Transient analysis of tapered distributed RC networks using finite elements A.J. Walton, M.Sc., Ph.D., and B.J. Marsden, B.Sc, Ph.D., Grad. M.I. Mech. E. Indexing term: Networks Abstract: The use of the finite-element method to determine the transient response of nonuniform distrib- uted RCnetworks is described. The method is compared with other available techniques and the relationship between the accuracy of solution and the element size and time step investigated. These results are then used to obtain the transient response of some exponentially and linearly tapered distributed RC structures. 1 Introduction The distributed RC network, when considered as an element in its own right, has been attracting interest since the late fifties [1, 2]. It has an infinite number of poles and may be used to obtain a large number of circuit functions many of which may not be achieved with lumped components. An example of some of the lumped passive circuits, which may be replaced by distributed RC structures, is shown in Fig. 1. It may be observed that their use also tends to reduce the total number of components, which in turn reduces the number of inter- connections, which should increase the reliability of circuits which contain them. The simplest distributed RC structure is the uniform net- -j_ _i_ _i_ T T T h Fig. 1 Some lumped circuits (LHS) with their distributed RC network equivalents (RHS) Paper 2133G, first received 21st April 1981 and in revised form 23rd June 1982 Dr. Walton is with the Department of Electrical Engineering, University of Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JL, Scotland, and Dr. Marsden is with the Department of Mechanical Engineering, Nottingham University, University Park, Nottingham, Notts., England IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 work shown in Fig. 2a, andthe performance of this structure may be improved upon by tapering. For example, the expo- nentially tapered network shown in Fig. 2b, when connected as a lowpass filter, has a sharper rate of cutoff and notch filters have a narrower rejection band when compared with their uniform counterparts [3]. If tapered networks are to be used in electronic circuits, then a knowledge of their transient response becomes desirable. Unfortunately, there is no simple analytical solution for these networks because the tapering causes 2-dimensional current flow. This paper describes the use of the finite-element technique to evaluate the transient response of nonuniform distributed RC net- works. This method takes account of the 2-dimensional current flow. 2 Transient response of tapered distributed RC networks The distributed RC structure is described by the following differential equation [4]: b 2 V(x,y,t) | b 2 V(x,y,t) bx 2 by 2 symmetry Fig. 2 RC networks a Uniform distributed b Exponentially tapered distributed 0143-7089/82/060295 + 06 $01.50/0 295
Transcript
Transient analysis of tapered distributed RCnetworks using finite elements
A.J. Walton, M.Sc., Ph.D., and B.J. Marsden, B.Sc, Ph.D., Grad. M.I. Mech. E.
Indexing term: Networks
Abstract: The use of the finite-element method to determine the transient response of nonuniform distrib-uted RC networks is described. The method is compared with other available techniques and the relationshipbetween the accuracy of solution and the element size and time step investigated. These results are then usedto obtain the transient response of some exponentially and linearly tapered distributed RC structures.
1 Introduction
The distributed RC network, when considered as an element inits own right, has been attracting interest since the late fifties[1, 2] . It has an infinite number of poles and may be used toobtain a large number of circuit functions many of which maynot be achieved with lumped components. An example ofsome of the lumped passive circuits, which may be replacedby distributed RC structures, is shown in Fig. 1. It may beobserved that their use also tends to reduce the total numberof components, which in turn reduces the number of inter-connections, which should increase the reliability of circuitswhich contain them.
The simplest distributed RC structure is the uniform net-
- j_ _i_ _i_
T T T
hFig. 1 Some lumped circuits (LHS) with their distributed RC networkequivalents (RHS)
Paper 2133G, first received 21st April 1981 and in revised form 23rdJune 1982Dr. Walton is with the Department of Electrical Engineering, Universityof Edinburgh, Kings Buildings, Mayfield Road, Edinburgh EH9 3JL,Scotland, and Dr. Marsden is with the Department of MechanicalEngineering, Nottingham University, University Park, Nottingham,Notts., England
IEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982
work shown in Fig. 2a, and the performance of this structuremay be improved upon by tapering. For example, the expo-nentially tapered network shown in Fig. 2b, when connectedas a lowpass filter, has a sharper rate of cutoff and notchfilters have a narrower rejection band when compared withtheir uniform counterparts [3]. If tapered networks are tobe used in electronic circuits, then a knowledge of theirtransient response becomes desirable. Unfortunately, thereis no simple analytical solution for these networks becausethe tapering causes 2-dimensional current flow. This paperdescribes the use of the finite-element technique to evaluatethe transient response of nonuniform distributed RC net-works. This method takes account of the 2-dimensionalcurrent flow.
