nonlinear circuit analysis using the method of harmonic

Nonlinear Circuit Analysis Using the Method of Harmonic Balance-A Review of the Art. Part I . Introductory Concepts

Rowan J. Gilmore' and Michael B. Steer2 'Compact Software, 483 McLean Boulevard, Paterson, New Jersey 07504 'High Frequency Electronics Laboratory and the Center for Communications and Signal Processing, Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, North Carolina 27695-791 1

Recen,ed Murch l 3 1990 reLised June 26. I990

ABSTRACT

The harmonic balance method is a technique for the numerical solution of nonlinear analog circuits operating in a periodic, or quasi-periodic, steady-state regime. The method can be used to efficiently derive the continuous-wave response of numerous nonlinear microwave components including amplifiers, mixers, and oscillators. Its efficiency derives from imposing a predetermined steady-state form for the circuit response onto the nonlinear equations representing the network, and solving for the set of unknown coefficients in the response equation. Its attractiveness for nonlinear microwave applications results from its speed and ability to simply represent the dispersive, distributed elements that are common at high frequencies. The last decade has seen the development and application of harmonic balance techniques to model analog circuits, particularly microwave circuits. The first part of this paper reviews the fundamental achievements made during this time. The second part covers the extension of the method to quasi-periodic regimes, optimization analysis, and practical application. A critical assessment of the various types of harmonic balance techniques is given. The different sampling and Fourier transform methods are compared, and numerical speed and precision results are given enabling a quantitative analysis of the merits of the major variants of the harmonic balance technique. Examples of designs which have been modeled using the harmonic balance technique and built both in hybrid and MMlC form are presented.

1. INTRODUCTION

The nonlinear analysis of electronic circuits has been an active area of study for many decades now. Nonlinear circuit analysis differs from linear circuit analysis in that the transfer characteristic between input and output is a function of signal level. The implications of this are enormous. and result in significant changes between the circuit

description of a linear circuit compared to a nonlinear one. First, all nonlinear components in the circuit must have large-signal models which describe their transfer function over the desired range of operating levels. Second, the analysis must ac- count for any new frequency components which are generated. Third, information on the bias

Internationd Journal ot Microwave and Millimeter-Wave Computer-Aided Enpeering. Vol 1, No. 1, 22-37 (1991) L 1991 John Wile). & Sons. Inc

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Nonlinear Analysis Using Harmonic Balance 23

sources and drive level must be accounted for in the analysis. Finally, the solution may not be unique, stable, or even exist.

SPICE is a general time-domain tool for the analysis of transient effects in a circuit, and although it can indeed analyze steady-state operation by extending its transient analysis out to steady-state, it is very inefficient when it does so. Although SPICE has been used for nonlinear analysis for over 20 years, it is generally unsuited to the design of microwave and distributed RF nonlinear circuits. This class of circuits is inefficient for SPICE-like analysis because they generally operate at steady-state. In addition, SPICE is not an eifective tool for optimization or statis- tical analysis, as this requires sensitivity calculation and repetitive analyses. Finally, many microwave circuit elements are best modeled in the frequency-domain (e.g., dispersive transmission lines and microstrip discontinuities) and cannot be easily incorporated into SPICE.

Prompted principally by the emergence of device models for the GaAs FET, and more readily available computing power, it was only in the early 1980s that effort was applied toward increasing simulation efficiency for steady-state analog circuits. The harmonic balance procedure has emerged as a practical and efficient tool for the design and analysis of steady-state circuits with sinusoidal excitation. Although these circuits are a subclass of the circuits which can be analyzed by SPICE, it includes most circuits of interest to the microwave engineer. The harmonic-balance simulator can be up to two orders of magnitude more efficient than SPICE, and lends itself well to optimization as well as to analysis of circuits including amplifiers, oscillators, mixers, frequency converters, and numerous types of control circuits such as limiters and switches, if transient effects are not of concern. Another major advantage of the harmonic balance method is that it can be formulated in such a way that linear circuits can be of practically any size, with no significant decrease in speed if additional internal nodes are added, or if elements of widely varying time-constants are used (such is not the case with SPICE).

