Nonlinear Dynamics (2006) 44: 167–172 DOI: 10.1007/s11071-006-1962-0
c Springer 2006
Numerical Investigation and Experimental Demonstration of Chaos from Two-Stage Colpitts Oscillator in the Ultrahigh Frequency Range 1,∗ ˇ ˇ S. BUMELIENE˙ 1 , A. TAMASEVI CIUS , G. MYKOLAITIS1,2 , A. BAZILIAUSKAS3 , 4 and E. LINDBERG 1 Semiconductor
Physics Institute, A. Goˇstauto 11, Vilnius, LT-01108, Lithuania; 2 Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul˙etekio 11, Vilnius, LT-10223, Lithuania; 3 Department of Signal Processing, Faculty of Telecommunications and Electronics, Kaunas University of Technology, Student 50, Kaunas, LT-51368, Lithuania; 4 Ørsted DTU Department, 348 Technical University of Denmark, Ørsted Plads, Kgs. Lyngby, DK-2800, Denmark; ∗ Author for correspondence (e-mail:
[email protected]; fax: +370-5-2627123)
(Received: 23 August 2004; accepted: 19 April 2005)
Abstract. A hardware prototype of the two-stage Colpitts oscillator employing the microwave BFG520 type transistors with the threshold frequency of 9 GHz and designed to operate in the ultrahigh frequency range (300–1000 MHz) is described. The practical circuit in addition to the intrinsic two-stage oscillator contains an emitter follower acting as a buffer and minimizing the influence of the load. The circuit is investigated both numerically and experimentally. Typical phase portraits, Lyapunov exponents, Lyapunov dimension and broadband continuous power spectra are presented. The main advantage of the two-stage chaotic Colpitts oscillator against its classical single-stage version is in the fact that operating in a chaotic mode it exhibits higher fundamental frequencies and smoother power spectra. Key words: Chaos, Colpitts oscillator, nonlinear dynamics, ultrahigh frequency
1. Introduction Chaotic oscillators have attracted much attention regarding their potential applications to secure and spread spectrum communications. Broadband chaotic oscillations at high megahertz and several gigahertz frequencies are of special interest [1]. Analysis shows that in a classical single-transistor √ Colpitts oscillator chaotic oscillations can be generated at the fundamental frequencies f ∗ = 1/2π LC (here L and C is the inductance and the capacitance of the LC resonator, respectively) approximately ten times lower than the threshold frequency fT of an employed bipolar junction transistor, i.e. f ∗ ≈ 0.1 f T . For example, the highest expected f ∗ is about 900 MHz for the BFG520 type microwave transistor with fT of 9 GHz. Indeed, only weak chaos with 20 to 30 dB height peaks at f ∗ also at its sub-harmonic and higher harmonic components is predicted in the case of f ∗ ≈ 1000 MHz [2]. Moreover, the PSpice simulations indicate that chaotic oscillations observed experimentally at f ∗ = 1060 MHz in [3] are due to the parasitic elements, like wiring inductance and wiring loss resistance that play an important role at ultrahigh frequencies. Practically it is difficult to achieve reliable and reproducible chaotic behaviour at fundamental frequencies higher than about 500 MHz in a single-transistor Colpitts oscillator [4]. To overcome this shortcoming a modified version, a two-stage circuit was suggested [2, 5]. The highest achievable fundamental frequency was expected to increase by a factor of 3, up to f ∗ ≈ 0.3 f T . As a first step the increase of f ∗ was already demonstrated in the very high frequency (VHF: 30 to 300 MHz) range using the general-purpose 2N3904 type transistors with the fT of about 300 MHz and
168 S. Bumelien˙e et al. setting f ∗ at 90 MHz [5]. However no experiments were carried out at that time with the two-stage Colpitts oscillator at even higher frequencies, say at f ∗ > 300 MHz. In the present work we refer to the two-stage Colpitts oscillator introduced several years ago and promising higher fundamental frequencies [2, 5]. We describe an example of a hardware implementation of the modified oscillator employing microwave transistors and give experimental evidence of its chaotic performance in the ultrahigh frequency (UHF: 300–1000 MHz) range.