2 Transient response of tapered distributed RC networks
The distributed RC structure is described by the followingdifferential equation [4]:
b2V(x,y,t) | b2V(x,y,t)
bx2 by2
symmetry
Fig. 2 RC networks
a Uniform distributedb Exponentially tapered distributed
0143-7089/82/060295 + 06 $01.50/0 295
where r and c are the resistance (£2/sq.) and capacitance(F/m2), respectively.
For a uniform network, as shown in Fig. 2a, eqn. 1 may besolved for a unit step input to give [4]
(2)(-1)"
where i>n are the poles of the network and are given by
Pn =-{(2n-l)n/2}2/RC (3)
where R and C are, respectively, the resistance (fl) and thecapacitance (F) of the structure.
Current flow in a uniform network is 1-dimensional, andso the transient reponse of the structure is an exact solution ofeqn. 1. Expressions for the transient response of exponentiallyand linearly tapered structures have been derived by Kaufmanand Garrett [3]. These were obtained assuming 1-dimensionalcurrent flow and, as is the case for the steady-state analysisof tapered distributed networks [5, 6] , these complicatedexpressions do not give the correct response. The only methodof accurately evaluating the transient response is to use anumerical technique. One method of approximating eqn. 1 isto use finite difference [7]. It is an iterative technique, theaccuracy of which is dependent on the size of the mesh usedto model the structure and the number of iterations taken.Curved boundaries are awkward to incorporate into finite-difference programs [8], and hence they tend to be applicableto only one geometry.
The finite-element method which consists of solving a setof equations, has no such problems with curved boundaries,and it is possible to use a single computer program to analysenumerous different geometries.
3 Finite element
The finite-element method models the structure by dividingit into smaller areas which are called elements. Each elementhas a number of nodal points, and these are used to connect itto other surrounding elements, thereby modelling the wholestructure. The continuum which the differential equationdescribes in effect becomes a set of discrete areas, over whichthe equation is approximated, and from this a set of simul-taneous equations result. The solution of these is the solutionof the differential equation.
The general equation which relates the nodal parameters tothe other points within the element is
V = ViNi (4)
where V is the voltage at some point inside the element, V{
the nodal voltage at node i, Nt the interpolation function ofnode i and n the number of nodes. The interpolation function
nodes
Fig. 3 8-noded isoparametric element
296
[9] describes the manner in which the voltage may vary withinthe element and also makes the voltage continuous across theboundaries of adjacent elements.
The element which is to be used in the following analysis isthe 8-noded isoparametric element [9], which has a quadraticinterpolation function. This element is shown in Fig. 3 and itmay be seen to have a node at each of the four corners andone on each side. The boundaries of this type of element maybe curved, making it useful for analysing almost any geometry.
4 Finite-element formulation
As previously mentioned, the differential equation whichdescribes the transient response of a distributed RC networkmay be approximated by a set of equations which relate thenodal voltages of an area (element) to one another. Eachelement models a small area of the total structure and theerror within it may be minimised using the method ofweighted residuals due to Galerkin [9]. This reduces eqn. 1 to
+ c[H] ie) (5)
where (e) denotes the element, / and V the nodal currentsand voltages, respectively, and components of [K] and [H]are given below:
(6)
(7)
For the 8-noded element, [K] and [H] are 8 by 8 matricesand eqns. 6 and 7 are evaluated numerically, using Gaussianquadrature [10]. For ease of computation, the integration foreach element is performed on a unit square, which is thenmapped to the global co-ordinates of the respective elements,thereby enabling the integration necessary to evaluate eqns. 6and 7 to be performed on irregular shapes.
Once the [H] and [K] matrices have been evaluated forevery element, they are all then combined to form a set ofsimultaneous equations. They relate the node voltages, thedifferential of the node voltages with respect to time and thenode currents. The node currents sum to zero (Kirchhoff law)except at the boundaries, where the voltage is fixed, whichresults in the number of equations equalling the number ofunknowns.
The added complications of the transient analysis, whencompared with the steady-state one, is the occurrence of[Kp e ) in eqn. 5. Using the lst-order forward differenceformula [8],
[V]tie) = At
where t is the time step between times t and t + At. Substi-tuting eqn. 8 into eqn. 5 gives:
(9)
The unknown voltages may now be solved because [V]t isknown from the initial conditions or the previous solution. Ifa suitable time step is chosen, then the transient response isobtained by stepping to the next point in time, using eqn. 9and repeating the process until the finishing time is reached.Unfortunately, the size of the time step must be kept small,otherwise inaccurate solutions are obtained.
