An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents

International Journal of Electronics and Communications

© Urban & Fischer Verlag http://www.urbanfischer.de/journals/aeue

An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents B. Sıddık Yarman, Ahmet Aksen and Ali Kılınç Dedicated to Professor Alfred Fettweis on the occasion of his 75th birthday. Abstract An immitance-based method is presented to model measured or computed data, obtained from a “passive one-port physical device” by means of its Darlington equivalent. In other words, the given data is modelled as a lossless two port terminated in a unit resistor. The basis of the new modelling tool rests on the numerical decomposition of the given immitance data into its Foster and minimum parts. Therefore, the proposed technique does not require any choice for the circuit topology to build the model. Rather, the optimum circuit topology that characterises the given data is the natural consequence of the modelling process proposed in this paper. A main algorithm is presented to construct the model from the given data. It is expected that the proposed modelling tool will find practical applications in the behaviour characterisation, simulation, and design of high speed/high frequency analog/digital mobile communication sub-systems manufactured on VLSI chips. An antenna-modelling example is included to systematically exhibit the implementation of the modelling technique. Keywords Passive Networks, Immitance Modelling

1. Introduction For many communications engineering applications, circuit models for measured data obtained from physical devices or sub-systems are inevitable. In practice, the class of problems, which demand circuit models for measured data, may be categorised as follows. Problems of Type-I “Characterisation”: The characterisation or assessment of the electrical behaviour of physical devices utilised in communication systems such as noise figure level, maximum power transfer capability etc. For this category, a model for the device or the system is essential. Problems of Type-II “Design”: The design of an analog/digital communication system such as antenna matching networks, microwave amplifiers for mobile or wireless communication systems etc. For these problems, depending on the design method, linear or non-linear circuit models for active and passive devices or sub-systems may be required. Received June 20, 2001. B. Sıddık Yarman, I¸sık University, 80670, Maslak ˙Istanbul, Turkey. A. Aksen, Department of Electronics Engineering, I¸sık University, 80670, Maslak, ˙Istanbul, Turkey. A. Kılınç, Nortel Networks – Neta¸s, ˙Istanbul, Turkey. Correspondenc to A. Aksen. E-mail: [email protected] ¨ 55 (2001) No. 6, 443−451 Int. J. Electron. Commun. (AEU)

Problems of Type-III “Simulation”: Fast simulation of high-speed, high-frequency analog/digital communication sub-systems to be manufactured on VLSI chips. In this category, commercially available simulation packages are employed. Depending on the complexity of the integrated circuits, simulation time may take several days. Therefore, in these type of problems, it is usually preferred to come up with circuit models for interconnects and active and passive devices to speed up the numerical computations. It will be proper to clearly identify the above categories by examples. In designing high-speed/high-frequency communication systems, one of the major issues is to determine the physical limitations of the commercially available devices for power transfer. The passive one-port device is regarded as a dissipative complex termination. This termination may be described by measured either immitance or reflectance data over the frequencies of interest. Precise theoretical power transfer limitations of a physical device may be determined by accessing the Analytic Gain Bandwidth Theory [1]. In this case, circuit models for the passive terminations are essential. This is a typical Type-I problem. There may be some other valid reasons to come up with circuit models for the given data obtained from physical devices. A typical example may be “the design of a single stage microwave amplifier which employs, per se, a GaAs FET as an active device”. In this problem, one has to determine the optimum source and the load impedances of the active device to optimise the performance of the amplifier. Performance optimisation may involve the maximisation of the flat transducer power gain while minimising the input voltage standing wave ratio V SWRmin and the noise figure over the prescribed frequency band. For this purpose, the optimum source and the load impedances are computed on the Smith Chart point by point as described in [2]. Eventually, one has to model the resulting source and the load impedances accordingly to end-up with the front-end and the back-end matching networks as shown in Fig. 1. Hence, the design of the microwave amplifier for optimum terminations is completed. Thus, a typical Type-II problem is described. Utilisations of circuit model tools are also very important to run fast simulation packages such as Spice, Touch Stones, Lab-View or Super-Compact etc. Simulation of very large networks, those consisting of a large number of nodes over 10,000, is a serious problem in the computer-aided design of integrated circuits. Circuits of 1434-8411/01/55/6-443 $15.00/0

444 B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents ents, employing realisable, two variable, driving point network functions [13, 14].

