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A theoretical and experimental study of injection-pulling for IL-PLL optoelectronic oscillator under RF signal injection
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
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Original research article
A theoretical and experimental study of injection-pulling for IL-PLL optoelectronic oscillator under RF signal injection
T
Abhijit Banerjeea,*, Larissa Aguiar Dantas de Brittob, Gefeson Mendes Pachecob a
Electronics and Communication Engineering Departament, Academy of Technology, Adisaptagram, Hooghly, 712121, West Bengal, India Aeronautical Institute of Technology (ITA), Microwave and Optoelectronic Department, Marechal Eduardo Gomes, 50 - Vila das Acácias, São José dos Campos, SP, 12228-900, Brazil
b
A R T IC LE I N F O
ABS TRA CT
Keywords: Injection-pulling dynamics Phase noise Optoelectronic oscillator Phase-lock loop Injection transfer function
A detailed study of injection-pulling behavior of an injection-locked and phase-locked loop (ILPLL) single-loop optoelectronic oscillator (OEO) under radio frequency (RF) signal injection is presented. From the locking equation, the expression for the output spectrum of the IL-PLL OEO under RF injection is derived in time domain, to evaluate the spurious performance of the system. The frequency response of the spurious outputs due to RF injection signal is investigated in terms of the injection transfer function. Also, a dual-loop model of the IL-PLL OEO is developed to study the phase noise performance of the system with co-frequency RF injection signal. The experimental results are given in partial support of the conclusions of the analysis.
1. Introduction A key subsystem common in the photonic systems in the field of microwave photonic is the RF source. The RF generation by optical methods can be done by mixing different optical wavelengths in a non-linear crystal or in a photodiode and by the use of a ring circuit formed by an optical link and a feedback RF loop named OEO [1,2]. Such oscillator is the RF source used in a wide variety of high-frequency signal processing systems, enables the generation of signals with frequencies from sub-GHz up to 70 GHz and phase noise down to -120 dBc/Hz at 1 kHz offset. The injection-locking (IL) technique in OEO becomes useful in a number of applications, including clock and carrier recovery, low-power RF signal detection, and low phase noise microwave signal generation. Also, PLL single-loop OEO is demonstrated to generate a microwave signal with long-term frequency stability and low-phase noise [3,4]. The IL-PLL OEO can provide ultra-low phase noise microwave signal with excellent long-term stability [5–7]. Injection-pulling, on the other hand, typically proves undesirable [8]. In a RF transceiver, the injection of unwanted spurious tones to the oscillator through the parasitic coupling path degrades the spectral purity of the output signal [9–11]. Thus, injection-locking and pulling dynamics of the OEO should be studied in detail in the unlocked-driven state to exploit its merit in intelligent potential applications and reduce its effects whenever undesirable. Few approaches [12–16] have been developed recently to forecast the phase dynamics and the frequency-pulling phenomena in single-loop OEO due to an independent RF sinusoidal signal injection. All the prior analytical model describes the injection-locking or pulling in nominally free-running single-loop OEO, a rare case of practical interest. Since the OEOs are usually phase-locked, the analysis must account for the correction produced by the PLL. Comprehensive theoretical and experimental works on the phase noise performance of IL-OEO, and IL-PLL OEO have been published. There is currently no dynamical frame-work of the IL-PLL OEO to study its dynamics. This lack of analytical insights into the dynamical properties of IL-PLL OEO does
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Corresponding author. E-mail address: [email protected] (A. Banerjee).
https://doi.org/10.1016/j.ijleo.2019.164059 Received 8 November 2019; Accepted 11 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 1. Electrical equivalent block diagram of a single-loop OEO under injection and its vector diagram.
not enable us to optimize its metrics. This paper develops an accurate phase dynamic model of an IL-PLL single-loop OEO under RF signal injection, and also, analyzes the injection-pulling effects in time and frequency domain. This paper also investigates the phase noise performance of an IL-PLL single-loop OEO under co-frequency sinusoidal RF signal injection. In this paper, the topics are organized as follows. In Section II, we derive the locking equation for the IL-PLL single-loop OEO under RF signal injection. Section III studies the injection-pulling behavior of the system in time domain. Section IV describes the frequency-domain model of the IL-PLL OEO to study the phase dynamics. Phase noise analysis is performed in Section V. Section VI presents the experimental results, and finally, conclusions are given in Section VII. 2. Locking equation derivation In this section, we first derive the injection-locking equation for a single-loop OEO under the influence of an independent RF sinusoidal signal injection. In order to generalize the analytical model, we then extend the locking equation for an IL-PLL single-loop OEO under RF injection. 2.1. Single-loop OEO under RF injection Fig. 1 shows a simplified electrical equivalent block diagram of a RF driven OEO. A summer is considered in the feedback path to allow the injection of RF signal. In this figure, VS is the injection RF signal with instantaneous frequency ωinj (t ) and constant amplitude Vinj , VO is the perturbed output RF signal with constant amplitude VOP ,and ϕ (t ) denotes the instantaneous phase difference between VS and VO . In the absence of injection signal, OEO oscillates with the free-running frequency ω0 . When an injection signal with frequency ωinj is applied to the OEO, the injection signal VS combines with the feedback signal VO to produce the signal VX . If ωinj is near to ω0 , the OEO will be injection-locked to the injection RF signal and the OEO will oscillate with the injection signal frequency ωinj . Under injection-locked condition, a phase difference between the injection signal and the oscillator signal is established to maintain oscillation at the injection frequency ωinj . The RF filter and the optical fiber introduces the necessary phase shift (θ − Ωτ ) to the signal, where Ω = (ωinj − ω0) denotes the frequency detuning between the free-running oscillation and the injection signal, τ is the time-delay provided by the optical fiber, and
θ = tan−1 ⎛ ⎝
⎜
2Q (ω0 − ω) ⎞ ω0 ⎠ ⎟
(1)
is the phase shift introduced by the first-order linear band pass filter (BPF) with transfer function H (s ) = where Q is the quality factor of the filter. From the vector diagram of Fig. 1, we obtain
tan(θ − Ωτ ) =
Vinj sinϕ VOP +Vinj cosϕ
=
KI sinϕ 1 + KI cosϕ
1 1 + Q ω0 s + s ω0
(
)
[17],
(2)
where KI = (Vinj / VOP ) is the injection ratio. Consider that injection signal vector VS is at rest and the oscillation output signal vector VO is rotating counterclockwise with instantaneous beat frequency is given by
ω = ωinj +
( ) with respect to V . Thus, the oscillator instantaneous frequency dϕ
dφ dt
dt
S
(3)
Substituting (1) into (2) and then using (3), we obtain the phase dynamics equation of the single-loop OEO under the influence of RF signal injection [16] 2
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 2. Electrical equivalent block diagram of an IL-PLL single-loop OEO under RF injection.
