# An Iterative Technique for the Synthesis of Active Antenna Oscillator Arrays

Authors: Theodoros N. Kaifas; John N. Sahalos
Journal: International Journal of Antennas and Propagation
 (2009)

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## Abstract

A design procedure for the synthesis of a coupled active antenna oscillator array is presented. Such an array is synthesized by deriving two sets of parameters: the radiators' positions and the oscillators' outputs. The outputs are used to excite the radiators. Minimization of the mean square error between the desired pattern and the resulting one is made. Synthesis starts from an initial array, which is perturbed iteratively by varying simultaneously the element excitations and positions. In the iteration, the first variation of the cost function is set equal to zero. The final array results from the last iteration, where the stopping criteria are met.
The procedure designs simultaneously both the antenna and the attached coupled oscillator array providing viable solutions. The  second by properly configuring the tuning parameters through the use of closed-form formulas. The resulting arrays are shown to comply with the desired pattern and the nonlinear dynamics thus proving the validity of our method.


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## Body

## 1. Introduction

It is well known that the angular
distribution of the radiated power from an antenna array depends on the
geometry and the excitation of the elements [1]. In a coupled active antenna oscillator linear array consisting of identical and parallel elements, the
pattern is expressed in the following form:  (1)F(θ)=∑k=1N(V˜k)fk(θ)ej(2π/λ)zkcos(θ),Vk=|V˜k|,ϕk=∠V˜k,
where the excitation V˜k is the complex voltage of the oscillator
output, (Vk=|V˜k| amplitude and ϕk=∠V˜k the respective phase), fk(θ) is the pattern, and zk is the abscissa of the kth element,
respectively. λ is the
free-space wavelength at the operating frequency.The two key sets of parameters that are important in the design
are the boxed terms of (1). The antenna array synthesis mainly focuses on the
excitation [2–4], while efforts aiming at extracting the positions of the array
elements are scarce [5]. In our present study, we cope either with one of the
above or with both sets simultaneously. A perturbation method that provides
iteratively the excitation and the placement of the elements in a quest of
matching the resulting far—field with the desired one is applied. The
whole synthesis is formulated as an optimization problem over a vector
containing the excitation and the element positions. The cost function is the
mean square error between the desired and the attained pattern.Apart from attaining the necessary parameters of the antenna
array, it is of equal importance the problem of feeding the array by a proper
network. In the literature, one can find several approaches to the feeding
networks [1], containing corporate feeds and feeding matrices [6]. Apart from
passive networks, there are others that utilize active circuitry. Among them,
the active feeding networks consisting of coupled oscillator arrays [7] are of
specific interest. It has been found that such devices allow for the
manipulation of the phase distribution without additional phase-shifting
circuitry, suggesting a potential for low-cost beam-scanning systems. In the
past, most of the studies on coupled oscillator arrays analyze the problem of
beam steering [8–10] and of null
control of the pattern, [11–13]. In the
current work, we extend the functionality of the coupled oscillator arrays to
more general pattern synthesis problems.It is noticed that there are cases
where a uniform excitation of a linear phased array and a nonuniform placement
is enough to solve the synthesis problem. In this case, constraining the
oscillators’ output voltage to a common value reduces the nonlinear dynamics of
the system. Alternatively, one can keep the phases equal and seek a design for
different amplitudes. The required excitation distribution for all the
simplifying cases, and for the general one (see the design examples), can be
readily implemented with proper tune of control parameters, such as the
varactor capacitance and the PIN conductance of the oscillators. The relative
closed-form formulas are presented for the case of the mutually synchronized
coupled oscillator array.The paper is organized as follows. In Section2, the coupled oscillator array (COA) theory is reviewed, and the key aspects
related to the present study are underlined. The system that contains both, the
antenna and the oscillators, is presented in Section 3, where closed-form
formulas are derived to design the mutually synchronized oscillator array. The
formulation and the solution of the radiation pattern shaping optimization
problem for the above active antenna array are given in Section 4. In Section 5, design
examples that verify the effectiveness of the method follow. Finally, some
concluding remarks are given in Section 6.