2. An immitance based tool to model the given data

Fig. 1. Amplifier design problem. (a) Optimum source and load impedances for an active device, (b) Darlington representation of the source and load impedances.

this size can typically require several days of CPU time on a workstation for an elegant simulation. However, in many applications, high accuracy in simulation is not required. For problems that require less precision, model order reduction methods have been found suitable. Several researchers have introduced algorithms and numerical techniques to approximate network transfer functions. These techniques include the waveform relaxation method [3, 4], the asymptotic waveform evaluation [5, 6] and the passive multi-point moment matching method [7, 8]. Lately, pre-assigned multi-port lumped circuit models gained importance in the simulation of high-density electronic packages [9–12]. In all these works, circuit models for interconnects and solid-state devices become essential to enhance the capability of the commercially available computer packages employed for analysis, design and simulation of Very Large Scale Integrated Circuits (a typical Type-III problem). Therefore, in this work, the problem is defined as “to model the given data as a lossless two port in resistive termination which is called the Darlington equivalent of the physical device” (Fig. 1b). This is the fundamental problem in designing, analysing and simulating the modern communications systems put on VLSI chips. Unfortunately, there is no well-established modelling technique available for measured passive immitance data in a Darlington sense in the current literature. The common exercise to model the given data starts with the choice of an appropriate circuit topology. Then, the element values of the chosen topology are determined to best fit the given data by means of an optimisation algorithm. Although this trial is straightforward, it presents serious difficulties. First, the optimisation is heavily non-linear in terms of the element values that may result in local minimas or may not converge at all. Secondly, there is no established process to initialise the element values of the chosen circuit topology. Worst of all “the optimum choice of the circuit topology which best describes the physical device is in question”. Fortunately, these problems are overcome employing the modelling technique introduced in this work. In the next sections, we present the proposed technique to model physical devices using lumped elements. However, the technique can easily be extended to handle models with distributed or with mixed, lumped and distributed elem-

In this approach, the measured data of the passive physical device is taken as a positive real (PR) immitance function F( jωi ) = R(ωi ) + jX(ωi ) over the frequencies ωi , where the subscript “i” designates the index of the test or sample frequencies. F( jω) could either be an impedance Z( jω) or an admittance Y( jω). In general, any PR-Rational immitance function F(s) can be written in terms of its minimum and the Foster part functions; F(s) = Fm (s) + Ff (s)

(1)

where s = σ + jω is the usual complex domain variable, Fm (s) is the minimum part of F(s), which is free of jω poles, and Ff (s) is the Foster part of F(s), which only includes jω poles. On the real frequency axis jω we have then, F( jω) = R(ω) + jX(ω), Fm ( jω) = Rm (ω) + jX m (ω), Ff ( jω) = jX f (ω).

(2a) (2b) (2c)

In the above representation, it is clear that R(ω) = Rm (ω), X(ω) = X m (ω) + X f (ω).

(3a) (3b)

Since Fm (s) is a PR minimum function which contains no poles on the jω axis, its imaginary part X m (ω) is related to the real part Rm (ω) by the Hilbert transformation relation; X m (ω) = H {R(ω)}

(4)

where H {.} designates the Hilbert Transformation Operation (HTO). That is, 1 X m (ω) = π


+∞ −∞

R(y) dy ω− y

(5)

The above form is also known as Bayard-Bode relation. In the immitance based modelling technique, the crux of the matter is to decompose the given data into its minimum part Fm ( jω) = Rm (ω) + jX m (ω) and Foster part Ff ( jω) = jX f (ω). Hence, the modelling process is carried out within two major steps: model for the minimum part and the Foster part (Figs. 2 and 3). In order to extract the Foster part Ff ( jω) = jX f (ω) from the original measured data, one has to generate X m (ω) using the Hilbert transformation relation of (5). Eventually, realisable analytical forms for the minimum

B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents 445

Step 3a: Find the realisable mathematical form for the given data R(ωi ) as simply as possible. It is well known that R(ω) must be a non-negative, even rational function. Therefore, on the jω axis, the general form of R(ω) may be selected as R(ω)= Fig. 2. Extraction of the Foster component Z f (s) from the impedance function Z(s) = Z m (s) + Z f (s).