dφ μsin(Ωτ ) − F sin(ϕ − Ωτ ) = Ω+ dt cos(Ωτ )+KI cos(ϕ − Ωτ )
(
(4)
)
where F = (μKI ) and μ = ω0 2Q is the half-bandwidth of the BPF. In the absence of time delay τ (τ = 0) and for small signal injection (KI ≪ 1) , (4) becomes identical to the familiar Adler’s equation [18]. 2.2. IL-PLL single-loop OEO under RF Injection In this sub-section, we are going to derive the generalized locking-equation for an IL-PLL OEO under RF signal injection shown in Fig. 2. By tuning the bias voltage of the MZM (which provides the non-linearity in the loop), the instantaneous frequency of the single-loop OEO can be varied and thus, acting as a voltage controlled oscillator (VCO) with tuning sensitivityKVCO . Let us assume that VO is now from an IL-PLL single-loop OEO controlled by both PLL and injection-locking mechanism. Thus, the inherent oscillation frequency of the oscillator is given by
ω0 (t ) = ω0 + KVCO VC (t ) + KRI φE (t )
(5)
where VC (t ) is the tuning voltage, and is given by
VC (t ) = KPD φE (t )* f (t )
(6)
φE (t ) = φR (t ) − φout (t )
(7)
where KPD is the gain of the phase-detector, f (t ) is the impulse response of the loop-filter, KRI is the injection ratio for reference signal and φE (t ) is the output phase-error of the phase-detector. φR (t ) and φout (t ) represent the reference phase and resultant output phase of the IL-PLL OEO, respectively. Combining (4), (5) and then using (3), the resultant output frequency of the IL-PLL OEO can be written as
ωout (t ) ≅ (ωO + μtan (Ωτ )) + KVCO VC (t ) + KRI φE (t ) − ωL β (t )
(8)
where ωL = μKI sec (Ωτ ) , and β (t ) = sin (φ − Ωτ ) . ωL β (t ) and KPD KRI φE (t ) represent the frequency modulation component due to RF injection signal and reference RF injection signal, respectively. KVCO VC (t ) is the frequency modulation of the oscillator for phaselocking mechanism. Integrating (8), the resultant output phase of the IL-PLL OEO can be found as
φout (t ) = (ωO + μtan (Ωτ )) t + KVCO
∫ VC (t ) dt + KRI ∫ φE (t ) dt − ωL ∫ β (t ) dt
(9)
Taken together (6), (7) and (9) describes a time-domain model of an IL-PLL OEO under RF signal injection shown in Fig. 3. 3. Injection – pulling analysis In this section, we formulate the behavior of an IL-PLL OEO under injection-pulling in time domain. We assume that the OEO is
Fig. 3. Time-domain model of an IL-PLL OEO under RF injection. 3
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injection-locked and phase-locked to the reference signal with frequency ωO and also the oscillator is injection-pulled by an independent RF signal at frequency ωinj . We also assume that the oscillator phase-locking control contains a small perturbation VC around a dc level. Substituting (5) into (4), phase perturbation can be expressed as
dφ = Ω+ μtan (Ωτ ) + KVCO VC (t ) + KRI φE (t ) − ωL sin (φ − Ωτ ) dt
(10)
Since the PLL is phase-locked to ωO and assuming φ is very small (φ ≪ 1rad ) , (10) may be written as
dφ ≈ KVCO VC (t ) + KRI φE (t ) + ωL sin (Ωt ) dt
(11)
We consider an active low-pass filter (LPF) with transfer function F (s ) = (1 + sτ2 ) sτ1 as the loop-filter of the PLL, where τ2 and τ1 are the filter time constants. Thus, in Laplace-domain, the control voltage is given by
VC (s ) = −KPD F (s ) φ (s ) = −KPD
(1 + sτ2 ) φ (s ) sτ1
(12)
where the negative sign represents the phase subtraction of the phase-detector. Taking the inverse Laplace Transform of (12), substituting for VC (t ) in (11) and then differentiating both sides with respect to time, one can easily show
dφ d 2φ + (2ξωn + KRI ) + ωn2 φ = ωL Ωcos (Ωt) dt dt 2
(13)
where ωn = K / τ1 , K = KPD KVCO , and ξ = ωn τ2/2 are the natural frequency, the loop-gain, and the damping factor of the PLL, respectively. Eq. (13) indicates that the IL-PLL single-loop OEO behaves as a second-order system in the presence of injection-pulling. The solution of (13) is given by
φ (t ) =
ωL Ωcos (Ωt−α ) (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
= φm cos (Ωt− α)
(14)
where α is the phase of the transfer function at a frequency of Ω, and
φm =
ωL Ω (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
(15)
The IL-PLL OEO output signal can be represented as
vO (t ) = VOut cos (ωO t + φ)
(16)
Substituting (14) into (16) and then using the following identities, ∞
cos (zcosθ) = JO (z ) + 2 ∑ (−1) k J2k (z ) cos (2kθ)
(17)
k=1 ∞
sin (zcosθ) = 2 ∑ (−1) k J2k + 1 (z ) cos ((2k + 1) θ)
(18)
k=1
the RF output signal of the IL-PLL OEO under injection-pulling can be found as ' ∞ ⎡ cos (ωO t − βk )+⎤ vO (t ) = JO (φm ) VOut cos (ωO t ) + VOut ∑ (−1) k J2k (φm ) ⎢ ⎥ '' k=1 ⎣ cos (ωO t + βk ) ⎦ ' ∞ ⎡ sin (ωO t − βk )+⎤ − VOut ∑ (−1) k J2k + 1 (φm ) ⎢ ⎥ '' k=1 ⎣ sin (ωO t + βk ) ⎦
ωO'
(19)
ωO''
= (ωO + 2k Ω) , = (ωO − 2k Ω) and βk = 2kα . From (14) and (16), we can see that the IL-PLL OEO output phase is where modulated sinusoid ally and also (12) indicates that the control voltage of the phase-locking mechanism varies sinusoidally with a frequency ofΩ. For weak RF injection and considering only the first harmonic with k = 1, from (19), it can be observed that the output RF signal-contains a carrier component at ωO and two symmetric sidebands at ωinj and (2ωO − ωinj ) . 4. Frequency domain IL – PLL OEO model In this section, we are going to present a simple frequency-domain model of an IL-PLL OEO to quantify the frequency dependence of the spurious outputs caused by the RF injection signal. Assume that the frequency difference between the injection signal and the IL-PLL OEO output signal is small. In that case, Laplace domain representations of Eq.s (6),(7), and (9), are given by
VC (s ) = KPD φE (s ) F (s )
(20)
φE (s ) = φR (s ) − φout (s )
(21) 4
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 4. Block diagram of the frequency-domain model of IL-PLL OEO under RF injection.