## 2. The Coupled Oscillator Array

### 2.1. Single Oscillator Description

A model that has been widely used for microwave oscillator
circuits is the canonical oscillator model in a parallel current-sum form which
leads to the Van der Pol equation for certain nonlinearities [14, 15]. The equivalent circuit of the oscillator
can be extracted from measurements [16] or from simulation [17], and is given
in Figure 1.Figure 1
The equivalent circuit of the oscillator in the current-sum form.The oscillator is assembled by three distinct parts: the active device,
the tuning tank, and the resonator tank. The active device provides the nonlinear
voltage dependent negative conductance. The tuning tank is responsible for the
adjustment of the tuning parameters in order to feed the attached antenna
element with the proper excitation. Last, the resonator tank includes the
parallel resonance circuit of the oscillator and the antenna elements.In detail, the active device is represented as a negative conductance as follows:(2)GD(|V|)=−a+b|V|2,where |V| is the amplitude of the output voltage, and a,b are constants characterizing the
nonlinear device.The voltage-dependent susceptance of the nonlinear device is
denoted byBD(|V|). Usually, the susceptance is independent on the device
characteristics and can be neglected. In this case, which is the one that we
follow, the current of the nonlinear device obeys (3)iD(t)=Y⋅V(t)=GD⋅V(t)=(−a+b⋅V2(t))⋅V(t).The reactive energy of the circuit is stored into theLC parallel circuit giving a
free-running circular frequency of  (4)ωo,k=1LrCo,k=1Lr(Cr+Co,t)for the kth oscillator.In our case, the capacitanceCt would be a varactor that is controlled by a voltage. This voltage will be the
tuning parameter which for the kth oscillator is Vkt. The varactor capacitance follows the well-known equation
[18]:  (5)Ct,k=2Co,t(1+Vkt/Vo)Γ, where Vo=1.2V (for a GaAs device), Co,t=1pF is the capacitance for Vo tuning voltage, and Γ is the varactor exponent (usually taken
equal to unity, a choice also followed here). Alternatively, other types of
voltage controlled capacitors [19, 20] or even
voltage controlled inductors [21] can be used instead.The conductanceGt represents additional losses in the oscillator circuit. This would be the second
(together with the Ct,k(Vkt)) tuning parameter. In our case, the conductance Gt would be a PIN diode
variable resistor controlled by its forward current Ikt. The PIN conductance follows the well-known equation [22, 23]: (6)Gt,k=voIkt.  Here,
we use vo=20mV. Alternatively, other types of variable resistors can be
used instead [24].To find the total conductance, one should add toGt the input conductance GA of the radiating element
of the array.Let us now compute the total admittanceY of the circuit:  (7)i(t)=(YD︸=−GD+YC︸=j(Ct+Cr)ω+YL︸=1/jLrω+YGt+YGA)V(t)⇒Y=−(GD(|V|)−Gt−GA)+j(Ct+Cr)ω2−ωo2ω.We assume that the resonant
circuit of the oscillator is of high Q. Thus, the output voltage follows,
nearly, a sinusoidal waveform with the following phasor representation:  (8)V(t)=V0(t)cos(ωt+ϕ(t))=Re{V0(t)ejϕ(t)ejωt}=Re{V˜(t)⋅ejωt}.


### 2.2. Coupled Oscillator Array Description

To complete the analysis, we suppose (see Figure2) that we have N
oscillators which are joined together using a linear coupling network [25].Figure 2
The coupled oscillator array.The governing equation reads(9)Yi⋅V˜i+∑j=1NYij⋅V˜j=0, where Yi is given by (7), and V˜i is defined in (8) as the complex output voltage
of the ith oscillator. Also, (10)Yij=Gij+jBij  is the
mutual admittance between the output nodes of the ith and jth
oscillators. Using the above, it can be shown [16–27] for the voltage
amplitude Vk of the kth oscillator that (11)dVkdt+ωo,k2QkVk[1−(−ak+bkVk2)Gt,k+GA]=12Ck(∑l=1NVl[Glkcos(ϕl−ϕk)−Blksin(ϕl−ϕk)]),  where (12)Qk=ωo,kCk(Gt,k+GA),  with Ck=Ct,k+Cr.Often the following substitution
is made:(13)1−(−ak+bkVk2)Gt,k+GA≈μk(ao,k2−|Vk|2),ao,k=ak+Gt,k+GAbk,μk=bkGt,k+GA,  where μk is a parameter
that characterizes the oscillator, while αo is the free-running
amplitude of the output voltage. This
weak quadratic nonlinearity is the simplest nonlinearity that approximates the
behavior of a wide range of active devices.Apart from the
time evolution of the voltage amplitudes given in (11) for the phase, we have (14)dϕkdt=−ω+ωo,k+12VkCk×(∑l=1NVl[Glksin(ϕl−ϕk)+Blkcos(ϕl−ϕk)]).  Equation (11) in conjunction with
(14) fully describes the coupled oscillator array. The above should provide a
framework for the design of an antenna oscillator array.


