal).
(a) Output C7
IV. CONCLUSION The graphical method described earlier is found to be simple, intuitive and fast means to determine limit cycle. It gives quick appreciation for the system behavior using MAT L A B simulation. The use of SIMULINK for oscillation prediction provides an ideal tool for comparing the accuracy of Describing Function solutions for more complicated system. The Technique can also be extended for the analysis of other electronic systems with non-linearities. V.ACKNOWLEDGEMENT The authors wish to thank the Department of Electronics Engineering, ABES Engg. College, Ghaziabad, India, for providing Computer Aided Design (CAD) Lab facilities for the M ATLAB/SIMULINK simulation.
(b) Output C8 Figure 11(a) Output C7 of BPF with Saturation Non-Linearity. (b) Output C8 of LPF with Saturation Non-Linearity.
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AKGEC INTERNATIONAL JOURNAL OF TECHNOLOGY, Vol. 3, No. 2 Table 3 Matlab/Simulink Results Type of Filter
Type of Non-Linearity& Parameters(δ=amplitude, m=slope) MATLAB/SIMULINKResults
G1
G2
N1
N2
LPF LPF LPF BPF BRF
HPF HPF HPF LPF LPF
Saturationδ=1.5, m=1 Dead Zoneδ=0.5 RelayD=1.0 Saturationδ=1.5, m=1 Saturationδ=1.5, m=1
Saturationδ=2.0, Saturationδ=2.0, Saturationδ=2.0, Saturationδ=2.0, Saturationδ=2.0,
m=1 m=1 m=1 m=1 m=1
ω
C1
C2
0.5969 0.6283 0.6201 0.6283 0.6283
0.5778 0.2556 0.4278 1.025 0.45
0.3125 0.1472 0.3333 0.6 0.2667
Table 4 Comparative Results of Graphical Method And Simulink Type of Filter
Type of Non-Linearity&Parameters
Graphical Method Results
MATLAB/SIMULINK Results
G1
G2
N1
N2
ω
C1
C2
ω
C1
C2
LPF LPF LPF BPF BRF
HPF HPF HPF LPF LPF
Sat. *δ=1.5, m=1 Dead Zoneδ=0.5 RelayD=1.0 Sat.*δ=1.5, m=1 Sat.*δ=1.5, m=1
Sat.*δ=2.m=1 Sat.*δ=2.m=1 Sat.*δ=2.m=1 Sat.*δ=2.m=1 Sat.*δ=2.m=1
.6183 .6358 .6442 .6253 .6412
.5835 .2656 .4305 1.030 .465
.3185 .1503 .3349 .589 .266
.5969 .6283 .6201 .6283 .6283
.5778 .2556 .4278 1.025 .45
.3125 .1472 .3333 .6 .2667
*Sat.-Saturation
VI. REFERENCES [1]
Malekzadeh, F.A.; Mahmoudi, R.; van Roermund, A.H.M.; “A new statistical approach for non-linear analysis of limit cycle amplifiers,” Radio and Wireless Symposium, 2009. RWS’ 09 IEEE 18-22 Jan. 2009, pp.47 – 50. [2] R.A.Roberts and C.T.Mullis, “Digital Signal Processing , Reading, Mass.”,:Addison Wesley, 1987 [3] L.B.Jackson, “Digital Filters and Signal Processing. Boston, Mass.”,: Kluwer, 1986 [4] M. E. Ahmed and P. R. Belanger, “Limit Cycles in fixed point implementation of control algorithms,” IEEE Trans .Ind. Electron., vol.IE- 31,pp.235-242,Aug.1984 [5] M.E.Ahmed and P.R.Belanger, “Scaling and roundoff in fixedpoint implementation of control algorithms,” IEEE Trans. Ind. Electron., vol. IE-31,pp.228-234, Aug.1984 [6] B.W.Bomar, “Low Roundoff-Noise limit-Cycle-Free Implementation of recursive Transfer functions on a Fixed point Digital Signal Processor,” IEEE Trans. Ind. Electron., vol.41, No.1 pp.70-78, Feb 1994 [7] S.R.Sanders, “On Limit Cycles and the Describing Function Method in Periodically switched circuits,” IEEE Trans. On Circuit and system, vol.40, No.9, pp.564-572, Sept 1993 [8] C.Eswaran, A.Antoniou and K.Manivannan, “Universal Digital Biquads which are free off limit cycles,” IEEE Trans. On Circuit and systems, vol.CAS-34, No.10, pp.1243-1248, Oct 1987 [9] M.D.Marco, M.Forti and Alberto Tesi, “Existence and Characterization of Limit Cycles in Nearly Symmetric Neural Networks,” IEEE Trans. On Circuit and systems ,vol.49, No.7, pp.979-992, July 2002 [10] Yasuyuki Matsuya, Y. Kado, J. Terada, T .Egauchi, M. Tamura and HFujiwara, “A 4th order Local-Feedforward D/A converter
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
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that Prevents Limit Cycle Oscillation,” Symposium on VLSI Digest of Technical Papers, pp.332-335, 2002 Patra, K.C., Swain,A.K., and Majhi,S.: “Application of neural network in he prediction of self oscillations and signal stabilization in non-linear multivariable systems,” Proc. of ANZIIS-93, pp.585-589, 1993 Singh,Y.P., and Subramanian,S.: “Frequency response identification structure of non-linear systems,” IEEE Proc.1980, 127, pp.77-82. Patra.K. C., and Singh,Y.P.: “Structural formulation and prediction of limit cycle for multivariable nonlinear system,” IETE, Tech. Rev. (India) ,1994,40,(5&6), pp 260. M.Kawamata and T.Higuchi, “A systematic approach to synthesis of limit cycles free digital filters,” IEEE Trans. Acoust., Speech, Signal Proc., vol.ASSP-31, pp.212-214, Feb.1983. B.W.Bomar, “On the design of second order state space digital filter sections,”IEEE Trans. On circuit. Syst., vol.CAS-36, pp.542-552, April.1989. B.W.Bomar, “New second order state space structures for realisin low round off noise digital filters,” IEEE Trans.Acoust., Speech, Signal Proc., vol.ASSP-33, pp.106-110, Feb.1985. Nikiforuk, P.N., and Wintontk, B. L.M.: “Frequency response analysis of two dimensional non-linear symmetrical and non symmetrical control systems,”Int.J.Control, 1968,7, pp.4962. K.C. Patra and Y.P. Singh, “Graphical Method of prediction of limit cycle for multivariable non-linear systems,” IEEE Proc.Control Thepry Appl., vol.143, No.5, pp.423-428, Sept.1996.