sφout (s ) = KVCO VC (s ) + KRI φE (s ) − ωL β (s )
(22)
where φE (s ) , φout (s ) , φR (s ) and β (s ) are the Laplace transform of φE (t ) , φout (t ) , φR (t ) and β (t ) , respectively. Combining (20), (21) and (22), one can easily calculate the closed-loop transfer function and the transfer function with respect to the injection, respectively as
HC (s ) = Hinj (s ) =
G (s ) 1 + G (s ) H (s )
(23)
ωL s [1 + G (s ) H (s )]
(24)
where G (s ) = [(KF (s )/ s ) + (KPD KRI /s )] and H (s ) = 1 are the forward-path transfer function and feedback-path transferfunction of the IL-PLL section, respectively.The output phase of the IL-PLL OEO can be written as
φout (s ) = HC (s ) φR (s ) − Hinj (s ) β (s )
(25)
Taken together (20)-(25) describes a frequency-domain model to analyze the phase dynamics of an IL-PLL OEO under RF signal injection shown in Fig. 4. Considering a loop filter with transfer function (s ) = (1 + sτ2 ) sτ1, the injection transfer function can be expressed as
Hinj (s ) =
sωL [s 2 + (2ξωn + KRI ) s + ωn2 ]
(26)
The absolute value of (26) is given by
|Hinj (s = jΩ)| =
ωL Ω (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
(27)
From the expression for absolute value of Hinj (jΩ) in (27), it can be seen that there exists an optimum frequency detuning Ω for which |Hinj (jΩ)| will be maximum, and thus representing the band-pass behavior of the injection transfer function. Taking the first derivative of |Hinj (s )| with respect to Ω, setting it equal to zero, and solving the resulting equation, the peak value of the injection transfer function can be derived as
HP =
ωO KI 2Q (2ξωn + KRI )(1 − 0.5ωn2 τ 2)
(28)
As can be inferred from (28), the natural frequency and damping factor of the PLL should be selected high to reduce the injectionpulling effect. But, the loop-delay provided by the optical fiber increases the pulling effects. Using (27) we plot in Fig. 5 the magnitude response of the injection transfer function |Hinj (jΩ)| against the frequency detuning. The injection signal power is set at - 30 dB m, which corresponds to the locking range ωL of 1 kHz. The loop natural frequency ωn and damping ratio ξ of the PLL section are 63.245 kHz and 0.707, respectively. From the Fig. 5, we can see that when Ω is equal to the naturalfrequency ωn of the PLL, the
Fig. 5. Absolute value of injection transfer function as a function of frequency detuning for the IL-PLL OEO under RF signal injection. 5
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
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absolute value of Hinj (jΩ) becomes maximum and is equal to 3.12 dB. 5. Phase noise analysis This section presents an equivalent dual loop model of an IL-PLL OEO under RF injection from the generalized locking Eq. (10) for phase noise analysis. We investigate the dependence of IL-PLL OEO output phase noise on the injection phase noise in terms of the noise transfer functions when the frequency detuning between the injection RF signal and the IL-PLL OEO output signal is zero. We assume that the IL-PLL OEO is injection-locked by an independent weak sinusoidal RF injection signal (KI ≪ 1) and the output signal is purely sinusoidal. Under locked condition, the generalized locking equation with the help of (6), and (7) can be written as
dφSS dt
= Ω+ μtan(Ωτ ) + K [φRSS − φoutSS ]* f (t ) + KRI [φRSS − φoutSS ] − ωLSS sin (φSS − Ωτ ) = 0
(29)
where φSS , φRSS , φoutSS and ωLSS are the steady state values of φ , φR , φout and ωL , respectively. Now we consider that the injection frequency, inherent oscillation frequency and reference frequency are disturbed by phase noise φn, i (t ) , φn, o (t ) , and φn, R (t ) , respectively. Thus, the phase perturbed steady-state phase difference is φSSn = φSS + φn , where φn is the phase noise induced by φn, i (t ) , φn, o (t ) , and φn, R (t ) . Similarly, the phase perturbed steady-state values of φRSS and φoutSS are φRSSn = φRSS + φn, R and φoutSSn = φoutSS + φn, out , respectively, where φn, out represents the phase noise of the output signal. In the steady-state, from (10), we obtain
dφSSn dt
= Ω+ μtan(Ωτ ) + K [φRSSn − φoutSSn]* f (t ) + KRI [φRSSn − φoutSSn] − ωLSS sin(φSSn − Ωτ ) +
dφn, o dt
−
dφn, i (30)
dt
Using the Taylor-series expansion of sin (φSSn − Ωτ ) around φSS and utilizing (29), one can easily show
dφn = K [φn, R − φn, out ]* f (t ) + KRI [φn, R − φn, out ] dt dφn, i dφn, o − φn ωLSS cos (φSSn − Ωτ ) + − dt dt
(31)
The output signal frequency can be rewritten as
ωout (t ) = ωinj + K [φn, R − φn, out ]* f (t ) + KRI [φn, R − φn, out ] − φn ωLSS cos (φSSn − Ωτ ) +
dφn, o (32)
dt
From (32), one can easily calculate the phase perturbation φn, out (t ) around ωinj in the following form
∫ ([φn,R − φn,out ]* f (t )) dt + KRI ∫ [φn,R − φn,out ] dt − ωLSS cos (φSSn − Ωτ ) ∫ φn dt + φn,o (t ) φn, out (t ) = K
(33)
Applying Laplace Transform to Eq.s (31),(33) and doing some mathematical manipulations, the overall phase noise in the frequency domain can be found as
φn, out (s ) = Hn, R (s ) φn, R (s ) + Hn, osc (s ) φn, o (s ) + Hn, inj (s ) φn, i (s )
(34)
where
Hn, R (s ) =
G (s ) Hn, o (s ) 1 + G (s ) H (s ) Hn, o (s )
(35)
Hn, osc (s ) =
Hn, o (s ) 1 + G (s ) H (s ) Hn, o (s )
(36)
Hn, inj (s ) =
Hn, i (s ) 1 + G (s ) H (s ) Hn, o (s )
(37)
are the transfer functions of reference noise, oscillation noise and injection noise, respectively. Hn, o (s ) =
1
1 + Gn, i (s ) ,and
Hn, i (s ) = Gn, i (s) 1 + Gn, i (s) , where Gn, i (s ) = ωLSS exp (−sτ ) cos (φSS ) s is the forward-path transfer function of the injection-locked loop only, as reported in [16]. Taken together (34) - (37) describes a frequency-domain model to study the phase-noise performance of an IL-PLL OEO under RF injection shown in Fig. 6. Thus, the power spectral density (PSD) of the overall phase noise φn, out at the offset angular frequency ωm becomes Sn, out (ωm) = |Hn, R (jωm )|2 S n, R (ωm) + |Hn, osc (jωm )|2 Sn, o (ωm) + |Hn, inj (jωm )|2 Sn, i (ωm)
(38)
where Sn, R (ωm) , Sn, o (ωm) , and Sn, i (ωm) are the PSDs of reference noise, oscillation noise and injection noise, respectively. A similar 6
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 6. Block diagram of the dual loop model for phase noise analysis of IL-PLL OEO under RF injection.
result for Gunn-effect oscillator was reported by Sugiura [19] using a different approach. Putting frequency offset equal to zero gives the same results as reported in [10]. 6. Experimental results This section presents the experimental results to validate the analytical results. A block diagram representation of the ILPLLsingle-loop OEO under study is shown in Fig. 7, and the various elements of the system architecture are listed below:
• An InGaAsP distributed feedback laser (DFB) laser source (Thorlabs 1546.92-20) operating at 1546 nm with a 3 mA threshold • • •
current and a maximum output optical power of 13 dB m at a current of 200 mA. In our present experiments a 100 mA injection current is used with an output optical power of about 8.633 dB m. The integrated LiNbO3Mach-Zehnder modulator (MZM) (Lucent 2623NA) with the transmission coefficient of 0.3 dB and a halfwave voltage Vπ = 2.99 V. The optical delay between the MZM and detector is provided by a 200 m, single-mode SMF-28 optical fiber. The delayed optical signal is detected by an InGaAs photo-detector (ThorlabsSIR5-FC) with a band-width of 5 GHz and responsivity of 1 A/W. A SAW filter of central frequency of 2.0175 GHz and a 3-dB bandwidth 15 MHz is used as RF filter to select the oscillation frequency of the OEO. The MZM, the optical fiber and the RF filter are kept in thermal enclosure boxes, as these components are very sensitive to temperature. Three cascaded RF amplifiers of overall gain of 71 dB amplify the power of the electrical signal at the output of the filter. Two analog signal generators (Agilent N5181A) are used to provide the RF injection signal the reference signal. The other components are two 50:50 power splitter and a power combiner. The loop-filter (for PLL section) is a first-order LPF with time constants τ1 = 16.5 μsec and τ2 = 0.165 μsec. The gain of the phase detector is 0.6365 V/rad, the tuning sensitivity of the OEO based VCO is 15.565 kHz/V and the loop natural frequency of the PLL is 63.2455 kHz.