## 2.1. Single Oscillator Description

A model that has been widely used for microwave oscillator
circuits is the canonical oscillator model in a parallel current-sum form which
leads to the Van der Pol equation for certain nonlinearities [14, 15]. The equivalent circuit of the oscillator
can be extracted from measurements [16] or from simulation [17], and is given
in Figure 1.Figure 1
The equivalent circuit of the oscillator in the current-sum form.The oscillator is assembled by three distinct parts: the active device,
the tuning tank, and the resonator tank. The active device provides the nonlinear
voltage dependent negative conductance. The tuning tank is responsible for the
adjustment of the tuning parameters in order to feed the attached antenna
element with the proper excitation. Last, the resonator tank includes the
parallel resonance circuit of the oscillator and the antenna elements.In detail, the active device is represented as a negative conductance as follows:(2)GD(|V|)=−a+b|V|2,where |V| is the amplitude of the output voltage, and a,b are constants characterizing the
nonlinear device.The voltage-dependent susceptance of the nonlinear device is
denoted byBD(|V|). Usually, the susceptance is independent on the device
characteristics and can be neglected. In this case, which is the one that we
follow, the current of the nonlinear device obeys (3)iD(t)=Y⋅V(t)=GD⋅V(t)=(−a+b⋅V2(t))⋅V(t).The reactive energy of the circuit is stored into theLC parallel circuit giving a
free-running circular frequency of  (4)ωo,k=1LrCo,k=1Lr(Cr+Co,t)for the kth oscillator.In our case, the capacitanceCt would be a varactor that is controlled by a voltage. This voltage will be the
tuning parameter which for the kth oscillator is Vkt. The varactor capacitance follows the well-known equation
[18]:  (5)Ct,k=2Co,t(1+Vkt/Vo)Γ, where Vo=1.2V (for a GaAs device), Co,t=1pF is the capacitance for Vo tuning voltage, and Γ is the varactor exponent (usually taken
equal to unity, a choice also followed here). Alternatively, other types of
voltage controlled capacitors [19, 20] or even
voltage controlled inductors [21] can be used instead.The conductanceGt represents additional losses in the oscillator circuit. This would be the second
(together with the Ct,k(Vkt)) tuning parameter. In our case, the conductance Gt would be a PIN diode
variable resistor controlled by its forward current Ikt. The PIN conductance follows the well-known equation [22, 23]: (6)Gt,k=voIkt.  Here,
we use vo=20mV. Alternatively, other types of variable resistors can be
used instead [24].To find the total conductance, one should add toGt the input conductance GA of the radiating element
of the array.Let us now compute the total admittanceY of the circuit:  (7)i(t)=(YD︸=−GD+YC︸=j(Ct+Cr)ω+YL︸=1/jLrω+YGt+YGA)V(t)⇒Y=−(GD(|V|)−Gt−GA)+j(Ct+Cr)ω2−ωo2ω.We assume that the resonant
circuit of the oscillator is of high Q. Thus, the output voltage follows,
nearly, a sinusoidal waveform with the following phasor representation:  (8)V(t)=V0(t)cos(ωt+ϕ(t))=Re{V0(t)ejϕ(t)ejωt}=Re{V˜(t)⋅ejωt}.


## 2.2. Coupled Oscillator Array Description

To complete the analysis, we suppose (see Figure2) that we have N
oscillators which are joined together using a linear coupling network [25].Figure 2
The coupled oscillator array.The governing equation reads(9)Yi⋅V˜i+∑j=1NYij⋅V˜j=0, where Yi is given by (7), and V˜i is defined in (8) as the complex output voltage
of the ith oscillator. Also, (10)Yij=Gij+jBij  is the
mutual admittance between the output nodes of the ith and jth
oscillators. Using the above, it can be shown [16–27] for the voltage
amplitude Vk of the kth oscillator that (11)dVkdt+ωo,k2QkVk[1−(−ak+bkVk2)Gt,k+GA]=12Ck(∑l=1NVl[Glkcos(ϕl−ϕk)−Blksin(ϕl−ϕk)]),  where (12)Qk=ωo,kCk(Gt,k+GA),  with Ck=Ct,k+Cr.Often the following substitution
is made:(13)1−(−ak+bkVk2)Gt,k+GA≈μk(ao,k2−|Vk|2),ao,k=ak+Gt,k+GAbk,μk=bkGt,k+GA,  where μk is a parameter
that characterizes the oscillator, while αo is the free-running
amplitude of the output voltage. This
weak quadratic nonlinearity is the simplest nonlinearity that approximates the
behavior of a wide range of active devices.Apart from the
time evolution of the voltage amplitudes given in (11) for the phase, we have (14)dϕkdt=−ω+ωo,k+12VkCk×(∑l=1NVl[Glksin(ϕl−ϕk)+Blkcos(ϕl−ϕk)]).  Equation (11) in conjunction with
(14) fully describes the coupled oscillator array. The above should provide a
framework for the design of an antenna oscillator array.