Fig. 3. Extraction of the Foster component Y f (s) from the admittance function Y(s) = Ym (s) + Y f (s).

immitance function Fm ( jω) = Rm (ω) + jX m (ω) and the Foster function X f (ω) = X(ω) − X m (ω) are obtained by means of an appropriate curve fitting or interpolation algorithms and they are synthesised to yield the desired device model under consideration. The approach used in the immitance based datamodelling algorithm is summarised in the next section.

3. An immitance based tool to model the data obtained from a passive complex termination: the main algorithm The steps for the algorithm to model the given data are described in the following. Step 1: Measure the physical device characteristics or compute the optimum immitance data for the problem under consideration and prepare the driving point immitance table F( jωi ) = R(ωi ) + jX(ωi ) over the test frequencies ωi , i = 1, 2, 3, · · · , N. The integer N designates the total number of samples. Step 2a: Decide whether to work with impedance or admittance functions to model the measured or computed data. Step 2b: Employing any numerical integration technique, generate the imaginary part X m (ω) of Fm ( jω) = Rm (ω) + jX m (ω) by means of (5). In this step, X m will be computed from the given real part data R(ω) point by point. However, this step may be omitted since the Gewertz procedure described in Step 3b produces the minimum immitance function from the given rational-realisable form of the real part. Thus, on the jω axis, the imaginary part of the immitance function yields the desired data for X m .

N(ω) A0+A1 ω2+A2ω4+· · ·+Aq ω2q = D(ω) B0+B1 ω2+B2 ω4 + · · ·+Bn ω2n

(6)

where n designates the total number of elements in the minimum part of the circuit model and q is the total number of transmission zeros such that q ≤ n. In this representation, q and n are related to the complexity of the circuit model associated with the minimum part of the driving point immitance. In many practical cases, the minimum part of the circuit model may be chosen as a lossless ladder with transmission zeros at DC, infinity, and may be at finite frequencies or any combination of these. No matter where the transmission zeros are, R(ω) must always be non-negative. Therefore, coefficients Ai and Bi must be determined accordingly. Step 3b: Once the realisable mathematical form of R(ω) is determined, generate the PR minimum function Fm (s). In this step, Bode or Gewertz or any other rocedure can be employed [15]. In order to ease the synthesis, the Gewertz technique may be preferred over the others which yields the following PR rational form of minimum function Fm (s). Fm (s)=

n(s) a0+a1 s+a2 s2+· · ·+an−1 sn−1 = d(s) b0+b1s+b2 s2+· · ·+bn sn

(7)

In (7), it is presumed that as frequency approaches to zero, the real part data goes to zero (i.e R(∞) = R∞ = 0 and in Step 3a, the integer q of (6) is set to q = n − 1). If this is not the case, then a constant term R∞ will be added to (7), which in turn effects (6) by having q = n. Step 4: Using (3b), obtain the Foster part of F( jωi ) point by point as, X f (ωi ) = X(ωi ) − X m (ωi );

i = 1, 2, · · · , N

(8)

where X m (ω) is either generated from the imaginary part of the minimum function Fm (ω) such that X m (ω) = Im {Fm (ω)} or computed from (5). Step 5: Find the mathematical realisable Foster form for the data X f (ωi ) by means of an appropriate curve-fitting algorithm as k 0  kr ω + + k∞ω ω ω2 − ωr2 p

X f (ω) = −

(9)

r=1

where the integer “ p” is the total number of non-zero finite pole pairs of the Foster function X f (ω). It should be noted that in the above representation, the residues k0 , kr and k∞ must be all non-negative. Once the mathematical form of X f (ω) is obtained, it is straightforward to end up

446 B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents with the realisable form of Ff (s); k 0  kr s + + k∞ s s s2 + ωr2

N(ω) is given by N(ω) = ω2k

p

Ff (s) =

(10)

r=1

Step 6: Synthesise the immitance function Ff (s) = Fm (s) + F f (s) as a lossless two-port in unit termination which in turn yields the desired model of the physical device.