A single-loop OEO injection-locked and phase-locked to a reference signal of frequency 2.013230 GHz and reference signal power – 15 dB m has been designed. The output RF signal of the IL-PLL OEO has 5.98 dB m in power. We use an independent RF sinusoidal signal as an injection signal of the IL-PLL OEO for investigating the injection-pulling effects. Fig. 8 shows the experimentally measured RF output signal spectra of the IL-PLL OEO under RF injection for two different values of frequency detuning Δf . The RF
Fig. 7. Experimental setup of an IL-PLL single-loop OEO under RF injection for injection-pulling experiments. 7
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
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Fig. 8. Measured output spectrum of the IL-PLL single-loop OEO under RF injection with various frequency detuning between the injection signal and the reference signal. The frequency detuning Δf is (a) 100 kHz, and (b) 50 kHz. Injection signal power is -30 dB m.
injection signal power Pinj is kept constant at - 30 dB m. We use an Agilent spectrum analyser N2090A to monitor the three different spectrums with the same setting 500 kHz span, 14 dB m reference level and 4.7 kHz resolution bandwidth. In both the cases, the injection-pulling effects become less significant and the output RF signal has two spurious tone levels at the frequencies finj , (2fO − finj ) and the carrier component fO . It should be noted that the right sideband located at finj is slightly larger than the left sideband at (2fO − finj ) . This is because that the RF injection signal also feeds through the IL-PLL OEO to its output. When Pinj is increased to -15 dB m, the injection-pulling effects become stronger as observed in Fig. 9. The output RF signal contains several sidebands with significant power both above and below the pulled carrier signal frequency fO and the spectral distribution is almost symmetric. It should be noted that Razavi experimentally reported a similar sideband distribution about the carrier for a 1 GHz charge pump phaselocked LC oscillator in [9]. Stover [20] experimentally reported similar spectrum for unlocked-driven tunnel diode oscillator. The phase noise is monitored by the Agilent N2090A spectrum analyzer. Fig. 10 shows the single-sideband (SSB) phase noise of the free-running OEO, injection RF signal, IL-OEO and IL-PLL OEO under RF signal injection with Δf = 0 . The phase noise of the IL-
Fig. 9. Measured output spectrum of the IL-PLL single-loop OEO under RF injection with various frequency detuning between the injection signal and the reference signal. The frequency detuning Δf is (a) 80 kHz, and (b) 70 kHz. Injection signal power is -15 dB m. 8
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Fig. 10. Measured SSB phase noise spectrum of the free-running OEO (black), injection RF signal (red), IL-OEO with Pinj= - 25 dB m (green), and IL-PLL OEO with RF injection signal having frequency detuning Δf = 0 and Pinj = -25 dB m (blue).
OEO is suppressed by 40 dB and 20 dB at 100 Hz and 1 kHz frequency-offset, respectively and then follows the phase noise of the freerunning OEOs over 10 kHz frequency-offset. For IL-PLL OEO under RF injection, the obtained phase noise is - 89 dBc/Hz, - 85 dBc/Hz, and - 90 dBc/Hz at frequency-offset of 100 Hz, 1 kHz and 10 kHz, respectively. Thus, the phase noise is further suppressed by 24 dB, 21 dB and 18 dB compared to the IL-OEO. It should be noted that over the frequency-offset range of 10 kHz–100 kHz, which is around the natural frequency of the PLL, the phase noise is suppressed by approximately 22 dB compared to that of free-running OEO. Thus, further phase noise suppression of 14 dB compared to the IL-OEO is obtained, but the phase noise at high offset-frequencies (> 100 kHz) remains unchanged. Table 1 present comparisons of the measured and calculated phase noise levels of IL-PLL OEO using (38) as a function of frequency-offset at three different injection powers Pinj . The comparisons show very good agreement. At Δf = 63 kHz, which is equal to the natural frequency of PLL, the phase noise decreases sharply as Pinj increases from -20 dBm to-10 dBm. To study the degradation of the signal quality due to the injection-pulling effect, we use a Gaussian minimum-shift keying (GMSK) signal, which imitate the interference from wireless direct conversion transmitter (DCT) for the global system for mobile communication (GSM), as an injection signal of the IL-PLL OEO. Frequency detuning between the GMSK signal center frequency and the reference signal frequency is zero, and the data rate of GMSK signal is set at 270.833 kb/s, which corresponds to a very narrow bandwidth. Fig. 11 shows the output spectral re-growth due to the injection-pulling effects in the IL-PLL OEO under GMSK signal injection. As injection signal power increases, the spectral broadening of the output signal also increases and the center frequency of the pulled IL-PLL OEO increases (i.e., pulled spectrum shifts right with respect to the reference signal frequency). Fig. 12 displays the effect of increasing the interfering signal strength on the overall phase noise of the IL-PLL OEO under injection pulling. The overall phase noise of the IL-PLL OEO under GMSK injection increases rapidly with the increase in injection signal power inside the lockrange of the oscillator. This is because that the modulated signal provides the extra phase noise to the oscillator due its dramatic phase variation. When injection signal power increases from – 40 dB m to – 30 dB m, the overall output phase noise significantly increases over the frequency-offset range of 300 Hz to 100 kHz and then follows the phase noise of free-running oscillator over 200 kHz frequency-offset. 7. Conclusion This paper has presented an accurate analytical model to capture the injection-pulling dynamics in an IL-PLL OEO under RF signal injection in time and frequency domain. Outcomes of the theoretical analysis have been verified by the experiments. We believe that our research ideas on the injection-pulling in IL-PLL OEO not only could stimulate the development on the new configuration of local oscillator (LO) with improved phase noise performance and long-term frequency stability but also would provide interesting directions to solve design challenges of LO for fifth generation (5 G) wireless and mobile networks. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to Table 1 Comparison of measured and calculated phase noise of IL-PLL OEO under RF signal injection. Injection Signal Power (Pinj)
−10 dBm −15 dB m −20 dBm
Phase Noise (dBc/Hz) Freq. Offset 1 kHz
Freq. Offset 10 kHz
Freq. Offset 63 kHz
Freq. Offset 100 kHz
Freq. Offset 1 MHz
Analyt.
Expt.
Analyt.
Expt.
Analyt.
Expt.
Analyt
Expt
Analyt
Expt.
−82.96 −81.54 −79.59
−86 −84 −83
−90.55 −85.50 −80.90
−94 −88 −84
−104.35 −94.68 −89.88
−110 −98 −91
−95.50 −92.80 −88.48
−99 −95 −92
−122.60 −119.68 −123.95
−124 −124 −127
9
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A. Banerjee, et al.
Fig. 11. Experimental results of the output spectrum for the IL-PLL OEO with a GMSK RF injection signal having no frequency offset and three different injection signal powers of – 30 dB m (red), - 40 dB m (green), and – 50 dB m (magenta). The black line is the spectrum of the IL-PLL OEO locked with reference RF injection signal of power – 15 dB m.
Fig. 12. Measured SSB phase noise of the IL-PLL OEO (black), and IL-PLL OEO with a GMSK RF injection signal having no frequency offset and for three different injection signal powers of – 30 dB m (green), - 40 dB m (magenta) and – 50 dB m (blue).
influence the work reported in this paper. References [1] X.S. Yao, L. Maleki, Optoelectronic oscillator for photonic systems, IEEE J. Sel. Top. Quantum Electron. 32 (July(7)) (1996) 1141–1149. [2] J. Tang, T. Hao, R. Banos, M. Li, Integrated optoelectronic oscillators, Opt. Exp. 26 (2018) 12257–12265. [3] A. Blustone, D.E. Spencer, S. Srinivasan, An ultra-low phase-noise 20-GHz PLL utilizing optoelectronic voltage controlled oscillator, IEEE Trans. Microw. Theory Technol. 63 (3) (2015) 1045–1052. [4] Y. Zhang, L. Hou, D. Zhao, Long term frequency stabilization of an optoelectronic oscillator using phase locked loop, J. Lightwave Technol. 32 (July(13)) (2014) 2408–2414. [5] R. Fu, X. Jin, X. Zhang, Frequency stability optimization of an OEO using phase lock loop and self injection locking, Opt. Commun. 386 (November) (2016) 27–30. [6] L. Zhang, A.K. Poddar, U.L. Rohde, A.S. Daryoush, Self-ILPLL using optical feedback for phase noise reduction in microwave oscillators, IEEE Photonics Technol. Lett. 27 (March) (2015) 624–627. [7] Z. Zhenghua, et al., An ultra-low phase noise and highly stable optoelectronic oscillator utilizing IL-PLL, IEEE Photonics Technol. Lett. 28 (February(4)) (2016) 516–519. [8] B. Razavi, Challenges in portable RF transmitter design, IEEE Circuits Devices Mag. 12 (September (5)) (1996) 12–25. [9] B. Razavi, A Study of injection locking and pulling in oscillators, IEEE J. SolidState Circuits 39 (September(9)) (2004) 1415–1424. [10] K. Kurokawa, Injection locking of oscillators, Proc. IEEE 61 (October(10)) (1973) 1386–1410. [11] P. Maffezzoni, D. D’Amore, Evaluating pulling effects in oscillators due to small signal injection, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 28 (January (1)) (2009). [12] A. Banerjee, B.N. Biswas, Analysis of phase locking in optoelectronic microwave oscillators due to small RF signal injection, IEEE J. Quantum Electron. 53 (May (10)) (2017). [13] A. Banerjee, J. Sarkar, N.R. Das, B.N. Biswas, Phase locking dynamics in optoelectronic oscillator, Optics Commun. 414 (January) (2018) 119–127. [14] J. Sarkar, A. Banerjee, B.N. Biswas, Analysis of frequency pulling phenomenon phenomenon in an optoelectronic oscillator, Opt. Eng. 57 (June(6)) (2018). [15] A. Banerjee, J. Sarkar, Improved analytical model of phase-locking dynamics in unlocked-driven optoelectronic oscillators under RF injection locking, Optics Commun. 437 (2019) 184–192. [16] A. Banerjee, L.A.D. Britto, G.M. Gefeson, Analysis of injection-locking and injection-pulling in single-loop optoelectronic oscillator, IEEE Trans. Microw. Theory Technol. (2019), https://doi.org/10.1109/TMTT.2019.2891595 Accepted for publication. [17] F.M. Gardner, Phase-lock Techniques, 3rd edition, Wiley, New York, 2005. [18] R. Adler, A study of locking phenomena in oscillators, Proc. IEEE 61 (October) (1973) 1380–1385. [19] T. Sugiura, S. Sugimoto, FM noise reduction of Gunn-effect oscillators by injection locking, Proc. IEEE 57 (January (1)) (1969) 77–78 1069. [20] H.L. Stover, Theoretical explanation for the output spectra of unlocked driven oscillators, Proc. IEEE (1965) 310–311.