2. Circuit Description Simplified circuit diagram of the two-stage Colpitts oscillator [2, 5] is sketched in Figure 1, while its specific hardware implementation is presented in Figure 2. The Q1–Q2-based stages compose the intrinsic two-stage Colpitts oscillator while the Q3-based one is an emitter follower inserted to buffer the influence of the measuring devices. The resonance tank combines the loss resistor R, the inductor L, and three series capacitors C1 , C2 , C3 . The C4 is a coupling capacitor. Small auxiliary capacitors of 300 pF (not shown in the circuit diagram) are connected in
Figure 1. Two-stage Colpitts oscillator.
Figure 2. Full circuit diagram of the two-stage Colpitts oscillator.
Chaos from Two-Stage Colpitts Oscillator in the UHF Range 169
Figure 3. Typical chaotic phase portrait, current through the inductive element I L versus collector voltage V Q 1 . R = 47 , L = 16 nH, C1 = 2.4 pF, V1 = 10 V, V2 = 24 V.
Figure 4. Experimental power spectra. f ∗ ≈ 600 MHz, R = 33 , L ext = 12 nH, C1 = 2 pF, V1 = 10 V, V2 = 25 V.
parallel with the main blocking capacitors C0 to improve filtering at high frequencies. The bias emitter current I0 can be tuned by varying the voltage source V2 . In the hardware prototypes the circuit elements were the following: R0 = 100 , R1 = 510 , R2 = 3 k, R3 = 5.1 k, R4 = 3 k, R5 = 200 , Re = 1.5 k, C0 = 47 nF, C2 = C3 = 10 pF, C4 = 1 pF, C5 = 270 pF. Other parameters of the tank elements, namely R, L, and C1 depend on the chosen fundamental frequency f ∗ and are given in the captions to Figures 3–5. We note, that in a real circuit the total tank inductance L consists of: (1) the Lext controlled by an external inductive element, (2) the parasitic inductance of the loss resistor LR , and (3) the parasitic inductance LC0 of the filter capacitor. Thus, L = L ext + L R + L C0 . The two latter values are approximately 2 nH each. The microwave BFG520 type transistors mentioned in the Introduction were employed in the circuits. The specific values of the supply voltages V1 and V2 were adjusted to achieve the desired chaotic performance of the oscillator.
Figure 5. Experimental power spectra. f ∗ ≈ 1100 MHz, R = 17 , L ext = 4 nH, C1 = 1 pF, V1 = 6.3 V, V2 = 27 V.
170 S. Bumelien˙e et al. 3. PSpice Simulation Simulations of the circuit in Figure 2 were performed by means of the Electronics Workbench Professional simulator, based on the PSpice software. The Gummel-Poon model of the transistors was employed. Chaotic performance of the oscillator is observed in a sufficiently wide range of the control parameters and is illustrated in Figure 3 with a typical phase portrait.
4. Lyapunov Exponents Dynamics of the two-stage oscillator in Figure 1 is given by the following set of four ordinary differential equations: ⎧ d VC1 ⎪ ⎪ = I L − IEQ1 (r, VC2 , VC3 ), ⎪ C1 ⎪ dt ⎪ ⎪ ⎪ ⎪ d IL ⎪ ⎪ = V0 − VC1 − VC2 − VC3 − R IL , ⎨L dt dV ⎪ ⎪ ⎪ C3 C3 = I L − IEQ2 (r, VC2 ), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ d VC2 ⎪ ⎩ C2 = IL − I0 . dt
(1)
We note, that by omitting in Equation (1) the third equation for VC3 and setting VC3 = 0 in the other equations one comes to the set describing the classical Colpitts oscillator. In Equation (1) the forward current gain of the both transistors in the common base configuration is assumed for simplicity to be α = 1, that is the base currents are neglected. By introducing the following dimensionless state variables x=
VC1 , ρ I0
y=
IL , I0
z=
VC2 , ρ I0
v=
VC3 , ρ I0
t=
t τ
(2)
and parameters ρ=
L , C1
τ=
√
LC1 ,
ε2 =
C2 , C1
ε3 =
C3 , C1
a=
ρ , r
b=
R ρ
(3)
Equations (1) are transformed into the form convenient for numerical integration: ⎧ dx ⎪ ⎪ = y − F1 (a, z, v), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dy ⎪ ⎪ = −x − z − v − by, ⎨ dt ⎪ ⎪ ε dv = y − F (a, z). ⎪ 3 2 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ dz ⎪ ⎩ ε2 = y − 1, dt
(4)
In Equations (4) the constant term V0 /ρI0 has been omitted since it does not influence the overall dynamics of the system. The nonlinear functions F1 and F2 describe the current-voltage characteristics