## 3. Design of the Mutually Synchronized Array

In Figure3 [28], an antenna array with the corresponding oscillators
is presented. In this mutually synchronized array, each oscillator is
bilaterally coupled to the neighboring array elements. The configuration of Figure
3 was first
proposed by Stephan [8, 15] and named as “interinjection-locked” oscillators.
Mutual coherence is achieved via the injection-locking process. The steady-state
phase relationships of each oscillator depend on its neighbors’ amplitude and
phase. A nonlinear system of simultaneous equations must be solved to determine
the self-consistent amplitude and phase relationships [26]. In such systems,
(11) and (14) take the form (15)dVkdt=Vk2Ck[Gt,k+GA+ak−bkVk2]+12Ck∑l=1N[Vlκlk⋅cos(ϕl−ϕk+Φlk)]dϕkdt=−ω+ωo,k+12Ck∑l=1N[VlVkκlk⋅sin(ϕl−ϕk+Φlk)],  where κlk are different from zero only when |l−k|≤1 since only the nearest neighbor coupling is
assumed. There are various sets of parameters in (15). Those are the oscillator
detuning, (ω−ωo,k), the un-normalized coupling strengths, (κlk=|Glk+jBlk|), and the coupling phases, (Φlk=∠(Glk+jBlk)). There are also the capacitance and the conductance of each one of the
oscillators. Since our aim is to manipulate the amplitude and the phase
distribution of the active antenna array, any set of parameters could be used
for control. Here, we eliminate the coupling phases and we use the conductance
of the PIN diodes and the capacitance of the varactors to control the steady
state of the amplitudes and phases.Figure 3
Bilateral injection-locking or mutual
synchronization coupled oscillator’s array.Before we
present the followed procedure, some issues relative to the coupling network
should be addressed. To couple the oscillators, we use resistively loaded lines [29], whose characteristic impedance Z0 and phase Φ of the electrical
length are (16)Zo=R,Φ=2π.  For such lines, it is (17)Yij={ni2R,i=j,−12R,|i−j|=1,0,otherwise,  where ni=2 except for the edge
elements where it is equal to one.Using this
coupling network, closed-form expressions for the tuning parameters can be
derived. So, forYij given
in (17), (15),
for the steady state, where the time derivatives vanish, reads (18)Vk[Gt,k+GA+ak−bkVk2]=−12R{V1−V2cos(ϕ2−ϕ1),k=1,−Vk−1cos(ϕk−1−ϕk)+2Vk−Vk+1cos(ϕk+1−ϕk),k≠k,N,VN−VN−1cos(ϕN−1−ϕN),k=N,(ω−1LrCk)4RCk={sin(ϕ2−ϕ1)V2,V1,i=1,+sin(ϕk−1−ϕk)Vk−1Vk−sin(ϕk+1−ϕk)Vk+1,Vk,k≠k,N,−sin(ϕN−1−ϕN)VN−1VN,i=N.  Using (19)Sk={1−cos(ϕ2−ϕ1)(V2V1),k=1,−cos(ϕk−1−ϕk)(Vk−1Vk)+2−cos(ϕk+1−ϕk)(Vk+1Vk),k≠1,N,1−cos(ϕN−1−ϕN)(VN−1VN),k=N,Uk={sin(ϕ2−ϕ1)(V2V1),i=1,+sin(ϕk−1−ϕk)(Vk−1Vk)−sin(ϕk+1−ϕk)(Vk+1Vk),k≠k,N,−sin(ϕN−1−ϕN)(VN−1VN),i=N,  the solution for the tuning
parameters, Ct,k(Vkt) and Gt,k(Ikt),
in terms of Vk,ϕk is (20)Ck=Co,k(12[1±1+UkRLrCo,k])2,Ct,k=Ck−Cr,Vkt=Vo[(Co,k(Ck−Cr))1/Γ−1],Gt,k=−Sk2R−ak+bkVk2−GA,Ikt=voGt,k.Before closing the section, the design procedure
and check of the stability of the solution for the coupled oscillator array are
listed. So, to design the mutually synchronized oscillator array in order to
produce a given output voltage distribution (amplitudes and phases), the
following steps are implemented.(i)
The given output amplitudes and phases,Vk,ϕk,
are inserted into (19) to produce the intermediate parameters Sk and Uk.(ii)
In turn,Sk and Uk are inserted into (20). This results
in computing the tuning parameters Ct,k(Vkt) and Gt,k(Ikt).(iii)
InsertingGt,k and Ct,k into the differential equations (15) provides
the evolution of the amplitudes and phases with time.The key step of
computing the properVk,ϕk is addressed next.