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Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
A theoretical and experimental study of injection-pulling for IL-PLL optoelectronic oscillator under RF signal injection
T
Abhijit Banerjeea,*, Larissa Aguiar Dantas de Brittob, Gefeson Mendes Pachecob a
Electronics and Communication Engineering Departament, Academy of Technology, Adisaptagram, Hooghly, 712121, West Bengal, India Aeronautical Institute of Technology (ITA), Microwave and Optoelectronic Department, Marechal Eduardo Gomes, 50 - Vila das Acácias, São José dos Campos, SP, 12228-900, Brazil
b
A R T IC LE I N F O
ABS TRA CT
Keywords: Injection-pulling dynamics Phase noise Optoelectronic oscillator Phase-lock loop Injection transfer function
A detailed study of injection-pulling behavior of an injection-locked and phase-locked loop (ILPLL) single-loop optoelectronic oscillator (OEO) under radio frequency (RF) signal injection is presented. From the locking equation, the expression for the output spectrum of the IL-PLL OEO under RF injection is derived in time domain, to evaluate the spurious performance of the system. The frequency response of the spurious outputs due to RF injection signal is investigated in terms of the injection transfer function. Also, a dual-loop model of the IL-PLL OEO is developed to study the phase noise performance of the system with co-frequency RF injection signal. The experimental results are given in partial support of the conclusions of the analysis.
1. Introduction A key subsystem common in the photonic systems in the field of microwave photonic is the RF source. The RF generation by optical methods can be done by mixing different optical wavelengths in a non-linear crystal or in a photodiode and by the use of a ring circuit formed by an optical link and a feedback RF loop named OEO [1,2]. Such oscillator is the RF source used in a wide variety of high-frequency signal processing systems, enables the generation of signals with frequencies from sub-GHz up to 70 GHz and phase noise down to -120 dBc/Hz at 1 kHz offset. The injection-locking (IL) technique in OEO becomes useful in a number of applications, including clock and carrier recovery, low-power RF signal detection, and low phase noise microwave signal generation. Also, PLL single-loop OEO is demonstrated to generate a microwave signal with long-term frequency stability and low-phase noise [3,4]. The IL-PLL OEO can provide ultra-low phase noise microwave signal with excellent long-term stability [5–7]. Injection-pulling, on the other hand, typically proves undesirable [8]. In a RF transceiver, the injection of unwanted spurious tones to the oscillator through the parasitic coupling path degrades the spectral purity of the output signal [9–11]. Thus, injection-locking and pulling dynamics of the OEO should be studied in detail in the unlocked-driven state to exploit its merit in intelligent potential applications and reduce its effects whenever undesirable. Few approaches [12–16] have been developed recently to forecast the phase dynamics and the frequency-pulling phenomena in single-loop OEO due to an independent RF sinusoidal signal injection. All the prior analytical model describes the injection-locking or pulling in nominally free-running single-loop OEO, a rare case of practical interest. Since the OEOs are usually phase-locked, the analysis must account for the correction produced by the PLL. Comprehensive theoretical and experimental works on the phase noise performance of IL-OEO, and IL-PLL OEO have been published. There is currently no dynamical frame-work of the IL-PLL OEO to study its dynamics. This lack of analytical insights into the dynamical properties of IL-PLL OEO does
⁎
Corresponding author. E-mail address: [email protected] (A. Banerjee).
https://doi.org/10.1016/j.ijleo.2019.164059 Received 8 November 2019; Accepted 11 December 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 1. Electrical equivalent block diagram of a single-loop OEO under injection and its vector diagram.
not enable us to optimize its metrics. This paper develops an accurate phase dynamic model of an IL-PLL single-loop OEO under RF signal injection, and also, analyzes the injection-pulling effects in time and frequency domain. This paper also investigates the phase noise performance of an IL-PLL single-loop OEO under co-frequency sinusoidal RF signal injection. In this paper, the topics are organized as follows. In Section II, we derive the locking equation for the IL-PLL single-loop OEO under RF signal injection. Section III studies the injection-pulling behavior of the system in time domain. Section IV describes the frequency-domain model of the IL-PLL OEO to study the phase dynamics. Phase noise analysis is performed in Section V. Section VI presents the experimental results, and finally, conclusions are given in Section VII. 2. Locking equation derivation In this section, we first derive the injection-locking equation for a single-loop OEO under the influence of an independent RF sinusoidal signal injection. In order to generalize the analytical model, we then extend the locking equation for an IL-PLL single-loop OEO under RF injection. 2.1. Single-loop OEO under RF injection Fig. 1 shows a simplified electrical equivalent block diagram of a RF driven OEO. A summer is considered in the feedback path to allow the injection of RF signal. In this figure, VS is the injection RF signal with instantaneous frequency ωinj (t ) and constant amplitude Vinj , VO is the perturbed output RF signal with constant amplitude VOP ,and ϕ (t ) denotes the instantaneous phase difference between VS and VO . In the absence of injection signal, OEO oscillates with the free-running frequency ω0 . When an injection signal with frequency ωinj is applied to the OEO, the injection signal VS combines with the feedback signal VO to produce the signal VX . If ωinj is near to ω0 , the OEO will be injection-locked to the injection RF signal and the OEO will oscillate with the injection signal frequency ωinj . Under injection-locked condition, a phase difference between the injection signal and the oscillator signal is established to maintain oscillation at the injection frequency ωinj . The RF filter and the optical fiber introduces the necessary phase shift (θ − Ωτ ) to the signal, where Ω = (ωinj − ω0) denotes the frequency detuning between the free-running oscillation and the injection signal, τ is the time-delay provided by the optical fiber, and
θ = tan−1 ⎛ ⎝
⎜
2Q (ω0 − ω) ⎞ ω0 ⎠ ⎟
(1)
is the phase shift introduced by the first-order linear band pass filter (BPF) with transfer function H (s ) = where Q is the quality factor of the filter. From the vector diagram of Fig. 1, we obtain
tan(θ − Ωτ ) =
Vinj sinϕ VOP +Vinj cosϕ
=
KI sinϕ 1 + KI cosϕ
1 1 + Q ω0 s + s ω0
(
)
[17],
(2)
where KI = (Vinj / VOP ) is the injection ratio. Consider that injection signal vector VS is at rest and the oscillation output signal vector VO is rotating counterclockwise with instantaneous beat frequency is given by
ω = ωinj +
( ) with respect to V . Thus, the oscillator instantaneous frequency dϕ
dφ dt
dt
S
(3)
Substituting (1) into (2) and then using (3), we obtain the phase dynamics equation of the single-loop OEO under the influence of RF signal injection [16] 2
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Fig. 2. Electrical equivalent block diagram of an IL-PLL single-loop OEO under RF injection.