4. Remarks on the numerical behaviour of the main algorithm In order to ease the implementation of the algorithm described in section 3, one should pay attention to the following remarks. •

The algorithm described above could be on impedance or admittance base. The choice may be arbitrary. However, one should bear in mind that, the structural model would differ with the initial choice. If the impedance-base approach is chosen, the model will start with a series arm which includes all the jω poles of the Foster Impedance Z f in series with the minimum reactance driving point input impedance Z m (Fig. 2). If one starts with admittance data, then the model will include a shunt arm, which contains all the jω poles of the Foster Admittance Y f in parallel with the minimum susceptance driving point input admittance Ym (Fig. 3). Practical choices can be made according to the physical behaviour of the device. Certainly, numerical stability of the algorithm depends on the initial choice of the unknown coefficients Ai and Bi of (6). • Evaluation of the Hilbert Transform Integral of (5) may create some problems. A simple numerical approach may be to consider R(ω) as a linear combination of line segments which connect the measured points Ri = R(ωi ) on the real part curve. With this representation of Rm (ω), X m (ω) is also expressed in terms of the linear combination of the same, so called, break points Ri = R(ωi ) as described in [16]. • If preferred, step 2 of the main algorithm can be omitted. Instead, the rational form of the minimum immitance function, which is constructed at step 3b, is utilised to generate the imaginary part X m (ω). Then, the Foster portion of the immitance function is obtained. This choice depends on the numerical behaviour of the problem. One can always employ both methods to generate X m (ω) and, compare the resulting shapes of the computed data for the Foster part X f . Then, the best reasonable choice, which results in a minimum number of circuit elements, is selected. • In Step 3, one simply utilises a linear curve-fitting algorithm, which in turn yields the coefficients of Ai and Bi of (6). Depending on the given data, there may be several practical forms for (6). For many applications,

(11)

•

This form of N(ω) is called the “Modelling form-A”. In (11), k = 0 is chosen if the data for R(ω) starts at a reasonably large fixed value; k > 0 is chosen if R(ω) tends to go zero at the lower end of the frequency band of measurement. The value of “k” depends on the rolloff skirt of the measured data at the lower-end of the frequency band. • In order to assure the physical realisability, step 3 must yield a non-negative real part R(ω). Unfortunately, a linear curve-fitting algorithm may not necessarily result in a non-negative real part. Therefore, in the following section a practical technique is presented to model the given real part data by means of a non-linear curve fitting approach assuring the realisable solution for R(ω).

5. A nonlinear curve fitting approach to model the given real part data In this method, the numerator N(ω) and the denominator D(ω) polynomials of (6) are expressed in terms of the auxiliary polynomials PN(orD) as described in [15]. That is, polynomials N(ω) and D(ω) may be expressed as,  2  PN (ω) + PN2 (−ω) ≥ 0; ∀ω , (12a) N(ω) = 2  2  P (ω) + PD2 (−ω) D(ω) = D > 0; ∀ω , (12b) 2 where PN (ω) = c0 + c1 ω + · · · + cq ωq , PD (ω) = d0 + d1ω + · · · + dn ωn ,

(13a) (13b)

such that d0 = 0. In this case, the curve-fitting algorithm of step 3 is no longer linear and it computes the unknown coefficient ci and di , which are related to coefficients Ai and Bi of (6). By simple manipulation, one obtains, Ai = c2i + 2

i 

ci+ j .ci− j ,

j=1

for i = 1, 2, · · · , q; i ≥ j and i + j ≤ q

(14)

and Bi = di2 + 2

i 

di+ j .di− j ,

j=1

for i = 1, 2, · · · , n; i ≥ j and i + j ≤ n (15) The general form of the numerator polynomial N(ω) given by (6) or (12a) unnecessarily complicates the modelling problem. As stated before, for many practical problems the simple form of N(ω) given by (11) is adequate.

B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents 447

However, some complicated problems, such as antenna modelling, demand the following form of N(ω). N(ω) = ω2k

m   2 2 ω − ω2p

(16)