The purpose of this paper is to provide a tutorial study of the method and to review the fundamental achievements made during the last decade, including the extension of the method to quasi-periodic regimes.

Part I of this review paper presents a brief overview of time-domain analysis and a conceptual example of harmonic balance analysis, so the dif-

ferences between the two methods are clear. A more mathematical overview of how the harmonic balance equations are fomulated and their different solving schemes is discussed. In part I1 [l] the various methods for performing time-to-frequency-domain conversions in the case of several incommensurate tones are evaluated, and quantitative comparisons made. The other major cat- egory of nonlinear analog circuit simulation techniques is frequency-domain spectral balance. Such a technique treats both the linear and nonlinear circuits in the frequency-domain and so avoids aliasing that can result from the time-to-frequency-domain conversions. One of the more promising techniques of frequency-domain spectral balance methods, which are reviewed in an accompanying paper [2], is included in the comparisons. Finally, some practical examples of how the method is useful in the design of MMIC circuits are given.

11. TIME-DOMAIN METHODS

Time-domain methods analyze the nonlinear circuit by solving the nonlinear differential equations governing the circuit in the time-domain. In this section we examine three such techniques: direct numerical integration of the state equations, associated discrete circuit modeling, and shooting methods. Specific programs are not considered but the reader is directed to ref. [3] for a good survey of the more popular time-domain programs.

A. Direct Integration of the State Equations The most direct method for analyzing nonlinear circuits is numerical integration of the differential equations describing the network. By applying Kirchoffs voltage and current laws and using the characteristic equations for the circuit elements (frequently using the tableau or modified nodal formulations), the state equations can often be written in normal form [4-71

x = f(X, t ) (1)

where, for example, the time derivative of a quan- tity such as voltage or current is a function of time and the voltages and currents in the circuit. More generally the state equations are rearranged and written in the implicit form

g(X, x, t ) = 0 (2)

24 Gilmore and Steer

where

x = [ X , . X ? , X, , . . . . X,,]

is a set of voltages and currents, typically at different nodes and different time instants.

The equations above impose topological constraints on the circuit. Each element in the circuit must then be described by its constitutive relation. For nonlinear resistors, these take the form [5]

Charge and flux are used as additional variables to describe nonlinear capacitors and inductors, respectively. Nonlinear capacitors are described by a pair of equations

and nonlinear inductors by a similar pair

The solution is found by numerical intergration of eq. ( 2 ) typically using a predictor-corrector scheme [4-91. In this approach. the nonlinear differential equations are converted into nonlinear algebraic equations through discretization of the time variable. This is often done using the back- ward difference formula which approximates the derivative at the next time step. X(t,,-i), in terms of X ( t , , _ and the k past values X ( t , > ) . X(t,,-l). . . . . X ( t , , - L - ,) (the predictor). The resulting algebraic equations are solved iteratively, often using Newton’s method (the corrector). This is repeated until the solution is found for all time points t,. The resulting time-domain solution can then be converted into a frequency domain representation if required using a Fourier transform.

While this method is quite general, it has some limitations when applied to microwave circuit analysis. The selection of an appropriate time step is one such problem. Microwave circuits typically have widely separated time constants resulting in a set of stiff state equations. Having widely varying time-constants. the stiff state equations are no- toriously difficult to solve [4]. The consequence is that a small time step must be chosen and a large

number of iterations may be required to reach steady-state, leading to excessive computation time. Similarly, it may be difficult to identify the steady-state solution when widely spaced frequencies are present.

6. Associated Discrete Circuit Modeling A related method involves the use of associated discrete circuit models (also called companion models) [4] and is the technique used in the popular simulator SPICE [lo].

SPICE is probably the most common of the time-domain methods used for nonlinear circuit analysis. Its ready availability and maturity make it a well-known candidate. In particular, numerous vendors have proprietary libraries of both digital and analog ICs which considerably simplify the modeling of circuits. The ability of SPICE to represent arbitrary time-domain waveforms, and model time delays, level shifts, and sinking and sourcing aspects of such circuits, has made it over- whelmingly popular among digital designers concerned with analog issues. Its use for purely analog circuits is also widespread. Only when coupling, transmission line effects, or multitone excitation must be considered, do the weaknesses of SPICE become apparent.