dφ μsin(Ωτ ) − F sin(ϕ − Ωτ ) = Ω+ dt cos(Ωτ )+KI cos(ϕ − Ωτ )
(
(4)
)
where F = (μKI ) and μ = ω0 2Q is the half-bandwidth of the BPF. In the absence of time delay τ (τ = 0) and for small signal injection (KI ≪ 1) , (4) becomes identical to the familiar Adler’s equation [18]. 2.2. IL-PLL single-loop OEO under RF Injection In this sub-section, we are going to derive the generalized locking-equation for an IL-PLL OEO under RF signal injection shown in Fig. 2. By tuning the bias voltage of the MZM (which provides the non-linearity in the loop), the instantaneous frequency of the single-loop OEO can be varied and thus, acting as a voltage controlled oscillator (VCO) with tuning sensitivityKVCO . Let us assume that VO is now from an IL-PLL single-loop OEO controlled by both PLL and injection-locking mechanism. Thus, the inherent oscillation frequency of the oscillator is given by
ω0 (t ) = ω0 + KVCO VC (t ) + KRI φE (t )
(5)
where VC (t ) is the tuning voltage, and is given by
VC (t ) = KPD φE (t )* f (t )
(6)
φE (t ) = φR (t ) − φout (t )
(7)
where KPD is the gain of the phase-detector, f (t ) is the impulse response of the loop-filter, KRI is the injection ratio for reference signal and φE (t ) is the output phase-error of the phase-detector. φR (t ) and φout (t ) represent the reference phase and resultant output phase of the IL-PLL OEO, respectively. Combining (4), (5) and then using (3), the resultant output frequency of the IL-PLL OEO can be written as
ωout (t ) ≅ (ωO + μtan (Ωτ )) + KVCO VC (t ) + KRI φE (t ) − ωL β (t )
(8)
where ωL = μKI sec (Ωτ ) , and β (t ) = sin (φ − Ωτ ) . ωL β (t ) and KPD KRI φE (t ) represent the frequency modulation component due to RF injection signal and reference RF injection signal, respectively. KVCO VC (t ) is the frequency modulation of the oscillator for phaselocking mechanism. Integrating (8), the resultant output phase of the IL-PLL OEO can be found as
φout (t ) = (ωO + μtan (Ωτ )) t + KVCO
∫ VC (t ) dt + KRI ∫ φE (t ) dt − ωL ∫ β (t ) dt
(9)
Taken together (6), (7) and (9) describes a time-domain model of an IL-PLL OEO under RF signal injection shown in Fig. 3. 3. Injection – pulling analysis In this section, we formulate the behavior of an IL-PLL OEO under injection-pulling in time domain. We assume that the OEO is
Fig. 3. Time-domain model of an IL-PLL OEO under RF injection. 3
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injection-locked and phase-locked to the reference signal with frequency ωO and also the oscillator is injection-pulled by an independent RF signal at frequency ωinj . We also assume that the oscillator phase-locking control contains a small perturbation VC around a dc level. Substituting (5) into (4), phase perturbation can be expressed as
dφ = Ω+ μtan (Ωτ ) + KVCO VC (t ) + KRI φE (t ) − ωL sin (φ − Ωτ ) dt
(10)
Since the PLL is phase-locked to ωO and assuming φ is very small (φ ≪ 1rad ) , (10) may be written as
dφ ≈ KVCO VC (t ) + KRI φE (t ) + ωL sin (Ωt ) dt
(11)
We consider an active low-pass filter (LPF) with transfer function F (s ) = (1 + sτ2 ) sτ1 as the loop-filter of the PLL, where τ2 and τ1 are the filter time constants. Thus, in Laplace-domain, the control voltage is given by
VC (s ) = −KPD F (s ) φ (s ) = −KPD
(1 + sτ2 ) φ (s ) sτ1
(12)
where the negative sign represents the phase subtraction of the phase-detector. Taking the inverse Laplace Transform of (12), substituting for VC (t ) in (11) and then differentiating both sides with respect to time, one can easily show
dφ d 2φ + (2ξωn + KRI ) + ωn2 φ = ωL Ωcos (Ωt) dt dt 2
(13)
where ωn = K / τ1 , K = KPD KVCO , and ξ = ωn τ2/2 are the natural frequency, the loop-gain, and the damping factor of the PLL, respectively. Eq. (13) indicates that the IL-PLL single-loop OEO behaves as a second-order system in the presence of injection-pulling. The solution of (13) is given by
φ (t ) =
ωL Ωcos (Ωt−α ) (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
= φm cos (Ωt− α)
(14)
where α is the phase of the transfer function at a frequency of Ω, and
φm =
ωL Ω (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
(15)
The IL-PLL OEO output signal can be represented as
vO (t ) = VOut cos (ωO t + φ)
(16)
Substituting (14) into (16) and then using the following identities, ∞
cos (zcosθ) = JO (z ) + 2 ∑ (−1) k J2k (z ) cos (2kθ)
(17)
k=1 ∞
sin (zcosθ) = 2 ∑ (−1) k J2k + 1 (z ) cos ((2k + 1) θ)
(18)
k=1
the RF output signal of the IL-PLL OEO under injection-pulling can be found as ' ∞ ⎡ cos (ωO t − βk )+⎤ vO (t ) = JO (φm ) VOut cos (ωO t ) + VOut ∑ (−1) k J2k (φm ) ⎢ ⎥ '' k=1 ⎣ cos (ωO t + βk ) ⎦ ' ∞ ⎡ sin (ωO t − βk )+⎤ − VOut ∑ (−1) k J2k + 1 (φm ) ⎢ ⎥ '' k=1 ⎣ sin (ωO t + βk ) ⎦
ωO'
(19)
ωO''
= (ωO + 2k Ω) , = (ωO − 2k Ω) and βk = 2kα . From (14) and (16), we can see that the IL-PLL OEO output phase is where modulated sinusoid ally and also (12) indicates that the control voltage of the phase-locking mechanism varies sinusoidally with a frequency ofΩ. For weak RF injection and considering only the first harmonic with k = 1, from (19), it can be observed that the output RF signal-contains a carrier component at ωO and two symmetric sidebands at ωinj and (2ωO − ωinj ) . 4. Frequency domain IL – PLL OEO model In this section, we are going to present a simple frequency-domain model of an IL-PLL OEO to quantify the frequency dependence of the spurious outputs caused by the RF injection signal. Assume that the frequency difference between the injection signal and the IL-PLL OEO output signal is small. In that case, Laplace domain representations of Eq.s (6),(7), and (9), are given by
VC (s ) = KPD φE (s ) F (s )
(20)
φE (s ) = φR (s ) − φout (s )
(21) 4
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
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Fig. 4. Block diagram of the frequency-domain model of IL-PLL OEO under RF injection.