p=1

This form of N(ω) is called the “Modelling form-B”. In the above formulation, integer k represents the order of the transmission zeros at DC (i.e., ω = 0) as in (11). ω p designates the transmission zeros on the real frequency axis. The integer “m” designates the total number of such zeros. Certainly, measured or computed shape of the real part curve suggests the appearance of N(ω). For simple problems, if R(ω) starts with a constant value at ω = 0 and rolls off smoothly, then N(ω) = 1 or a constant A0 is chosen (Fig. 4a). If it starts from zero level at DC, raises gently up to a point and rolls-off to zero as frequency goes to infinity, then N(ω) = ω2k is selected (Fig. 4b). For more complicated problems, the supplied real part data may hit the real frequency axis ω, several times; say m times. In this case, if R(ω1 ) = R(ω2 ) = R(ω3 ) = · · · = R(ωm ) = 0, then the full form of N(ω) given by (16) is selected (Fig. 4c). For more difficult problems, the numerator polynomial “N” may be chosen in such a way that the transmission zeros include real and/or quadruple-mirror image complex conjugate pairs. A typical example may be “the modelling problem of a very short physical length mono-pole antenna”. In this case, the general form of N is given in the complex “s” plane by N(s) = (−1)k s2k

mt m  2   (σt − s)(σt + s) s2 + ω2p

p=1

.

mr 

t=1

[s + (αr + jβr )] [s + (αr − jβr )]

r=1

. [s − (αr + jβr )] [s − (αr − jβr )]

(17)

• s = ± jω p represents the total m pairs of complex conjugate real frequency transmission zeros, • s = ±(αr ± jβr ) represents total m r pairs of the quadruple-mirror image complex conjugate transmission zeros • s = ±σt represents total m t number of the real transmission zeros. On the jω axis, (17) takes the following form: mt  m    2    N ω2 = ω2k ω2p − w2 σt2 + ω2 p=1

t=1

mr    2

 αr4 + 2αr2 ω2 + βr2 + ω2 − βr2

(18)

r=1

This form of N(ω) is called the “Modelling form-C”. It should be noted that (18) corresponds to the open form of (12a). As stated earlier, in many practical cases, forms of R(ω) given by (11) or (16) are adequate. At most, an RHP zero σt may be inserted in to (16) to provide further flexibility for the optimisation. The initial guess for the non-linear curve-fitting problem is important. However, the following approach is practised to be useful for generating the initial parameters for the non-linear curve fitting of R(ω). Having selected the proper form for R(ω), one carries out the curve-fitting process. At this point, a fixed form for N(ω) is favoured as in (16), and then the coefficients Bi of (6) are computed as if the problem is “linear curve fitting”. Later, these coefficients are used to start the non-linear curvefitting problem employing the initial guess for di of (15) as follows.   d0 = 2|B0 |, dn = 2|Bn | i |Bi | di = (−1) ; i = 1, 2, 3, · · · , (n − 1) (19) i

In the above formulation, • s2k represents the transmission zeros at DC,

Fig. 4. Possible shapes of the given real part data. (a) R(w) starts with a fixed values at DC and smoothly approaches to zero as frequency goes to infinity, (b) R(w) = 0 at DC and approaches to zero as frequency goes infinity, (c) R(w) posses several transmission zeros at finite frequencies.

6. A numerical method to model the foster part of the given immitance data: an estimation procedure In this work, the Foster part of the immitance data, which is generated at the fourth step of the main algorithm, is modelled by means of an efficient subroutine. Eventually, this module yields the required parameters of (9) such as number of poles, location of poles, and the residues associated with the poles for the Foster data. At the first step of this module, the existence of a DC pole is found by tracing the generated data X f point by point. Clearly, if X f increases monotonically at low frequencies starting from the negative values as shown in Fig. 5a, then we presume that X f poses a pole at DC (ω = 0). Obviously, at the lower frequencies designated by ωi , the Foster function X f (ωi ) of (9) is dominated by the

448 B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents DC pole and it is approximated by X f (ωi ) ∼ =−

k0 ωi

(20)

Let ω1 and ω2 be the given two consecutive frequen2 cies around ωi such that ωi ≈ ω1 +ω . Then, the derivative 2 of X f (ω) is roughly estimated in the neighbourhood of ωi as ∂X f X 1 X2 − X1 , = k0 2 ≈ = ∂ω ω ω2 − ω1 ωi

(21)

where X 1 = X f (ω1 ) and X 2 = X f (ω2 ). Hence, if a DC pole exists the residue k0 is initialised as k0 ≈ ω2i

X X2 − X1 = ω2i ≥ 0. ω ω2 − ω1

(22)