SPICE models a circuit by representing the differential equations with which the components are modeled (the constitutive relations) as finite difference equations (hence, the term “associated discrete circuit modeling”); and solving the resulting set of nonlinear algebraic equations at each time sample iteratively. For instance, a capacitor with the constitutive relation

du i = c - dt

may be modeled in discrete time by

where the subscript n refers to the nth time sample. The sampling interval is At = r, - r n - L . A series of constitutive relations for each component in the circuit can then be derived, which when used in the topological constraints for the network (Kirchoffs voltage and current laws), form a set of algebraic equations in the component values and circuit variables. As for eq. (7), the right-hand side depends only on the component values and


the circuit variables at the previous time step. For transmission lines with a delay T, which is some multiple of the sampling interval t, - t n - l , the circuit variables must be kept over several sampling instants. Obviously, a set of initial conditions must be imposed on the network to define the initial values of the iteration at to.

Because the components are typically nonlin- early dependent on the circuit variables, the system of topological equations is an implicit function of the circuit variables and must therefore be solved numerically, typically using the Newton- Raphson method. For instance, if a nonlinear resistor is modeled functionally as i = f ( ~ ) , then at the kth step of the Newton iteration

from which a linear equivalent circuit can be derived

where g ( k ) is the differential conductance d f l d v and f i k ) a current source of value f ( v ( " ) - g( ' )dk) . This linear constitutive relation can be solved to satisfy the topological constraints and determine the electrical variables at the kth iteration. The linearization is repeated until the electrical variables satisfy the nonlinear circuit equations.

It should be apparent from the above that a problem can result from convergence difficulties, particularly when the initial conditions are poorly chosen. In the case of distributed elements with time delay, the voltage and current samples must be kept at all time instants back to the delay, T ,

of the longest line. The state of the circuit may be represented at any time by its state variables, which define the memory of previous events. This set typically includes the capacitor voltages, in- ductor currents, and the consecutive time samples of the voltages and currents at the ports of the distributed elements. Although the number of state variables may be minimized when Kirchoffs laws force some interdependence of the members of the set on each other, it is apparent that each distributed element requires 2 ~ / A t state variables to describe its state. Given that At must be chosen as small as possible to retain accuracy and prevent divergence, this can rapidly become very large for microwave circuits. There is no practical way to handle losses, parasitics, and dispersion for dis-

tributed elements, although convolution methods in the frequency-domain to determine the corresponding time-domain response are possible [ 11- 141.

When the initial conditions selected correspond to the physical turn on of the circuit, SPICE enables the build-up of transients to be readily ex- amined. However, when only the steady-state response is desired, this can be a burden, particularly when the differential equations representing the network are stiff. This arises in the case of high Q circuits in which the transients are not rapidly dissipated, or in circuits with blocking capacitors or bias chokes with time constants much longer than the period of excitation. A similar problem arises whenever the excitation is quasi-periodic, such as in a mixer, where the sampling instant must be related to the highest frequency component present, but the response is typically desired over the difference or lowest frequency component, which implies many sampling steps are needed for such circuits.

However, SPICE has been successfully used to simulate nonlinear microwave circuits [15-201. In ref. [15], for example, a microwave amplifier was simulated using SPICE. Valid results were obtained only after simulating the circuit for 10 periods of the 2 GHz fundamental. More complicated circuits require simulation of as many as 30 periods in order to reach the steady-state solution. In addition, a small time step must be used. The result is that a significant amount of computing power is required to solve a relatively simple problem. With multifrequency excitation it is particularly difficult to determine when steady-state has been achieved so that only relatively low dynamic range (i.e., the ratio of the largest frequency component to the minimum detectable component) can be practically achieved. Despite these difficulties, this method and the direct integration method have the unrivaled ability to calculate the transient or steady-state response for a complex nonlinear circuit.