sφout (s ) = KVCO VC (s ) + KRI φE (s ) − ωL β (s )
(22)
where φE (s ) , φout (s ) , φR (s ) and β (s ) are the Laplace transform of φE (t ) , φout (t ) , φR (t ) and β (t ) , respectively. Combining (20), (21) and (22), one can easily calculate the closed-loop transfer function and the transfer function with respect to the injection, respectively as
HC (s ) = Hinj (s ) =
G (s ) 1 + G (s ) H (s )
(23)
ωL s [1 + G (s ) H (s )]
(24)
where G (s ) = [(KF (s )/ s ) + (KPD KRI /s )] and H (s ) = 1 are the forward-path transfer function and feedback-path transferfunction of the IL-PLL section, respectively.The output phase of the IL-PLL OEO can be written as
φout (s ) = HC (s ) φR (s ) − Hinj (s ) β (s )
(25)
Taken together (20)-(25) describes a frequency-domain model to analyze the phase dynamics of an IL-PLL OEO under RF signal injection shown in Fig. 4. Considering a loop filter with transfer function (s ) = (1 + sτ2 ) sτ1, the injection transfer function can be expressed as
Hinj (s ) =
sωL [s 2 + (2ξωn + KRI ) s + ωn2 ]
(26)
The absolute value of (26) is given by
|Hinj (s = jΩ)| =
ωL Ω (Ω2 − ωn2 )2 + (2ξωn + KRI )2Ω2
(27)
From the expression for absolute value of Hinj (jΩ) in (27), it can be seen that there exists an optimum frequency detuning Ω for which |Hinj (jΩ)| will be maximum, and thus representing the band-pass behavior of the injection transfer function. Taking the first derivative of |Hinj (s )| with respect to Ω, setting it equal to zero, and solving the resulting equation, the peak value of the injection transfer function can be derived as
HP =
ωO KI 2Q (2ξωn + KRI )(1 − 0.5ωn2 τ 2)
(28)
As can be inferred from (28), the natural frequency and damping factor of the PLL should be selected high to reduce the injectionpulling effect. But, the loop-delay provided by the optical fiber increases the pulling effects. Using (27) we plot in Fig. 5 the magnitude response of the injection transfer function |Hinj (jΩ)| against the frequency detuning. The injection signal power is set at - 30 dB m, which corresponds to the locking range ωL of 1 kHz. The loop natural frequency ωn and damping ratio ξ of the PLL section are 63.245 kHz and 0.707, respectively. From the Fig. 5, we can see that when Ω is equal to the naturalfrequency ωn of the PLL, the
Fig. 5. Absolute value of injection transfer function as a function of frequency detuning for the IL-PLL OEO under RF signal injection. 5
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A. Banerjee, et al.
absolute value of Hinj (jΩ) becomes maximum and is equal to 3.12 dB. 5. Phase noise analysis This section presents an equivalent dual loop model of an IL-PLL OEO under RF injection from the generalized locking Eq. (10) for phase noise analysis. We investigate the dependence of IL-PLL OEO output phase noise on the injection phase noise in terms of the noise transfer functions when the frequency detuning between the injection RF signal and the IL-PLL OEO output signal is zero. We assume that the IL-PLL OEO is injection-locked by an independent weak sinusoidal RF injection signal (KI ≪ 1) and the output signal is purely sinusoidal. Under locked condition, the generalized locking equation with the help of (6), and (7) can be written as
dφSS dt
= Ω+ μtan(Ωτ ) + K [φRSS − φoutSS ]* f (t ) + KRI [φRSS − φoutSS ] − ωLSS sin (φSS − Ωτ ) = 0
(29)
where φSS , φRSS , φoutSS and ωLSS are the steady state values of φ , φR , φout and ωL , respectively. Now we consider that the injection frequency, inherent oscillation frequency and reference frequency are disturbed by phase noise φn, i (t ) , φn, o (t ) , and φn, R (t ) , respectively. Thus, the phase perturbed steady-state phase difference is φSSn = φSS + φn , where φn is the phase noise induced by φn, i (t ) , φn, o (t ) , and φn, R (t ) . Similarly, the phase perturbed steady-state values of φRSS and φoutSS are φRSSn = φRSS + φn, R and φoutSSn = φoutSS + φn, out , respectively, where φn, out represents the phase noise of the output signal. In the steady-state, from (10), we obtain
dφSSn dt
= Ω+ μtan(Ωτ ) + K [φRSSn − φoutSSn]* f (t ) + KRI [φRSSn − φoutSSn] − ωLSS sin(φSSn − Ωτ ) +
dφn, o dt
−
dφn, i (30)
dt
Using the Taylor-series expansion of sin (φSSn − Ωτ ) around φSS and utilizing (29), one can easily show
dφn = K [φn, R − φn, out ]* f (t ) + KRI [φn, R − φn, out ] dt dφn, i dφn, o − φn ωLSS cos (φSSn − Ωτ ) + − dt dt
(31)
The output signal frequency can be rewritten as
ωout (t ) = ωinj + K [φn, R − φn, out ]* f (t ) + KRI [φn, R − φn, out ] − φn ωLSS cos (φSSn − Ωτ ) +
dφn, o (32)
dt
From (32), one can easily calculate the phase perturbation φn, out (t ) around ωinj in the following form
∫ ([φn,R − φn,out ]* f (t )) dt + KRI ∫ [φn,R − φn,out ] dt − ωLSS cos (φSSn − Ωτ ) ∫ φn dt + φn,o (t ) φn, out (t ) = K
(33)
Applying Laplace Transform to Eq.s (31),(33) and doing some mathematical manipulations, the overall phase noise in the frequency domain can be found as
φn, out (s ) = Hn, R (s ) φn, R (s ) + Hn, osc (s ) φn, o (s ) + Hn, inj (s ) φn, i (s )
(34)
where
Hn, R (s ) =
G (s ) Hn, o (s ) 1 + G (s ) H (s ) Hn, o (s )
(35)
Hn, osc (s ) =
Hn, o (s ) 1 + G (s ) H (s ) Hn, o (s )
(36)
Hn, inj (s ) =
Hn, i (s ) 1 + G (s ) H (s ) Hn, o (s )
(37)
are the transfer functions of reference noise, oscillation noise and injection noise, respectively. Hn, o (s ) =
1
1 + Gn, i (s ) ,and
Hn, i (s ) = Gn, i (s) 1 + Gn, i (s) , where Gn, i (s ) = ωLSS exp (−sτ ) cos (φSS ) s is the forward-path transfer function of the injection-locked loop only, as reported in [16]. Taken together (34) - (37) describes a frequency-domain model to study the phase-noise performance of an IL-PLL OEO under RF injection shown in Fig. 6. Thus, the power spectral density (PSD) of the overall phase noise φn, out at the offset angular frequency ωm becomes Sn, out (ωm) = |Hn, R (jωm )|2 S n, R (ωm) + |Hn, osc (jωm )|2 Sn, o (ωm) + |Hn, inj (jωm )|2 Sn, i (ωm)
(38)
where Sn, R (ωm) , Sn, o (ωm) , and Sn, i (ωm) are the PSDs of reference noise, oscillation noise and injection noise, respectively. A similar 6
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 6. Block diagram of the dual loop model for phase noise analysis of IL-PLL OEO under RF injection.
result for Gunn-effect oscillator was reported by Sugiura [19] using a different approach. Putting frequency offset equal to zero gives the same results as reported in [10]. 6. Experimental results This section presents the experimental results to validate the analytical results. A block diagram representation of the ILPLLsingle-loop OEO under study is shown in Fig. 7, and the various elements of the system architecture are listed below:
• An InGaAsP distributed feedback laser (DFB) laser source (Thorlabs 1546.92-20) operating at 1546 nm with a 3 mA threshold • • •
current and a maximum output optical power of 13 dB m at a current of 200 mA. In our present experiments a 100 mA injection current is used with an output optical power of about 8.633 dB m. The integrated LiNbO3Mach-Zehnder modulator (MZM) (Lucent 2623NA) with the transmission coefficient of 0.3 dB and a halfwave voltage Vπ = 2.99 V. The optical delay between the MZM and detector is provided by a 200 m, single-mode SMF-28 optical fiber. The delayed optical signal is detected by an InGaAs photo-detector (ThorlabsSIR5-FC) with a band-width of 5 GHz and responsivity of 1 A/W. A SAW filter of central frequency of 2.0175 GHz and a 3-dB bandwidth 15 MHz is used as RF filter to select the oscillation frequency of the OEO. The MZM, the optical fiber and the RF filter are kept in thermal enclosure boxes, as these components are very sensitive to temperature. Three cascaded RF amplifiers of overall gain of 71 dB amplify the power of the electrical signal at the output of the filter. Two analog signal generators (Agilent N5181A) are used to provide the RF injection signal the reference signal. The other components are two 50:50 power splitter and a power combiner. The loop-filter (for PLL section) is a first-order LPF with time constants τ1 = 16.5 μsec and τ2 = 0.165 μsec. The gain of the phase detector is 0.6365 V/rad, the tuning sensitivity of the OEO based VCO is 15.565 kHz/V and the loop natural frequency of the PLL is 63.2455 kHz.