At the end of this step, the DC pole is removed point by point from X f leaving

 k0 X r f (ωj ) = X f (ωj ) − − , j = 1, 2, · · · , N (23) ωj At the second step, finite jω poles, which are designated by ωr , are removed from X r f if they exist. While X r f is increasing monotonically, these poles are detected with sudden sign change as depicted in Fig. 5b. In the neighbourhood of a finite pole ωr , X r f is dominated by this pole and it is approximated as X r f (ωi ) ≈

kr ωi ωr2 − ω2i

(24)

In the neighbourhood of the pole ωr , let X 1r and X 2r be the values of X r f at ω1r and ω2r with opposite signs respectively (Fig. 5b). Then, X 1r and X 2r are given by means of (24) as kr ω1r ωr2 − ω21r kr ω2r ≈ 2 ωr − ω22r

X 1r ≈ X 2r

Solving (22) for ωr and kr one obtains   X 1r  X ω1r − ω1r ω2r ωr =  2r X ω1r 1r X 2r − ω2r kr =

ω22r − ω21r ω1r ω2r X 1r − X 2r

(26a) (26b)

Once, ωr and kr are determined, the pole is removed from X r f point by point. The above process continues until the remaining Foster part does not change its sign but increases monotonically. Hence, the remaining Foster part, which is designated by X ∞ is left only with a pole at infinity. At this step, the total number of poles “ p” is registered. Finally, the remaining Foster function X ∞ is modelled as X ∞ = k∞ .ω

(27)

Ultimately, the residue k∞ is estimated by taking the numerical derivative of X ∞  employing, perhaps, the last two pairs X ∞(N−1) , ω(N−1) , X ∞N , ω N as depicted in Fig. 6c. k∞ =

X ∞ X ∞N − X ∞(N−1) ∂X ∞ ≈ = ∂ω ω ω N − ω(N−1)

(28)

Once the initial values for the unknown parameters of (9) are estimated, the realisable form of the Foster data is determined by means of an efficient non-linear, constraint curve-fitting algorithm. Thus, the values for residues k0 , k∞ and kr and the location of finite poles ωr are determined.

7. Example

(25a) (25b)

Fig. 5. Point by point estimation of the Foster data. (a) Extraction of DC pole, (b) Extraction of finite j pole, (c) Extraction of a pole at infinity.

In the present section, we consider an example to demonstrate the application of the novel modelling technique introduced in this paper. This example was taken from [17]. It is desired to build a model for a monopole antenna employing the new technique. It should be noted that, as far as the modelling process is concerned, there is no difference to model the given data for any type of problems mentioned in Section 2. Certainly, the generation of data for characterisation, design and simulation type of problems is different in nature and it is beyond the scope of this paper. However, once the data is provided, and, if it is modelled in the Darlington sense, then, the main algorithm presented in Section 3 is employed in a straightforward manner. Therefore, in this section, we will just present one example to demonstrate the utilisation of the new modelling tool. However, the reader is encouraged to handle different modelling cases, which fall into her or his area of expertise. Model for a monopole antenna: In this example, we will construct a model for the data that belongs to a short monopole antenna usually employed for

B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents 449

military communication purposes. Data for the antenna is provided over 20 MHz to 100 MHz. The ultimate goal is to design a matching network. The model is required to determine the theoretical gain-bandwidth limitations of the antenna over the measured frequencies. One can implement the main algorithm step by step as follows. Step 1: The measured data is normalised with respect to the high-end of the frequency band, which is 100 MHz, and the termination resistance 50 Ω as depicted in Fig. 6a– b and shown in Table 1. Step 2a: The numerical behaviour of the problem demands the impedance approach. Step 2b: In this step, there was no need to evaluate the imaginary part of the minimum reactance impedance by the Hilbert transformation as explained in the main algorithm. Rather, it is produced in Step 3b. Step 3a: In this step, the measured real part data is fitted to the rational form R(ω) = N(ω)/D(ω) by means of a non-linear optimisation algorithm. Here, N(ω) is set to a constant A0 (the “Modelling form-A” with k = 0). The degree of D(ω) is set to n = 6, which describes a six element ladder network. Thus, the following result is obtained.

where, a5 = 0.865, a4 = 2.03, a3 = 4.882, a2 = 4.685, a1 = 4.085, a0 = 0.258, b6 = 0.337, b5 = 0.79, b4 = 2.07, b3 = 2.219, b2 = 2.519, b1 = 0.96, b0 = 0.699

