Analysis of Chirped Oscillators Under Injection Signals Franco Ramírez, Sergio Sancho, Mabel Pontón, Almudena Suárez Dept. of Communications Engineering, University of Cantabria, Santander, Spain Abstract—An in-depth investigation of the response of chirped reach a locked state during the chirp-signal period. The possi- oscillators under injection signals is presented. The study con- ble system application as a receiver of frequency-modulated firms the oscillator capability to detect the input-signal frequen- signals with carriers within the chirped-oscillator frequency cies, demonstrated in former works. To overcome previous anal- ysis limitations, a new formulation is presented, which is able to band is investigated, which may, for instance, have interest in accurately predict the system dynamics in both locked and un- sensor networks. locked conditions. It describes the chirped oscillator in the enve- lope domain, where two time scales are used, one associated with the oscillator control voltage and the other associated with the II. FORMULATION OF THE INJECTED CHIRPED OSCILLATOR beat frequency. Under sufficient input amplitude, a dynamic Consider the oscillator in Fig. 1, which exhibits the free- synchronization interval is generated about the input-signal frequency. In this interval, the circuit operates at the input fre- running oscillation frequency and amplitude at the quency, with a phase shift that varies slowly at the rate of the control voltage . A sawtooth waveform will be intro- control voltage. The formulation demonstrates the possibility of duced at the the control node and one or more input signals detecting the input-signal frequency from the dynamics of the will be injected, modeled with their equivalent currents ,, beat frequency, which increases the system sensitivity. All the where 1 to . The whole system will be modeled with an results have been validated with both full circuit-level simula- tions and measurements. envelope-domain outer-tier equation at the fundamental fre- Index Terms—Chirp signal, injection locking, beat frequency quency, considering the higher harmonic terms in an inner tier [6]. Initially, a single input signal will be assumed. The first harmonic of the output voltage is expressed as: I. INTRODUCTION Δ (1) The interesting works [1]-[2] successfully investigated the possible application of chirped oscillators under the injection where is the input frequency. The outer-tier equation, in of wireless signals in cognitive radio, for spectrum sensing terms of the admittance function Y at i, at the output node, is: [3]-[4] and frequency demodulation. This conclusion relies on the capability of the chirped oscillator to get locked to an input Δ, /,Δ, , Δ 0 (2) signal having a frequency comprised within its dynamic oscil- where the superscripts indicate real and imaginary parts, is lation-frequency interval, which is actually achieved with a the time-derivative operator, is the middle point of the certain input-signal power. The theoretical analysis in [2] chirp-voltage excursion, /2 and Δ is the predicts a linear variation of the chirp-oscillator frequency time-varying increment with respect to . Assuming a quasi- within the synchronization interval, though this frequency should agree with that of the input source. This analysis is linear oscillator response versus (as expected in the chirped performed in terms of the oscillation frequency, instead of the oscillator) and low input amplitude , the oscillator can be chirp-signal voltage, that constitutes the actual control varia- linearized about , and . Neglecting the time-varying ble. term Δ one obtains: Here an in-depth investigation of the dynamics of the chirp- Δ sin cos , signal oscillator under wireless injection is presented. It is (3) based on an envelope-domain, semi-analytical formulation of ≡,Δ ≡ Δ the injected circuit, using an oscillator model extracted from harmonic balance (HB) [5]-[6]. The variations of the dynamic where is the input-signal phase and the coefficients , oscillation under both locked and unlocked conditions are and correspond to those given in [10]-[11]. In each cycle analyzed using two time scales, one associated with the con- of the chirp signal, the tuning voltage grows linearly as Δ trol voltage of the chirped oscillator, and the other associated . Taking into account the slow oscillation-frequency modu- with the beat frequency. This avoids the computational diffi- lation due to and the possible existence of a beat frequency culties of a full envelope-domain formulation [7]-[9], which , the nonlinear equation (3) can be solved, expressing the must cover the slow sawtooth-signal period (on the order of phase variable (t) in a Fourier series with two time scales: ms) at a small integration time step, accounting for the beat frequency outside the synchronization time