A single-loop OEO injection-locked and phase-locked to a reference signal of frequency 2.013230 GHz and reference signal power – 15 dB m has been designed. The output RF signal of the IL-PLL OEO has 5.98 dB m in power. We use an independent RF sinusoidal signal as an injection signal of the IL-PLL OEO for investigating the injection-pulling effects. Fig. 8 shows the experimentally measured RF output signal spectra of the IL-PLL OEO under RF injection for two different values of frequency detuning Δf . The RF
Fig. 7. Experimental setup of an IL-PLL single-loop OEO under RF injection for injection-pulling experiments. 7
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 8. Measured output spectrum of the IL-PLL single-loop OEO under RF injection with various frequency detuning between the injection signal and the reference signal. The frequency detuning Δf is (a) 100 kHz, and (b) 50 kHz. Injection signal power is -30 dB m.
injection signal power Pinj is kept constant at - 30 dB m. We use an Agilent spectrum analyser N2090A to monitor the three different spectrums with the same setting 500 kHz span, 14 dB m reference level and 4.7 kHz resolution bandwidth. In both the cases, the injection-pulling effects become less significant and the output RF signal has two spurious tone levels at the frequencies finj , (2fO − finj ) and the carrier component fO . It should be noted that the right sideband located at finj is slightly larger than the left sideband at (2fO − finj ) . This is because that the RF injection signal also feeds through the IL-PLL OEO to its output. When Pinj is increased to -15 dB m, the injection-pulling effects become stronger as observed in Fig. 9. The output RF signal contains several sidebands with significant power both above and below the pulled carrier signal frequency fO and the spectral distribution is almost symmetric. It should be noted that Razavi experimentally reported a similar sideband distribution about the carrier for a 1 GHz charge pump phaselocked LC oscillator in [9]. Stover [20] experimentally reported similar spectrum for unlocked-driven tunnel diode oscillator. The phase noise is monitored by the Agilent N2090A spectrum analyzer. Fig. 10 shows the single-sideband (SSB) phase noise of the free-running OEO, injection RF signal, IL-OEO and IL-PLL OEO under RF signal injection with Δf = 0 . The phase noise of the IL-
Fig. 9. Measured output spectrum of the IL-PLL single-loop OEO under RF injection with various frequency detuning between the injection signal and the reference signal. The frequency detuning Δf is (a) 80 kHz, and (b) 70 kHz. Injection signal power is -15 dB m. 8
Optik - International Journal for Light and Electron Optics 203 (2020) 164059
A. Banerjee, et al.
Fig. 10. Measured SSB phase noise spectrum of the free-running OEO (black), injection RF signal (red), IL-OEO with Pinj= - 25 dB m (green), and IL-PLL OEO with RF injection signal having frequency detuning Δf = 0 and Pinj = -25 dB m (blue).
OEO is suppressed by 40 dB and 20 dB at 100 Hz and 1 kHz frequency-offset, respectively and then follows the phase noise of the freerunning OEOs over 10 kHz frequency-offset. For IL-PLL OEO under RF injection, the obtained phase noise is - 89 dBc/Hz, - 85 dBc/Hz, and - 90 dBc/Hz at frequency-offset of 100 Hz, 1 kHz and 10 kHz, respectively. Thus, the phase noise is further suppressed by 24 dB, 21 dB and 18 dB compared to the IL-OEO. It should be noted that over the frequency-offset range of 10 kHz–100 kHz, which is around the natural frequency of the PLL, the phase noise is suppressed by approximately 22 dB compared to that of free-running OEO. Thus, further phase noise suppression of 14 dB compared to the IL-OEO is obtained, but the phase noise at high offset-frequencies (> 100 kHz) remains unchanged. Table 1 present comparisons of the measured and calculated phase noise levels of IL-PLL OEO using (38) as a function of frequency-offset at three different injection powers Pinj . The comparisons show very good agreement. At Δf = 63 kHz, which is equal to the natural frequency of PLL, the phase noise decreases sharply as Pinj increases from -20 dBm to-10 dBm. To study the degradation of the signal quality due to the injection-pulling effect, we use a Gaussian minimum-shift keying (GMSK) signal, which imitate the interference from wireless direct conversion transmitter (DCT) for the global system for mobile communication (GSM), as an injection signal of the IL-PLL OEO. Frequency detuning between the GMSK signal center frequency and the reference signal frequency is zero, and the data rate of GMSK signal is set at 270.833 kb/s, which corresponds to a very narrow bandwidth. Fig. 11 shows the output spectral re-growth due to the injection-pulling effects in the IL-PLL OEO under GMSK signal injection. As injection signal power increases, the spectral broadening of the output signal also increases and the center frequency of the pulled IL-PLL OEO increases (i.e., pulled spectrum shifts right with respect to the reference signal frequency). Fig. 12 displays the effect of increasing the interfering signal strength on the overall phase noise of the IL-PLL OEO under injection pulling. The overall phase noise of the IL-PLL OEO under GMSK injection increases rapidly with the increase in injection signal power inside the lockrange of the oscillator. This is because that the modulated signal provides the extra phase noise to the oscillator due its dramatic phase variation. When injection signal power increases from – 40 dB m to – 30 dB m, the overall output phase noise significantly increases over the frequency-offset range of 300 Hz to 100 kHz and then follows the phase noise of free-running oscillator over 200 kHz frequency-offset. 7. Conclusion This paper has presented an accurate analytical model to capture the injection-pulling dynamics in an IL-PLL OEO under RF signal injection in time and frequency domain. Outcomes of the theoretical analysis have been verified by the experiments. We believe that our research ideas on the injection-pulling in IL-PLL OEO not only could stimulate the development on the new configuration of local oscillator (LO) with improved phase noise performance and long-term frequency stability but also would provide interesting directions to solve design challenges of LO for fifth generation (5 G) wireless and mobile networks. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to Table 1 Comparison of measured and calculated phase noise of IL-PLL OEO under RF signal injection. Injection Signal Power (Pinj)
−10 dBm −15 dB m −20 dBm
Phase Noise (dBc/Hz) Freq. Offset 1 kHz
Freq. Offset 10 kHz
Freq. Offset 63 kHz
Freq. Offset 100 kHz
Freq. Offset 1 MHz
Analyt.
Expt.
Analyt.
Expt.
Analyt.
Expt.
Analyt
Expt
Analyt
Expt.
−82.96 −81.54 −79.59
−86 −84 −83
−90.55 −85.50 −80.90
−94 −88 −84
−104.35 −94.68 −89.88
−110 −98 −91
−95.50 −92.80 −88.48
−99 −95 −92
−122.60 −119.68 −123.95
−124 −124 −127
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Optik - International Journal for Light and Electron Optics 203 (2020) 164059
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Fig. 11. Experimental results of the output spectrum for the IL-PLL OEO with a GMSK RF injection signal having no frequency offset and three different injection signal powers of – 30 dB m (red), - 40 dB m (green), and – 50 dB m (magenta). The black line is the spectrum of the IL-PLL OEO locked with reference RF injection signal of power – 15 dB m.
Fig. 12. Measured SSB phase noise of the IL-PLL OEO (black), and IL-PLL OEO with a GMSK RF injection signal having no frequency offset and for three different injection signal powers of – 30 dB m (green), - 40 dB m (magenta) and – 50 dB m (blue).
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