8.5 Microwave Simulation-Driven Design Exploiting Physically-Based Surrogates
8.5.2 Simulation-Based Tuning and Tuning Space Mapping
Tuning is ubiquitous in engineering practice. It is usually associated with the process of manipulating free or tunable parameters of a device or system after that device or system has been manufactured. The traditional purpose of permitting tunable elements is (8.1) to facilitate user-flexibility in achieving a desired response or behavior from a manufactured outcome during its operation, or (8.2) to correct inevitable postproduction manufacturing defects, small due perhaps to tolerances, or large due perhaps to faults in the manufacturing process [54].
Tuning of an engineering design can be seen, in essence, as a user- or robot- directed optimization process.
Tuning space mapping (TSM) [50] combines the concept of tuning, widely used in microwave engineering [55], [56], and space mapping. It is an iterative op- timization procedure that assumes the existence of two surrogate models: both are less accurate but computationally much cheaper than the fine model. The first model is a so-called tuning model Rt that contains relevant fine model data (typi- cally a fine model response) at the current iteration point and tuning parameters (typically implemented through circuit elements inserted into tuning ports). The tunable parameters are adjusted so that the model Rt satisfies the design specifica- tions. The second model, Rc is used for calibration purposes: it allows us to trans- late the change of the tuning parameters into relevant changes of the actual design variables; Rc is dependent on three sets of variables: design parameters, tuning pa- rameters (which are actually the same parameters as the ones used in Rt), and SM parameters that are adjusted using the usual parameter extraction process [7] in order to have the model Rc meet certain matching conditions. Typically, the model Rc is a standard SM surrogate (i.e., a coarse model composed with suitable trans- formations) enhanced by the same or corresponding tuning elements as the model Rt. The conceptual illustrations of the fine model, the tuning model and the cali- bration model are shown in Fig. 8.5.
The iteration of the TSM algorithm consists of two steps: optimization of the tuning model and a calibration procedure. First, the current tuning model Rt
(i) is built using fine model data at point x(i). In general, because the fine model with in- serted tuning ports is not identical to the original structure, the tuning model re- sponse may not agree with the response of the fine model at x(i) even if the values of the tuning parameters xt are zero, so that these values must be adjusted to, say, xt.0(i)
, in order to obtain alignment [50]:
( ) ( ) ( )
.0 arg min ( ) ( )
t
i i i
t = f − t t
x
x R x R x (8.5) In the next step, one optimizes Rt(i) to have it meet the design specifications. Op- timal values of the tuning parameters xt.1(i)
are obtained as follows:
( )
( ) ( )
.1 arg min ( )
t
i i
t = U t t
x
x R x (8.6) Having xt.1(i), the calibration procedure is performed to determine changes in the design variables that yield the same change in the calibration model response as that caused by xt.1(i)
– xt.0(i)
[50]. First one adjusts the SM parameters p(i) of the calibration model to obtain a match with the fine model response at x(i)
( ) ( ) ( ) ( )
arg min ( ) ( , , .0) .
i i i i
f c t
= −
p
p R x R x p x (8.7) The calibration model is then optimized with respect to the design variables in or- der to obtain the next iteration point x(i+1)
( 1) ( ) ( ) ( ) ( )
.1 .0
arg min ( ) ( , , ) .
i i i i i
t t c t
+ = −
x x R x R x p x (8.8) Note that xt.0
(i) is used in (8.7), which corresponds to the state of the tuning model after performing the alignment procedure (8.5), and xt.1(i)
in (8.8), which
corresponds to the optimized tuning model (cf. (6)). Thus, (8.7) and (8.8) allow finding the change of design variable values x(i+1) – x(i) necessary to compensate the effect of changing the tuning parameters from xt.0(i)
to xt.1(i)
.
Design Variables
Response Fine
Model
Rf(x) x
Tuning Model
B t
? ıcD
?0 ıcB yD J- jy ıã ?/E
E D?g
H B?o
ıã ?Hj
Response Rt(xt)
Tuning Parameters xt
(a) (b)
Tuning Parameters xt
Space Mapping
Space Mapping Parameters
p Design
Variables x
Response Rc(x,p,xt) Calibration Model
(c)
Fig. 8.5 Conceptual illustrations of the fine model, the tuning model and the calibration model: (a) the fine model is typically based on full-wave simulation, (b) the tuning model exploits the fine model “image” (e.g., in the form of S-parameters corresponding to the cur- rent design imported to the tuning model using suitable data components) and a number of circuit-theory-based tuning elements, (c) the calibration model is usually a circuit equiva- lent dependent on the same design variables as the fine model, the same tuning parameters as the tuning model and, additionally, a set of space mapping parameters used to align the calibration model with both the fine and the tuning model during the calibration process.