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Larger time steps may be taken using the central differenceformula, and this is known as the Crank Nicholson method[10, 11]. This gives
(10)
and the resulting equations are solved in a manner similar tothat described above.
5 Construction and evaluation of finite element model
An exact solution exists for the uniform network and is tobe used to compare with the finite-element solution. A uni-form network with unit width and length and an RC productof unity is to be modelled with various numbers of squareelements, in order to give some indication of the number ofelements that should be used to obtain an accurate solution.An example of the uniform network being modelled with fourelements is given in Fig. 4. The nodes and elements are allnumbered as indicated and the connection of the elementsto one another at their nodal points may be observed.
The boundary conditions are now inserted. The edges ofthe network where no current flows across are automaticallytaken care of by the finite-element formulation. The twocontact pads constrain the voltage at the nodes 1 to 5 to beequal and also nodes 17 to 21 to be equal. The initial con-
dielectric\
resistance/
}<.. .<\<./.
input pad conductor output pad
14V
13
12
10
16 21
20"
15 19
6 9 14 17
Fig. 4 Uniform distributed RC network and its finite element model
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ditions for a unit step input require that, at time t = 0, all theinput nodes (1 —5) be at 1 volt and all the others at zero volts.
Table 1 compares the analysis of the square distributed RCnetwork mentioned, for different numbers of elements. It canbe seen that the accuracy of solution increases as the size ofthe elements are reduced. It should, however, be noted thatthe size of the elements is not the only factor which influencesthe accuracy. The size of the time step will obviously influencethis smaller time steps, leading to a more accurate model.
Table 1: Comparison of exact transient response of square distributedRC network with that obtained using different numbers of finite
There is a relationship between the time step and the elementsize which if not observed, can lead to instability in thesolution. This relationship is given below [12]:
At
red'(11)
where t is the time step and d is the closest distance betweenadjacent nodes where the electrical field has its steepestgradient. The timesteps used in Table 1 were obtained usingeqn.11.
6 Analysis of some tapered distributed RC networks
6.1 Construction of modelThe distributed RC network is divided up into a number ofelements, bearing in mind the results given in Section 5. Itshould be remembered that, although the elements may takeirregular shape, care should be taken that their aspect ratioshould not be too large (5:1) or the angle of the corners tooacute (30°), otherwise the element will be unable to accuratelyapproximate the electric field.
input
output
Fig. 5 Finite-element mesh used to model exponentially tapered(B = + 1) distributed RC network (Only half the network is usedbecause of symmetry)
297
In many cases, the size of a problem may be reduced bytaking advantage of symmetry and, in both the structuresanalysed, this was the case. The line of symmetry is throughthe centre of the network, as shown in Fig. 2b, and this maybe used to reduce the size of the problem by half. A typical
mesh is shown in Fig. 5, and all the networks analysed weremodelled using ten elements in the length and four in width.
6.2 Exponential taperA distributed RC network with an exponential taper has itsresistance and capacitance varying along its length as follows:-
1.0
0.8
0.6
B = 2
0.2
=-2
r(x) =
If X is the length of the structure, then
a = —X r(0)
The degree of taper is defined as
(12)
(13)
(14)
05)
0.5 1.0 1.5 2.0 2.5time, s
3.0 3.5 4.0
Some exponential networks with tapers between B — + 2 and— 2 were analysed by dividing the structure into elements in asimilar manner to that shown in Fig. 5. Their transient re-sponse is shown in Fig. 6 with electric field plots for bothpositive and negative tapers shown in Fig. 7 — clearly showingthe effect of the 2-dimensional current flow.
Fig. 6 Transient response of some exponentially tapered distributedRC networks
Fig. 7 Electrical field plot for exponentially tapered distributed RC network 0.11 s after a unit step input
(a) B = - 1 (&) B = + 1
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6.3 Linear taperThe linear tapered network has its resistance and capacitancevarying along its length in the following manner:
r(x) =
c(x) =
/•(0) XT
XT-X(|T-1|)C(0)(XT-X(|T-1|))
XT
(16)
(17)
With X again the length of the network and T the taper factor,which is defined as
T =r(x)r(0)
(18)
The transient response of some linearly tapered networks withtapers between 10 and 0.1 were evaluated, using a similarmesh to that shown in Fig. 8. Fig. 9 gives these responses.