## 4. The Synthesis Method

The system of
Figure3 with the restriction of nonvarying interelement distance, (Δz=zk−zk−1=const), has been used to control linear phased arrays in the
limited cases of beam steering and pattern null in certain directions [8–13]. In the
current study, we attempt a more general (not limited in certain cases) pattern
synthesis. The system is designed by providing the proper values for the
positions of the radiators and for the voltage outputs of the oscillators. It
is obvious that the proposed design offers specific advantages due to the
degrees of freedom that make
use.To start the design of the array, we derive the cost functionC of the following form:  (21)C=∥Fd−F∥2,where  (22)∥Fd−F∥2=∫0π|Fd(θ)−F(θ)|2sin(θ)dθ.We define  (23)A∘B=∫θ=0πA(θ)⋅B*(θ)⋅sin(θ)dθ.From (23), (22) becomes  (24)∥Fd−F∥2=(Fd−F)∘(Fd−F),where Fd(θ) is the desired pattern, F(θ) is the resulting
pattern, and ∥Fd−F∥2 is the mean square
error. By taking into account the output voltage of the coupled oscillators, the
resulting pattern can be expressed as  (25)F(θ)=∑k=1N[Vkejϕk]︸Fk,where Fk=fk(θ)ej(2π/λ)zkcos(θ).We suppose a
perturbation of the excitation of the oscillator outputs and of the position of the elements of
the array. Due to the perturbation of the excitation, the new output of thekth
oscillator becomes Vk+ΔVk;ϕk+Δϕk. Also the new position of the kth element is zk+Δzk. The pattern of the array results in  (26)F(θ)=F0(θ)+δF(θ)≈∑k=1NFkVkejϕk+∑k=1NFkejϕk⋅ΔVk+j∑k=1NFkVkejϕk⋅Δϕk+∑k=1N∂Fk∂zk⋅Vkejϕk⋅Δzk,where F0(θ) is the initial pattern of the array.It is shown in the Appendix that
after assembling the perturbed values of the output voltage, (ΔVk,Δϕk), of the oscillators and the position, (Δzk), of the elements in an unknown vector [Δ],
the solution of the optimization problem leads to  (27)[Δ]=[B]−1[A],where [A] and [B] are proper matrices
that include information relative to the specific problem. Expression (27)
gives the appropriate perturbed values of the output voltage of the oscillators
and the position of the elements. It is supposed that the desired pattern is
not much different than the resulting one. In this case, the perturbation will
produce the necessary changes in the excitation and the position of the
elements. If∥Fd−F∥2 is large, then the
perturbation technique is applied iteratively until the mean square error
becomes less than a given threshold. For further details on the functionality
and the problems relative to the convergence of the proposed iterative method,
the reader is referred to [5].


## 5. Design Examples

In the design
examples that follow, the array can be derived by varying the excitation
(amplitude and phase) simultaneously with the element positions. The desired far-field
pattern is a Chebyshev one with HPBW = 15
degrees and SLL = −25 dB. We start from
an array of 9 elements with uniform initial excitation and0.45λ interelement distance. We apply the
synthesis method for three different cases. In case A, the excitations and the positions are to be derived. In case B, the phase of the excitations is
set equal to zero, and in case C, the amplitude of the excitation is equal to
unit.For each one of
the cases A, B, C, three different oscillators are employed to produce the
respective far-field. The Van der Pol parameters of those oscillators are taken
from published data (see Table1).Table 1
Van
der Pol parameters of the employed oscillators.
Osc. #I (~1 GHz)II (~4.4 GHz)III (~10 GHz)Reference[18][26][15]a20.96570.02320.0036b34.86690.03820.01Lr0.214 pH0.24 nH10 pHCr105 nF5.45 pF23 pFRA=1/GA50Ω58Ω32ΩThe parameters of the coupling network are the same
for all the examples as
follows:(28)Zo=R=270Ω.


### 5.1. General Design Example

In this first example, both the element placement
and the excitation are allowed to vary. After 20 iterations, the designed
antenna array produces a pattern that is given in Figure4.Figure 4
The attained pattern for the general design
case.The respective element placement and the excitations
are given in Table2.Table 2
Details on
the general design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.780.5340.0342−1.170.842−0.8063−0.730.628−0.5154−0.3610.516500.6240.72460.3610.51670.730.628−0.51581.170.842−0.80691.780.5340.034The tuning
parameters,Vkt and Ikt,
of the oscillators are computed after inserting the last two columns of Table
2 into (19)-(20). For the
present case, the relative results are presented in Table 3.Table 3
The tuning parameters for the general design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91231.203141207.790.4401203.150.2002, 81182.6814021195.612.2161198.221.4553, 71171.235931192.680.9201197.030.5474, 61212.1021481203.043.4601201.232.34251215.635811203.920.9101201.590.542The evolutions ofVk(t) and ϕk(t) versus t are produced after inserting into (15):(1)
the parameters of Table1 and (28),(2)
the tuning parameter values,Ct,k(Vkt) and Gt,k(Ikt), and(3)
the initial, att=0,
amplitudes and phases:
Vk(t=0) = random number uniformly distributed into 10−2+(0,1)⋅10−3V,
ϕk(t=0)  = random number uniformly distributed into (0,2π)⋅10−3.  In Figure 5, it is demonstrated that the
designed coupled oscillator array can produce the necessary solution (see Table
2 and Figure 4) and furthermore, this solution is stable. In Figure 5(a), the
voltage amplitudes of the nine oscillators versus time are given. Also, in Figure
5(b), the time evolution of the successive phase differences (output voltage
phase of the kth oscillator minus the output oscillator phase of the (k−1)th
oscillator) are given. It is clear that the amplitudes and phases converge to
the correct values (continuous horizontal lines).The coupled oscillator array response evolution with time for the general
design case. (a) The evolution of the amplitudes. (b) The evolution of the
phases.
(a)(b)


