Modeling the dynamics of Voltage-Controlled Oscillators by ... - wseas

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

Modeling the dynamics of Voltage-Controlled Oscillators by combining modified Volterra series with an envelope transient formulation Jacobo Domínguez, Sergio Sancho, Almudena Suárez Dpto. Ingeniería de Comunicaciones ETSIIT, Universidad de Cantabria Av. Los Castros s/n, Santander 39005 SPAIN http://www.dicom.unican.es/ Abstract: - In this paper, a behavioral model for frequency-modulated microwave voltage-controlled oscillators (VCO) is presented. The so-called “Deviation-Reduction” Volterra series are applied to model the VCO dynamics, building a black box model for the VCO when applying modulation signals to the tuning voltage. The output signal is obtained by making use of an envelope transient formulation. The model can be built from simulation or measured data. Key-Words: - microwave oscillators, behavioral model, Volterra series, envelope transient phenomena which are present in the VCO, such as memory effects and nonlinear dynamical behavior. Several works have been presented for power amplifier modeling [1], using Volterra series as the mathematical tool that allows modeling nonlinear functions with memory effects. In this work, the dynamics of the VCO will be simulated by applying the so-called “DeviationReduction” Volterra series [2] to the envelope transient model of the VCO state variables. The technique allows identifying the Volterra kernels [3] both using simulations in commercial software or measurements from real VCO, which would more interesting for practical applications. This technique will be applied to simulate the dynamics of a microwave VCO in presence of modulation signals.

1 Introduction Microwave oscillators are an important part of communication systems. The design and implementation of this kind of circuits is usually a difficult topic due to their autonomous behavior. A wide range of accurate design techniques have been developed. However, these techniques usually involve complex simulations with different optimization procedures that may not find the stable solution for a set of the oscillator parameters. Further design approximations include more complexity in the oscillator circuit model, e.g. bias stubs, more accurate transistors models… Although the simulation time is not prohibitive, taking into account the autonomous behavior makes the optimization procedure to find stable solutions quite challenging. Once the design is finished it must be integrated into the main system that usually includes more oscillators, mixers, amplifiers… This makes very difficult to do system simulations using oscillator circuit models. To avoid this situation one of the main objectives of oscillators design is to decouple all their characteristics from its load, in order to find fewer problems while integrating phase. This allows testing the whole system by developing models for different abstraction levels: physical, circuital, system… One solution is to build the simplest model, or the simplest simulation model, which reproduce all the possible behavior of the original circuit. A first approach to a voltage controlled oscillator (VCO) model could be a look-up table providing the output frequency for each voltage. Although it is extremely easy to use this model, it does not predict some

ISSN: 1790-2769

2 VCO behavioral modeling One of the main parameters of the VCO is the free-running frequency f0. This frequency is changed by varying the tuning voltage vt. Nevertheless, due to the autonomous nature of the VCO, any change in the circuit components may modify the overall solution of the VCO. Here, the dynamics of the frequencymodulated VCO will be modeled, when a slow timevarying signal vt(t) is used as tuning voltage. This signal consists on a DC component vt0, corresponding to the free-running case, together with a slow-varying component, which provides the frequency modulation. In this case, the output voltage of the VCO is:

227

ISBN: 978-960-474-007-9

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

v(t ) = A(t ) cos( wot + Φ (t )) dΦ = h[vt (t )] dt

p =1

(1) P

p =1

M

p =1 i1 = 0

M

p

i p =0

j =1

)



 r =1 

p −r

M

(n)∑ L i1 =1

(5)

r   h p ,r (0,...,0, i1 ,..., i r )∏ x (n − i j )   i r = i r −1 j =1   M

∑

where the new parameter r is the possible number of product terms of the delayed input in the input items [7]. The introduction of this parameter increases the control over the model, allowing an efficient model order reduction [7]. In [7], (5) is known as the Dynamic Deviation Reduction-Based Volterra series, which divide the original Volterra series in a static and a dynamical part. This makes it easy to obtain the kernel parameters using any optimization method choosing an upper limit for the parameter r, which was not available in the classic expression. We will apply this modified Volterra series to the envelope transient model (3). The main parameters that will be analyzed are time waveform, and output spectrum. Issues as phase noise, and extreme values for output frequency, tuning voltage and input frequency will not be address by this case. Here, the modified Volterra series low-pass approach detailed in [8] will be used to build a black box model that relates the input baseband tuning voltage with the time-varying harmonics of the output voltage:

0

P



p

∑ ∑  x

where both the amplitude and phase have a dynamic relation with the input signal vt. The change in amplitude is common in any VCO with no output buffers. The input tuning voltage modulates the output frequency. Making use of the envelope transient formulation, the VCO output voltage waveform is expressed as: N  v ( t ) = Real  ∑ VK ( t ) e j2 πkf t  (2)  k=0  where f0 is the VCO free-running frequency, corresponding to the DC value vt0 of the tuning voltage vt . The time-varying harmonic components Vk(t) provide a slow-varying frequency shifting around the carrier f0, so f0 agrees with the mean value of the VCO output frequency. Taking this into account we can avoid building a model that covers all the VCO frequency range, and focus our behavioural model in the small band around any particular value of the free-running frequency f0. The memory dependency of the harmonic components of the VCO output voltage can be expressed as: && (t) = g(x ,..., x , V , V & ,..., v , v& ,...) (3) V k 1 n i i t t where xn are the circuit parameters, Vi are the harmonic components of the state variables, and vt the tuning voltage. The maximum order two has been used for the time-derivatives. A similar approach is presented in [4] using up to first order derivatives and a nonlinear mapping function. This kind of model represents the memory dependences that are present in the VCO between the output and tuning voltages. This approach is similar to [6], which uses Volterra series to deal with the non-linearity memory effects. Volterra series are more commonly used in power amplifier models as a black box model that relates input and output signals. In the general case, the discrete expression that models a nonlinear system with memory effects using the Volterra series theory is:

y ( n) = ∑ ∑L ∑ h p (i1 ,K, i p )∏ x(n − i j )

(

P

y ( n) = ∑ h p , 0 0, K ,0) x p (n) +

Vk ( t ) = H ( vt ( t ) )

In (6) we assume that the envelope frequency basis is set to average value of the input tuning voltage, hence, a constant tuning voltage vt0 will provide a constant frequency value f0. However during the extraction of the model we will have to discard any VCO output frequency/envelope base mismatch.

3 Application to a microwave VCO The technique has been applied to model the dynamics of the VCO shown in Fig. 1. This circuit has been designed using a similar architecture that in [9]:

(4)

where P is the nonlinear order, M the memory order and hp is called the pth-order Volterra kernel. In the work [7], the following expression is obtained by taking into account the static and dynamical parts of (4), and applying some mathematical transformations : Fig. 1 VCO Schematic.

ISSN: 1790-2769

(6)

228

ISBN: 978-960-474-007-9

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

In order to show the VCO free-running performance, some simulations have been developed in commercial software (ADS of Agilent©). The freerunning frequency versus tuning voltage characteristic of the designed VCO is shown in Fig. 2:

Fig. 4. VCO time waveforms. Transient to oscillation and free-running steady-state.

Now, in order to build the Volterra models, several modulation signals were used. In order to illustrate the technique, the Volterra model has been applied to the simplest case of one harmonic component (k=1). The procedure is described in the following (see Figs. 5 and 6): In the first place an input signal was applied to the VCO input. Making use of commercial software, the magnitude and phase of the first time-varying harmonic component V1(t) of the output voltage was obtained. Then, an optimization procedure was carried out to obtain the Volterra kernels in (5) up to some degree. Once the model (5) is built, it can be used to predict the VCO response to other modulation signals. To test the model, a different signal was applied to the VCO and the output was obtained both using the Volterra model and commercial software. After several tests, the memory order M=2 and the nonlinear order P=5 was chosen, because it was a model complex enough to fit our VCO dynamical behavior. Moreover, larger models introduce too much sensitivity to input signals that may overdrive the output response. A dynamical order r=2 was chosen, since higher orders provide a high increment of the model complexity and did not show great improve in the model accuracy. Several types of input signal were tested to excite all the possible states, with the aim to build the most complete model: Single tone, multi-harmonic, filtered white noise, sawtooth waveform, triangular waveform, squared signal… However complex input signals were not suitable to obtain a stable set of Volterra kernels. Indeed, the Volterra kernels obtained when using complex input signals reproduced the original signal but failed to model the output for simpler input signals. To summarize, the overall procedure to obtain and test the models is : 1. Apply an input signal complex enough to obtain the main characteristics of the VCO dynamics that the model should represent. 2. Correct the constant shift in the output envelope phase to reduce envelope base mismatch with the mean output frequency. 3. Solve the linear equation (5), to obtain the

Fig. 2 Frequency vs. Vt

In Fig. 3, the power of the first six harmonic components has been shown, in terms of the freerunning frequency and the tuning voltage. There is over 15dB difference between the first and second harmonic components in all the frequency range.

Fig. 3 Harmonic Output Power vs. Vt anf f0.

Finally, the VCO transient to oscillation is shown in Fig. 4:

ISSN: 1790-2769

229

ISBN: 978-960-474-007-9

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

of V1, (c) Time-Varying phase of V1

Volterra kernels for different memory, nonlinearities, and dynamic orders. 4. Select the smaller optimal M, P and r orders for the desired accuracy. 5. Test the validity of the model with a different set of signals that the ones used to generate it.