7 Discussion and conclusions
The accuracy of the finite-element technique as a method toevaluate the transient, response of distributed RC networkshas been demonstrated by comparison of results with theexact solution of the uniform network.
The two tapered structures were analysed using the sameprogram which may be used to give the transient response ofany nonuniform distributed RC network. This is a distinctadvantage over the finite-difference method, where programsare usually written for use with a limited number of geom-etries.
There are numerous different geometric tapers [13, 14, 15]whose transient response may easily be analysed using finiteelements. Some synthesis techniques give distributed RCstructures which cannot be defined by continuous functions[16]. Once again, the finite-element method may be used toevaluate their transient response.
A knowledge of transient response may be used to calculatethe dominant poles of a non-uniform distributed RC network
element
output
input
Fig. 8 Finite-element mesh used to model linearly tapered (r = 3)distributed RC network
Only half the network used, because of symmetry
1.0
0.8
0.6
0.4
0.2
• •
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0t i m e s
Fig. 9 Transient response of some linearly tapered distributed RC networks with various tapersIEEPROC, Vol. 129, Pt. G, No. 6, DECEMBER 1982 299
[17]. The usual solutions for tapered networks assume 1-dimensional current flow, which must lead to inaccurate polelocations, whereas trimmed uniform structures have no ana-lytical solution [18]. If the transient response is known, itmay be used to calculate the dominant poles of a distributedRC network which are necessary in the design of active filters.
8 References
1 SMITH, A.B., and COOPER, G.: 'Printed distributed RC networks',Electr. Manuf., Nov. 1956, pp. 121-125
9 HUEBNER, K.H.: The finite element method for engineers'(John Wiley & Sons, New York, 1975)
10 ZIENKIEWICZ, O.C.: The finite element method' (McGraw-Hill,London, 1979, 3rd expanded edn.)
11 CRANK, J., and NICHOLSON, P.. 'A practical method fornumerical evaluation of solutions of partial differential equations ofthe heat conduction type', Proc. Camb. Phil. Soc, 1943, 43,pp. 50-67
12 ELLIMAN' D.G.: 'Experimental and numerical studies of anaxisymmetric combustor'. Ph.D. thesis, Nottingham University,1977
13 SU, K.L.: 'The trigonometrical RC line', IEEE Int. Conf. Rec,1963, 11, Pt. 2, pp. 43-45
14 SU, K.L.: 'Hyperbolic RC transmission line', Electron. Lett., 1965,1,(3), pp. 59-60
15 WATKINS, J.: 'Distributed RC notch filters based on circulargeometry', IEEE Trans., 1974, CAS-21, pp. 271-274
16 WYNDRUM, R.W.: 'The exact synthesis of distributed RCnetworks'. TR-400-76, New York University Laboratory forElectroscience, May 1963
17 WALTON, A.J., MORAN, P.L., and BURROW, N.G.: 'The domi-nant poles of trimmed uniform distributed RC networks obtainedfrom their transient response', IEEE Trans., 1982, CHMT-5, pp.267-270
18 WALTON, A.J., MORAN, P.L., and BURROW, N.G.: 'The fre-quency response of some trimmed passive distributed RC lowpassnetworks', ibid., 1978, CHMT-1, pp. 309-315
Anthony John Walton received his B.Sc.in electrical engineering from the Univer-sity of Newcastle-upon-Tyne in 1974. In1976 he obtained his M.Sc. by researchinto thin-film lumped elements at micro-wave frequencies. From 1977 until 1979he worked as a research assistant atManchester Polytechnic, during whichtime he was awarded a Ph.D. for researchinto the fabrication and characterisationof thick-film distributed RC networks.
During 1980 he was a research fellow at LoughboroughUniversity working on hybrid active filters. Since 1981 he hasbeen working at Edinburgh University in the EdinburghMicrofabrication Facility.
Barry John Marsden served an apprentice-ship between 1969 and 1972 with theDrayton Kiln Company of Stoke-onTrent. He subsequently obtained a B.Sc.degree in electromechanical engineeringat the North Staffordshire Polytechnicin 1976. He then worked as a researchassistant at Manchester Polytechnic,where in 1980 he obtained a Ph.D. forresearch in the field of stress analysis. Heis now a research fellow at NottinghamUniversity.
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