### 5.2. Placement and Amplitude Design Example-Constant Phases

From Table2, it is shown that the phase of the
elements is close to zero. Also, the final array is nonuniform in the
excitation and the geometry. Namely, our solution is more complicated than the
classical one for Chebyshev patterns [1, 2]. Since we try
to have a technically simple feeding network, we will solve the same problem
with no phase difference between the elements and with only three different
excitations. In Table 4, the final placement and the excitation of the
elements, after 20 iterations, are given.Table 4
Details on
the constant phase design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.7790.402−1.3860.403−0.9610.804−0.46410501060.4641070.9610.8081.3860.4091.7790.40Also in Figure6, the pattern of the array is presented. In the pattern,
a difference of less than 0.5 dB between the desired and the resulting one is
observed.Figure 6
The attained pattern for the constant phase design
case.Comparing Table2 with Table 4, one can see that
the second solution gives simpler results.For the present case, the tuning
parameters are given in Table5.Table 5
The tuning parameters for the constant phase design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91200314.112000.44512000.2032, 81200313.412000.48312000.2403, 71200251512004.19612002.8924, 61200416612007.01912004.91851200416612007.02812004.924In Figure7, the time evolution curves for the amplitudes show
the stability of the solution.Figure 7
The coupled oscillator array response evolution with time for the constant
phase design case. The evolution of the amplitudes.Note that comparing Figure7 with Figure 5(a), different
converging times are observed. In general, this fact is to be expected since
the converging times, apart from the initial conditions Vk(t=0),ϕk(t=0),
depend on the tuning parameter values that are different for the two design
examples.


### 5.3. Placement and Phase Design Example-Constant Amplitudes

In this last case, we provide a design where the amplitudes are
equal. The attained solution is given in Table6.Table 6
Details on
the constant amplitude design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.591−0.0432−0.981−4.1213−0.701+4.9094−0.261−6.975501−5.93660.261−6.97570.701+4.90980.981−4.12191.591−0.043The produced far-field pattern is given in Figure8.Figure 8
The attained pattern for the constant amplitude design case.It is noted that in this case, the SLL criterion
of −25 dB is not fulfilled, (SLL = −21.5 dB). The above shows that the phases of
the elements are not enough to solve the problem.For the present case, the tuning
parameters are given in Table7.Table 7
The tuning
parameters for the constant amplitude design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91298.56314.01224.130.4011209.730.2032, 8930.27314.01125.810.4011169.320.2033, 71805.51314.01328.560.4001250.490.2024, 6934.48314.01127.070.4001169.850.20351249.32314.01212.250.4011204.950.203For the resulting pattern, the respective evolution curves are
given in Figure9.Figure 9
The coupled oscillator array response evolution with time for the constant
amplitude design case. The evolution of the phases.It is obvious that in the array synthesis, one
must tradeoff the simplicity of networks, the size of the array, and the number
and type of the different excitations. From the presented cases, it is assessed
that the general one is rather complex, while the constant amplitude case fails
to comply with the desired pattern. It seems that the constant phase case
presents a better compromise between complexity and performance of the device
under study.Before closing the section, to evaluate the
validity of the presented method and the viability of the listed results, some
comments are in order. Both phase and amplitude control should be implemented
with a relatively easy way. Varactor and PIN diodes are simple electronic elements
that should be used. That is because varactor diodes are extensively used as
tuning elements in conventional VCOs, and also PIN diodes are employed as
variable resistors for vector modulator circuits. The issue that raises concern
is the imprecise setting of the values of the tuning parameters and the effect
that manufacturing tolerances will have on the overall performance of the
system. Fabrication and tuning inconsistencies among individual oscillators in
an array lead to differences in the free-running frequencies and amplitudes.
This randomness causes phase and amplitude errors that are expected to result
in pattern deformation manifested in beam-pointing and SLL errors [1, 30].To provide understanding of the relationships
among random tuning error distribution and pattern deformation, a statistical
study, focused on the sensitivity analysis, is provided through Monte Carlo
simulation. The element placement, the tuning voltage, and the tuning current
are treated as random vectors following normal probability distribution with
means given by the previous analysis and varying values of variance. In each
statistical experiment, after choosing the placement and tuning parameters,Ct,k(Vkt) and Gt,k(Ikt) are used to extract the time evolution of Vk(t) and ϕk(t).
At the steady state, the far field of the array is computed. The results show
that when only Ikt is varied and if two or three dBs degradation
on SLL performance are
acceptable, then the variance should be no more than 10% of the parameters
nominal value. This is shown in Figure 10 when all the patterns resulted from a
100-run relative Monte Carlo experiment are depicted. Nearly, for one and two orders of magnitude, less variation is needed for
attaining the same results when position and phase are varied, respectively.
This means that position and amplitudes setting problems are expected to be
less harsh compared to the problems that have already encountered in setting
the phase [31], in conventional VCOs. If the previously-stated performance
degradation is unacceptable, properly chosen calibration procedure should be
considered [32, 33].Figure 10
The attained amplitude sensitivity pattern
for the general design case.