Chaos from Two-Stage Colpitts Oscillator in the UHF Range 171 of the base-emitter junctions of the Q1 and Q2 transistors, respectively. They can be presented in the form of two-segment piece-wise linear functions: 1 − a(z + v), a(z + v) < 1, F1 (a, z, v) = (5) 0, a(z + v) ≥ 1. 1 − az, az < 1, F2 (a, z) = (6) 0, az ≥ 1. Parameter a in expressions (5) and (6) depends on r, the differential resistance of the base-emitter junction in the forward-active mode. In this simplified mathematical model the r is considered to be a constant parameter. In experiment it can be controlled by the emitter dc bias current I0 . To characterize the two-stage Colpitts oscillator quantitatively we made use of the Matlab Lyapunov Exponent’s Toolbox (LET) and estimated the full spectrum of the Lyapunov exponents. The LET requires the Jacobian matrix of the corresponding state-equations. Since Equations (4) contain two piece-wise linear functions F1 and F2 , each two-segments ones, the Jacobian matrix has four different forms. For the specific parameter set (a = 10, b = 0.4, and ε2 = ε3 = 3) the LET provides the following spectrum of the Lyapunov exponents (LE): λ1 = 0.072,
λ2 = 0,
λ3 = −0.152,
λ4 = −0.322.
(7)
Study of the LE dependence on the parameters a, b and ε is in progress and will be published elsewhere. Preliminary results show that in addition to wide chaotic domains there are narrow windows of periodic behaviour. Though Equations (4) describe a four-dimensional system it possesses only one positive LE, λ1 > 0. Thus, despite the enlarged dimensionality of the system the two-stage Colpitts oscillator remains simply chaotic one in contrast to a coupled system of two classical Colpitts oscillators that has two positive LE, therefore exhibits hyperchaotic behaviour [6]. Employing the Kaplan–Yorke conjecture j dL = j +
λi , |λ j+1 | i
j i
λi > 0,
j+1
λi < 0
(8)
i
we evaluated the Lyapunov dimension d L ≈ 2.5, that is essentially larger than the corresponding measure of the classical single-stage Colpitts oscillator d L = 2.08 estimated for a similar set of the parameter values.
5. Experimental Results To illustrate the performance of the oscillator experimentally several power spectra were taken at different fundamental frequencies f ∗ , all with spectral resolution of 120 kHz. Two of them are presented in Figures 4 and 5. The spectra are broadband continuous ones with only rather flat rises at the f ∗ and near its subharmonics f ∗ /2 (Figure 4) or the f ∗ /3, 2f ∗ /3 (Figure 5). In comparison with a classical single-stage oscillator investigated recently in this frequency range [3, 4] the modified version exhibits essentially better spectral features. For example, in Figure 4 the lower and upper frequency band limits are f 1 = 250 and f 2 = 700 MHz within the spectral unevenness S = 10 dB, while f 1 = 150 MHz and f 2 = 1000 MHz within S = 20 dB. Thus, for the central
172 S. Bumelien˙e et al. frequency f c = ( f 2 + f 1 )/2 of about 500 MHz the relative bandwidth f = ( f 2 − f 1 )/( f 2 + f 1 ) is 0.47 and 0.74 for S = 10 and S = 20 dB, respectively. Finally, the spectrum shown in Figure 5 covers the full UHF range (300– 1000 MHz) with the following spectral parameters: f c = 650 MHz, S = 6 dB, f = 0.54.
6. Conclusions A hardware prototype of the two-stage Colpitts oscillator has been designed for the UHF range and described in details. It has been demonstrated numerically and experimentally to generate broadband chaotic oscillations. The highest fundamental frequency that can be reliably achieved is about 1000 MHz using the BFG520 type transistors. The two-stage chaotic Colpitts oscillator in the UHF range has considerably better spectral characteristics than the classical single-transistor oscillator [3, 4]. Evidently the power spectrum can be shifted to the VHF and lower frequency ranges by simple change of the resonance tank parameters L, C1 , C2 , and C3 .
Acknowledgment This work was supported in part by the Lithuanian State Science and Studies Foundation under contract No. T-62/04.
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