R(ω) = A0 /D(ω), D(ω) = B0 + B1 ω2 + B2ω4 + B3ω6 + B4ω8 + B5ω10 + B6 ω12   where A0 = c20 and D(ω) = PD2 (ω) + PD2 (−ω) /2 > 0 with PD (ω) = d0 + d1ω + d2 ω2 + d3 ω3 + d4 ω4 + d5 ω5 + d6 ω6 . Coefficient di , which in turn yields Bi , and A0 = c20 by means of (14–15), is determined via a non-linear optimisation package. Thus, we ended up with the following result. A0 = 0.18, d0 = 0.699, d1 = 0.135, d2 = −1.872, d3 = −0.592, d4 = 1.166, d5 = 0.127, d6 = −0.337. Hence the coefficients of the denominator polynomial are B0 = 0.488, B1 = −2.6, B2 = 4.974, B3 = −4.451, B4 = 2.47, B5 = −0.769, B6 = 0.113 . In Fig. 6, measured and fitted R(ω) characteristics are shown. Step 3b: Using the Gewertz procedure, the minimum reactance function Z m is computed. Z m (s) =

a5 s 5 + a4 s 4 + a3 s 3 + a2 s 2 + a1 s + a0 b6 s 6 + b5 s 5 + b4 s 4 + b3 s 3 + b2 s 2 + b1 s + b0

Fig. 6. Plot of the measured impedance data and the computed impedance data obtained from the model for the monopole antenna. (a) Real part data plots, (b) Imaginary part data plots, (c) Plots of the Foster part data. Table 1. Normalised impedance data for the monopole antenna. Frequency (ω)

R(ω) = Real(Z)

X(ω) = Imag(Z)

0.20 0.30 0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.90 1.00

0.6 0.8 0.8 1.0 2.0 3.4 7.0 15.0 22.4 11.0 5.0 1.6 1.0

−6.0 −2.2 0.0 1.4 2.8 4.6 7.6 8.8 −5.4 −13.0 −10.8 −6.8 −4.4

450 B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents Step 4: Employing the above rational form, real and imaginary parts of Z m are computed on the jω axis. That is, Z m ( jω) = R(ω) + jX m (ω) Then, using the measured data, the Foster part of the impedance is obtained as X f (ω) = X(ω) − X m (ω) Step 5: Employing the estimation technique described in Section 6 it is found that the Foster data includes no finite pole. However, we have detected a pole at DC and a pole at infinity yielding a capacitance in series with an inductor. Hence, the following result is obtained. Z F (s) = 1.75s +

1.51 s

In Fig. 6c, the computed and fitted Foster parts are shown. Step 6: Finally, the PR impedance Z(s) = Z F (s) + Z m (s) is synthesised as a lossless two port in resistive termination as depicted in Fig. 7. It should be noted that result of the Darlington synthesis might not be unique. Alternative circuit topologies for the same driving point impedance may be obtained. It should be mentioned that the above example was solved utilising the Immitance Based Computer Aided Modelling Package or Tool, which is called “IB-CAMOT” developed in the course of this research program. IBCAMOT was written on Math Lab 5.1 by implementing the steps of the main algorithm in detail. Each step of the algorithm was considered as a separate module. These modules were then, combined under a shell or so called the main program. The IB-CAMOT is run automatically, by entering the input quantities of the modelling tool as given below. Inputs to IB-CAMOT: • •

• •

The measured immitance data of the physical device to be modelled. The complexity of the circuit model. That is, the total number of the circuit elements “n” and an appropriate form of N(ω) (Form A, B or C) must be selected among (11), (16) and (17,b). Initial values for the coefficients of the numerator polynomial h(s). ε : A criterion to terminate the non-linear curve fitting process. (For many applications, it was sufficient to choose ε less than 10−5 .)

Fig. 7. Synthesis of the impedance function obtained for the monopole antenna.

Within the non-linear curve fitting part of the package, a built-in Math-Lab 5.1 routine, so called “LevenbergMarquardt” optimisation technique, is employed. At the output of IB-CAMOT, the measured and the computed immitance characteristics are compared with relevant graphics and tables; and the element values of the desired circuit model are printed. All the computations were carried out on a 450 MHz, Pentium II – PC platform.