## 5.1. General Design Example

In this first example, both the element placement
and the excitation are allowed to vary. After 20 iterations, the designed
antenna array produces a pattern that is given in Figure4.Figure 4
The attained pattern for the general design
case.The respective element placement and the excitations
are given in Table2.Table 2
Details on
the general design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.780.5340.0342−1.170.842−0.8063−0.730.628−0.5154−0.3610.516500.6240.72460.3610.51670.730.628−0.51581.170.842−0.80691.780.5340.034The tuning
parameters,Vkt and Ikt,
of the oscillators are computed after inserting the last two columns of Table
2 into (19)-(20). For the
present case, the relative results are presented in Table 3.Table 3
The tuning parameters for the general design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91231.203141207.790.4401203.150.2002, 81182.6814021195.612.2161198.221.4553, 71171.235931192.680.9201197.030.5474, 61212.1021481203.043.4601201.232.34251215.635811203.920.9101201.590.542The evolutions ofVk(t) and ϕk(t) versus t are produced after inserting into (15):(1)
the parameters of Table1 and (28),(2)
the tuning parameter values,Ct,k(Vkt) and Gt,k(Ikt), and(3)
the initial, att=0,
amplitudes and phases:
Vk(t=0) = random number uniformly distributed into 10−2+(0,1)⋅10−3V,
ϕk(t=0)  = random number uniformly distributed into (0,2π)⋅10−3.  In Figure 5, it is demonstrated that the
designed coupled oscillator array can produce the necessary solution (see Table
2 and Figure 4) and furthermore, this solution is stable. In Figure 5(a), the
voltage amplitudes of the nine oscillators versus time are given. Also, in Figure
5(b), the time evolution of the successive phase differences (output voltage
phase of the kth oscillator minus the output oscillator phase of the (k−1)th
oscillator) are given. It is clear that the amplitudes and phases converge to
the correct values (continuous horizontal lines).The coupled oscillator array response evolution with time for the general
design case. (a) The evolution of the amplitudes. (b) The evolution of the
phases.
(a)(b)


## 5.2. Placement and Amplitude Design Example-Constant Phases

From Table2, it is shown that the phase of the
elements is close to zero. Also, the final array is nonuniform in the
excitation and the geometry. Namely, our solution is more complicated than the
classical one for Chebyshev patterns [1, 2]. Since we try
to have a technically simple feeding network, we will solve the same problem
with no phase difference between the elements and with only three different
excitations. In Table 4, the final placement and the excitation of the
elements, after 20 iterations, are given.Table 4
Details on
the constant phase design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.7790.402−1.3860.403−0.9610.804−0.46410501060.4641070.9610.8081.3860.4091.7790.40Also in Figure6, the pattern of the array is presented. In the pattern,
a difference of less than 0.5 dB between the desired and the resulting one is
observed.Figure 6
The attained pattern for the constant phase design
case.Comparing Table2 with Table 4, one can see that
the second solution gives simpler results.For the present case, the tuning
parameters are given in Table5.Table 5
The tuning parameters for the constant phase design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91200314.112000.44512000.2032, 81200313.412000.48312000.2403, 71200251512004.19612002.8924, 61200416612007.01912004.91851200416612007.02812004.924In Figure7, the time evolution curves for the amplitudes show
the stability of the solution.Figure 7
The coupled oscillator array response evolution with time for the constant
phase design case. The evolution of the amplitudes.Note that comparing Figure7 with Figure 5(a), different
converging times are observed. In general, this fact is to be expected since
the converging times, apart from the initial conditions Vk(t=0),ϕk(t=0),
depend on the tuning parameter values that are different for the two design
examples.