It should be noted that the calibration procedure described here represents the most generic approach. In some cases, there is a formula that establishes an ana- lytical relation between the design variables and the tuning parameters so that the updated design can be found simply by applying that formula [50]. In particular, the calibration formula may be just a linear function so that x(i+1) = x(i) + s(i)∗(xt.1(i) – xt.0(i)
), where s(i) is a real vector and ∗ denotes a Hadamard product (i.e., component-wise multiplication) [50]. If the analytical calibration is possible, there is no need to use the calibration model. Other approaches to the calibration process can be found in the literature [50], [57]. In some cases (e.g., [57]), the tuning parameters may be in identity relation with the design variables, which simplified the implementation of the algorithm.
The operation of the tuning space mapping algorithm can be clarified using a simple example of a microstrip transmission line [50]. The fine model is imple- mented in Sonnet em [17] (Fig. 8.6(a)), and the fine model response is taken as the inductance of the line as a function of the line’s length. The original length of the line is chosen to be x(0) = 400 mil with a width of 0.635 mm. The goal is to find a length of line such that the corresponding inductance is 6.5 nH at 300 MHz. The Sonnet em simulation at x(0) gives the value of 4.38 nH, i.e., Rf(x(0)) = 4.38 nH.
1 2
1 3 4 2
(a) (b)
Term 1 Z=50 Ohm
Term 2 Z=50 Ohm S4P
SNP1
1 2
3 4
Ref L L1 L=LnH
(c)
Term 1 Z=50 Ohm
Term 2 Z=50 Ohm MLIN
TL1 W= 0.635 mm L=x/2 mm
L L1 L=LnH
MLIN TL2 W= 0.635 mm L=x/2 mm
(d)
Fig. 8.6 TSM optimization of the microstrip line [50]: (a) original structure of the micro- strip line in Sonnet, (b) the microstrip line after being divided and with inserted the co- calibrated ports, (c) tuning model, (d) calibration model.
The tuning model Rt is developed by dividing the structure in Fig. 8.6(a) into two separate parts and adding the two tuning ports as shown in Fig. 8.6(b). A small induc- tor is then inserted between these ports as a tuning element. The tuning model is im- plemented in Agilent ADS [47] and shown in Fig. 8.6(c). The model contains the fine model data at the initial design in the form of the S4P element as well as the tuning element (inductor). Because of Sonnet’s co-calibrated ports technology [56], there is a perfect agreement between the fine and tuning model responses when the value of the tuning inductance is zero, so that xt.0(0)
is zero in this case.
Next, the tuning model should be optimized to meet the target inductance of 6.5 nH. The optimized value of the tuning inductance is xt.1(0) = 2.07 nH.
The calibration model is shown in Fig. 8.6(d). Here, the dielectric constant of the microstrip element is used as a space mapping parameter p. Original value of this parameter, 9.8, is adjusted using (8.7) to 23.7 so that the response of the calibration
model is 4.38 nH at 400 mil, i.e., it agrees with the fine model response at x(0). Now, the new value of the microstrip length is obtained using (8.8). In particular, one optimizes x with the tuning inductance set to xt.0(0)
= 0 nH to match the total in- ductance of the calibration model to the optimized tuning model response, 6.5 nH.
The result is x(1) = 585.8 mil; the fine model response at x(1) obtained by Sonnet em simulation is 6.48 nH. This result can be further improved by performing a second iteration of the TSM, which gives the length of the microstrip line equal to x(2) = 588 mil and its corresponding inductance of 6.5 nH.
Simulation-based tuning and tuning space mapping can be extremely efficient as demonstrated in Chapter 12. In particular, a satisfactory design can be obtained after just one or two iterations. However, the tuning methodology has limited applications. It it well suited for structures such as microstrip filters but it can hardly be applied for radiating structures (antennas). Also, tuning of cross- sectional parameters (e.g., microstrip width) is not straightforward [50]. On the other hand, the tuning procedure is invasive in the sense that the structure may need to be cut. The fine model simulator must allow such cuts and allow tuning elements to be inserted. This can be done using, e.g., Sonnet em [17]. Also, EM simulation of a structure containing a large number of tuning ports is computation- ally far more expensive than the simulation of the original structure (without the ports). Depending on the number of design variables, the number of tuning ports may be as large as 30, 50 or more [50], which may increase the simulation time by one order of magnitude or more. Nevertheless, recent results presented in [58] indi- cate possibility of speeding up the tuning process by using so-called reduced structures.