Three different models are shown in Fig. 7, comparing the model spectrum output with the original circuit model. In the cases of Fig. 7(a-b), the signal vt(t) used to build the model is periodic, containing one harmonic component. In the case of Fig. 7(c), vt(t) is periodic containing four harmonic components. Once the models are built, a multiharmonic test signal has been introduced to the VCO tuning voltage. In these figures, the frequencydomain comparison of the VCO response to the test signal, obtained both with the Volterra model and in commercial software has been shown. As can be seen, the models of the cases (a) and (b) present more accuracy than the model (c).

Fig. 5 Volterra Model Building

1

a)

0.8 0.6 0.4 0.2 0

8.5

9

9.5

10

V1 (º)

|V 1 |

Time (us)

Fig. 7 (a) Vt: 10MHz tone 1Vpp. Volterra orders M=2, P=5. Test signal 1,3,5,7 MHz, 0.25 Vpp each. (b) Vt: 10MHz, 1Vpp. Volterra order M=2, P=5. Test signal 3,7 MHz, 0.5 Vpp each. (c) Vt 1,3,5,7MHz, 0.25Vpp each. Volterra order M=2, P=5. Test signal 3,7 Mhz, 0.5 Vpp each. Fig. 6 (a) Large amplitude single tone Vt , (b) Time-Varying magnitude

ISSN: 1790-2769

230

ISBN: 978-960-474-007-9

8th WSEAS International Conference on SIMULATION, MODELLING and OPTIMIZATION (SMO '08) Santander, Cantabria, Spain, September 23-25, 2008

Microwave Theory And Techniques, Vol. 55, No. 5, May 2007. [8] Anding Zhu, John Dooley, Thomas J. Brazil, “Simplified Volterra Series Based Behavioral Modeling of RF Power Amplifiers Using Deviation-Reduction”, Microwave Symposium Digest, 2006. IEEE MTT-S International, June 2006 pp. 1113 – 1116. [9] C.-H. Lee, S. Han, B. Matinpour, and J. Laskar, “A Low Phase Noise X-Band MMIC GaAs MESFET VCO”, IEEE Microwave And Guided Wave Letters, Vol. 10, No. 8, August 2000.

4 Conclusion A Volterra model for frequency modulated VCO using envelope transient formulation has been presented. This black box model is based on the use of modified Volterra series to relate the harmonic components of the output voltage to the input modulation signal, applied to the tuning voltage. Main handicaps are finding convenient Volterra orders and choosing adequate signals to reproduce the dynamical behavior of our VCO. Good agreement has been found between the circuit model and the Volterra model for a set of different large periodic signals. Acknowledgement: The authors would like to thank Prof. T. J. Brazil, Dr. A. Zhu and Dr. J. Dooley from University College Dublin, and J. C. Pedro from University of Aveiro, for technical support and guidance, and for their works [2,7] which had a strong influence in this work. References: [1] J. C. Pedro and S.A. Maas, “A Comparative Overview of Microwave And Wireless PowerAmplifier Behavioral Modeling Approaches”, IEEE Trans. Microw. Theory Tech., vol.53, no.4, pp. 1150-1163, Apr. 2005 [2] Anding Zhu José C. Pedro, Thomas J. Brazil, “Dynamic Deviation Reduction-Based Volterra Behavioral Modeling of RF Power Amplifiers”, IEEE Transactions On Microwave Theory And Techniques, , Vol. 54, No. 12, December 2006. [3] V. Volterra, “The Theory of Functionals and of Integral-Diferential Equations”. New York: Dover, 1959. [4] Rohan Batra, Peng Li, Lawrence T. Pileggi, Wan-ju Chiang, “A Behavioral Level Approach for Nonlinear Dynamic Modeling of VoltageControlled Oscillators”, IEEE 2005 Custom Integrated Circuits Conference. [5] Rafael Cignani, Alberto Costantini, Giorgio Vannini, “VCO Behavioral Model Based on the Nonlinear-Discrete Convolution Approach”, 11th GAAS Symposium, Munich 2003. [6] Alberto Constantini, Corrado Florian, Giorgio Vannini, “VCO Behavioral Modeling Based on the Nonlinear Integral Approach”, Circuits and Systems, 2002. ISCAS 2002. IEEE International Symposium on, Volume 2, 26-29 May 2002 Page(s):II-137 - II-140 vol.2. [7] Anding Zhu, José Carlos Pedro, Telmo Reis Cunha, “Pruning the Volterra Series for Behavioral Modeling of Power Amplifiers Using Physical Knowledge”, IEEE Transactions On

ISSN: 1790-2769

231

ISBN: 978-960-474-007-9