## 5.3. Placement and Phase Design Example-Constant Amplitudes

In this last case, we provide a design where the amplitudes are
equal. The attained solution is given in Table6.Table 6
Details on
the constant amplitude design case.
Element numberElement position (λ): zkRelative excitation amplitude:VkExcitation phase (degrees):ϕk1−1.591−0.0432−0.981−4.1213−0.701+4.9094−0.261−6.975501−5.93660.261−6.97570.701+4.90980.981−4.12191.591−0.043The produced far-field pattern is given in Figure8.Figure 8
The attained pattern for the constant amplitude design case.It is noted that in this case, the SLL criterion
of −25 dB is not fulfilled, (SLL = −21.5 dB). The above shows that the phases of
the elements are not enough to solve the problem.For the present case, the tuning
parameters are given in Table7.Table 7
The tuning
parameters for the constant amplitude design case.
Element numberOSC. I (1.06 GHz)OSC. II (4.21 GHz)OSC. III (10.38 GHz)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)Vkt(mV)Ikt(mA)1, 91298.56314.01224.130.4011209.730.2032, 8930.27314.01125.810.4011169.320.2033, 71805.51314.01328.560.4001250.490.2024, 6934.48314.01127.070.4001169.850.20351249.32314.01212.250.4011204.950.203For the resulting pattern, the respective evolution curves are
given in Figure9.Figure 9
The coupled oscillator array response evolution with time for the constant
amplitude design case. The evolution of the phases.It is obvious that in the array synthesis, one
must tradeoff the simplicity of networks, the size of the array, and the number
and type of the different excitations. From the presented cases, it is assessed
that the general one is rather complex, while the constant amplitude case fails
to comply with the desired pattern. It seems that the constant phase case
presents a better compromise between complexity and performance of the device
under study.Before closing the section, to evaluate the
validity of the presented method and the viability of the listed results, some
comments are in order. Both phase and amplitude control should be implemented
with a relatively easy way. Varactor and PIN diodes are simple electronic elements
that should be used. That is because varactor diodes are extensively used as
tuning elements in conventional VCOs, and also PIN diodes are employed as
variable resistors for vector modulator circuits. The issue that raises concern
is the imprecise setting of the values of the tuning parameters and the effect
that manufacturing tolerances will have on the overall performance of the
system. Fabrication and tuning inconsistencies among individual oscillators in
an array lead to differences in the free-running frequencies and amplitudes.
This randomness causes phase and amplitude errors that are expected to result
in pattern deformation manifested in beam-pointing and SLL errors [1, 30].To provide understanding of the relationships
among random tuning error distribution and pattern deformation, a statistical
study, focused on the sensitivity analysis, is provided through Monte Carlo
simulation. The element placement, the tuning voltage, and the tuning current
are treated as random vectors following normal probability distribution with
means given by the previous analysis and varying values of variance. In each
statistical experiment, after choosing the placement and tuning parameters,Ct,k(Vkt) and Gt,k(Ikt) are used to extract the time evolution of Vk(t) and ϕk(t).
At the steady state, the far field of the array is computed. The results show
that when only Ikt is varied and if two or three dBs degradation
on SLL performance are
acceptable, then the variance should be no more than 10% of the parameters
nominal value. This is shown in Figure 10 when all the patterns resulted from a
100-run relative Monte Carlo experiment are depicted. Nearly, for one and two orders of magnitude, less variation is needed for
attaining the same results when position and phase are varied, respectively.
This means that position and amplitudes setting problems are expected to be
less harsh compared to the problems that have already encountered in setting
the phase [31], in conventional VCOs. If the previously-stated performance
degradation is unacceptable, properly chosen calibration procedure should be
considered [32, 33].Figure 10
The attained amplitude sensitivity pattern
for the general design case.


## 6. Conclusion

A
design procedure for the synthesis of active antenna arrays is presented. Those
arrays are sets of radiator oscillator couples with each element connecting
through a coupling network to the others. Such a system presents the designer
with two sets of parameters: the radiators’ positions and the oscillators’
outputs, the second used to excite the first.The
procedure designs simultaneously both the antenna and the attached coupled
oscillator array providing viable solutions. In this way, the current work
demonstrates how to control the dynamics of coupled oscillator arrays in order
to achieve pattern shaping. Whereas previous work has been almost exclusively
devoted to beam steering and null formation, this technique extends the
versatility of coupled oscillator arrays through manipulation of the aperture
amplitude and phase distribution. Furthermore, simultaneously an amplitude
space taper distribution, (Δz≠0), is attained to produce the
desired far-field pattern.The implementability
of our method is based first on the actually monitoring of the time evolution
of the oscillators’ outputs (27).
Afterward, the existence of the proper varactors and PINs by using (20) is addressed.
It is obvious that the limited range of their values should set some
constraints on the resultant pattern. So, in some cases, a proper iterative
combination between the capacitance and conductance and the desired pattern is
important.For all design
cases, and independently on the complexity of the specific problem, the
required excitation distribution can be readily implemented with proper tune of
control parameters, such as the varactor capacitance and the PIN conductance of
the oscillators. The relative closed-form formulas are presented for the case
of the mutually synchronized coupled oscillator array.



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*Source: 208706-2009-04-23.xml*