8. Conclusion In this paper, an immitance based novel tool is presented to model the measured or computed data obtained from a passive-one port device by means of its Darlington equivalent. Unlike the other available techniques, the proposed technique does not require any choice for the circuit topology; rather it is the natural consequence of the modelling process. The crux of idea for the new modelling tool is the point by point decomposition of the given immitance data in to its minimum and Foster parts. First, the real part of the given immitance data is modelled as a non-negative, even rational function which in turn yields the realisable positive-real minimum immitance. Then, the Foster data is modelled. Finally, the minimum and the Foster functions are synthesised as a losslesss two-port network in unit termination, which is called the Darlington representation of the given immitance data. A main algorithm is outlined to ease the understanding of the modelling process introduced in this paper. Subroutines to model real part data and the Foster data are also provided. It should be noted that model of a physical device, generated employing the new technique presented in this work, by no means, is unique. In the first place, it depends on the selected mathematical form of the real part, which will be fitted to the given data. Furthermore, there may be several alternatives to construct the Foster part of the immitance data. No matter what the final model is, the location of the real frequency transmission zeros of the resulting models must be verified experimentally. Obviously, the success of the model depends on the match between the theoretical and experimental results obtained from the model and the physical device respectively. An antenna-modelling example is presented to exhibit the implementation of the modelling process systematically. It is expected that the proposed modelling technique will be useful to assess the electrical behaviour, such as gain-bandwidth limitations or attainable optimum performance, of the physical devices under consideration. The new modelling tool will also enhance the analysis, design and simulation capability of the commercially available Computer Aided Design packages when employed to manufacture the high-speed/high frequency analog/digital communication sub-systems put on VLSI chips

B.S. Yarman et al.: An Immitance Based Tool for Modelling Passive One-Port Devices by Means of Darlington Equivalents 451

Acknowledgement The authors wish to extend their sincere appreciations to Prof. Dr. Yılmaz Tokad and Prof. Dr. Ergül Akçakaya for their valuable comments and editing efforts of this paper.

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B. Sıddık Yarman received the B.Sc. degree in electrical engineering from the Technical University of Istanbul (I.T.U.), Istanbul, Turkey, in 1974 and the M.E.E.E from Stevens Institute of Technology (S.I.T) in NJ, in 1977, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1982. He was a member of the faculty of the ITU in 1974 and a teaching and research associate for SIT and Cornell from 1976–1982. He was a member of the technical staff with David Sarnoff Research Center, NJ, from 1982 to 1984 and associate professor with Anadolu University and Middle East Technical University in 1985–1986. From 1987 to 1989 he was a visiting professor at Ruhr University, Bochum, Germany as an Alexander von Humboldt Fellow. He was with STFA SAVRONIK, a defense electronic corporation and was its Deputy General manager in 1989. He was a full professor at Istanbul University until 1996. Since 1996 he is the President of I¸sık University, Istanbul, Turkey. Dr. B.S. Yarmann holds four U.S. patents, recipient of research and technology award of the National Research Council of Turkey; selected as the International man of year in Science and Technology by Cambridge Biography Center of U.K. in 1989. He is the member Academy of Science of New York, senior member of IEEE.

Ahmet Aksen received the B.Sc. and M.Sc. degrees in electronics engineering from the Middle East Technical University, Ankara, Turkey in 1981 and 1985 respectively and the Ph.D. degree from the Ruhr University, Bochum, Germany in 1994. In 1989–1994, he has been with the “Lehrstuhl für Nachrichtentechnik, Ruhr Universität” Bochum under the sponsorship of “Deutscher Akademischer Austauschdienst.” He was an associate professor with Istanbul University in 1995–1997 and since 1997 he is with I¸sık University, Istanbul Turkey chairing the department of computer science and engineering. His current research interests are in the area of multivariable network theory, computer aided circuit design and in the design of microwave filters, broadband matching networks, amplifiers and MMIC.

Ali Kılınç received B.Sc. and M.Sc. degrees in electronic engineering from the Uludag University, Bursa, Turkey in 1986 and 1989 respectively. He completed his Ph.D. in the area of impedance modeling at Istanbul University, Turkey in 1995. Until 1998 he was teaching as a lecturer in Istanbul University. Then he joined Nortel Networks-Neta¸s, Turkey.