interval. By means , , (4) of the envelope-domain formulation with two time scales, it is , demonstrated that the chirp oscillator is able to detect a wire- less signal even when its power is below the one required to The slow scale , associated with the time-varying tuning The analysis method has been initially validated through voltage Δ, modulates the components of expression (4). comparison with circuit-level simulations at a chirp frequency The faster scale is associated with the beat frequency , of 10 kHz. This is the lowest value that could be considered in which varies with . Introducing (4) in (1), the instantaneous the circuit-level analysis, due to the huge computational cost frequency shift can be determined: of a long simulation time interval under relatively high . The input power is 8 dBm. Fig. 2 shows the dynamic , , , variation of the output amplitude shift Δ during the period of the chirp signal for f = 5.45 GHz, obtained through circuit- (5) i level simulations in (a), and with the formulation (3)(6) in , (b). In the synchronization interval the output amplitude ap- proaches the ellipsoidal curve that would be obtained in static In the time intervals with unlocked behavior, the partial de- conditions, that is, when varying in the interval rivative with respect to can be neglected since and ,. This static curve has also been represented as a are slowly-varying components. Then, equation (5) reference. predicts that at each t1, is a periodic signal centered at with period 2/. 4 3 Circuit level simulation 2 1 0 -1 -2 -3 Amplitude shift (mV) shift Amplitude -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Tuning voltage shift (V) (a) 4 3 Proposed model 2 1 0 -1 -2 -3 Amplitude shift (mV) shift Amplitude -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 Tuning voltage shift (V) (b) Fig. 2 Dynamic variation of the oscillation magnitude at the chirp frequency 10 kHz. (a) Circuit-level simulations. (b) Equations (3)(6). Within the synchronization interval, Δ ap- proaches the phase relationship that would be obtained in static conditions, so the oscillation frequency only exhibits a slow time variation, as predicted by (6). The synchronization interval is always centered about the tuning voltage that Fig. 1. Chirped oscillator based on the transistor NE3210S01. It is controlled would provide an oscillation at the input frequency in static by the tuning voltage , corresponding to a sawtooth waveform. (a) Sche- conditions. Fig. 3 shows the dynamic frequency variation matic of the oscillator circuit. (b) Photograph. obtained through circuit-level simulations and with (3)(6). In Now, operation under locked conditions will be assumed. In the unlocked time intervals, the instantaneous frequency is this case, 0 and , . For each , given by (5). The magnitude of decreases as Δ equations (3) and (4) predict a single-tone spectrum at , and shifts towards the synchronization band. In the unlocked in- the instantaneous frequency shift becomes: tervals, the average frequency increases with time due to the positive slope of Δ . , , ≡ (6) In the presence of input carriers , injection-locking in- , tervals may still be detected. Equation (2) can be extended to where is a very small frequency, just due to the slow this situation in the form: variation of the tuning voltage Δ. Δ the models of the circuit elements, and, as a result, in the de- ΔΔ Δ‐ 0 rivatives of the admittance function. Note that the new method Δ exhibits an excellent agreement with circuit-level simulations, as shown in Fig. 2 and Fig. 3. (7) ,, In the case of a single frequency-modulated input signal, it , , is possible to demodulate this signal during the synchroniza- tion time interval. Fig. 5 compares analysis and measurement Δ, Δ results after passing the oscillator output through a frequency where , and are the amplitude, phase and frequency discriminator and a high-pass filter [2]. The detection of fast of each carrier. With two (or more) input carriers, the instan- modulations requires a low chirp frequency. The detection taneous frequency will vary in a nearly periodic (or quasi- method can be applied to both analog and digital frequency periodic) manner within the synchronization intervals. modulated signals. 2.5 2 0.08 1.5 Measurement 1 0.04 0.5 0 0 -0.5 (V) voltage shift shift (MHz) -0.04 -1 Demodulator output Demodulator -1.5 Circuit level simulation -0.08 Instantaneous frequency -2 Pr opos ed model 0.45 0.5 0.55 0.6 -2.5 Time (ms) -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 (a) Tuning voltage shift (V) 0.08 Fig. 3. Dynamic variation of the oscillation frequency at the chirp frequency Proposed model 10 kHz. Comparison between equations (3)(6) and circuit-level simulation. 0.04 0 0.05 voltage (V) voltage -0.04 0.04 Demodulator output Demodulator 0.03 -0.08 0.45 0.5 0.55 0.6 0.02 Time (ms) (b) 0.01 Fig.