C. Shooting Method For strictly periodic excitation, shooting methods are often used to bypass the transient response altogether [21-27]. An attempt to achieve this is made by correctly choosing the initial conditions so that transients are not excited. If x ( t ) is the set of state variables obtained by a time-domain analysis, the boundary value constraint for periodicity is that x ( t ) = ~ ( t + T ) where T is the period. A

26 Gilniore and Steer

series of iterations at time points between t and t + T can be performed for a given set of initial conditions, and the condition for periodicity checked. A new set of initial conditions can then be determined using a gradient method based upon the error in achieving a periodic solution. Once the sensitivity of the circuit to the choice of initial conditions is established in this way. a set of initial conditions that establishes steady-state operation can be determined; this set is, of course, the desired solution. The computation becomes further complicated when transmission lines are present, because functional initial conditions are then required to establish the initial conditions at every point along the line (corresponding to the delayed instants in time seen at the ports of the line).

Shooting methods are attractive for problems that have small periods. Unlike the direct integration methods. the circuit equations are only integrated over one period (per iteration). They are. therefore. more efficient provided that the initial state can be found in a number of iterations that is smaller than the number of periods that must be simulated before steady-state is reached in the direct methods. Unfortunately. shooting methods can only be applied to find periodic solutions. Also, shooting methods become less attractive for cases where the circuit has a large approximate period. for example when several nonharmonic signals are present.

II. MIXED FREQUENCY-AND TIME DOMAIN SIMULATION

SPICE is an example of a technique that operates entirely in the time-domain: Volterra series and power series methods, reviewed in an accompanying paper [2]. treat the entire circuit in the frequency-domain. The harmonic balance method formulates the system of nonlinear equations in either domain (although more typically the frequency-domain), with the linear contributions calculated in the frequency-domain and the nonlinear contributions in the time-domain. This is a distinct advantage for microwave circuits, in that distributed and dispersive elements are then much more readily modeled analytically or using alternative electromagnetic techniques based in the frequency-domain.

Harmonic balance methods are variations of Galerkin’s method [28-291 applied to nonlinear circuits. Galerkin’s method, first described in 1915, assumes a solution containing unkown coefficients

[30]. The assumed solution is substituted into the governing equations and the unknown coefficients adjusted so that the governing equations are satisfied as accurately as possible.

When the assumed solution is a sum of sinusoids, this procedure has been referred to as harmonic balance. The name appears as early as 1937 in the work of the Ukrainian scientists Kryloff and Bogoliuboff (translated into English in 1943 [31]) [32]. More recently, the method has been developed and applied to nonlinear circuits by Baily [33] and Lindenlaub [34] in the 1960s. The modern version of the harmonic balance method was presented by Nakhla and Vlach in 1976 [35]. They reduced the number of variables to be optimized by partitioning the network into smaller subnetworks composed of either linear circuit elements or nonlinear elements. The linear subnetworks are solved in the frequency-domain. Only the variables associated with the connection of the subnetworks need to be optimized. They called the resulting technique piecewise harmonic balance. In recent years, their method has been adopted and the adjective “piecewise” is usually dropped.

The harmonic balance method is an efficient method for the simulation of steady-state response because the form of the solution is imposed a priori onto the equations. A linear combination of sinusoids is chosen as the basis and the coefficients of those sinusoids then become the circuit unknowns. This differs from the time-domain method, where a set of time samples is chosen and tied to adjacent time samples only through the circuit equations. The advantage of making no assumption a priori about the functional form of the solution is that the waveform can have any spectral content (subject of course to the bandlim- iting effect of sampling). However, now a set of (time-invariant) state variables describes the circuit only at a single time instant.

In harmonic balance, imposing the condition that the circuit is already operating in steady-state enables a time-invariant set of variables to identify the circuit at all time instants. The set typically consists of the coefficients of the sinusoids for each circuit variable, i.e., their spectral or phasor content, although an alternative representation is a set of time points within a single period, for each variable.

A simple example of the harmonic balance method is given to illustrate conceptually the idea of mixed domain simulation. Although the example uses the relaxation method to solve for the solution, this is of historical interest as much of


A

NONLINEAR

2 ’ SUBCIRCUIT

Figure 1. Analysis of an FET by the harmonic balance method, showing the partitioning of the circuit into linear and nonlinear subcircuits, and the definition of the variables at the linear-nonlinear interface.

the original work in harmonic balance used such techniques [36-45).

In this example, the harmonic balance technique is presented in an iterative form. The method seeks to match the frequency components (harmonics) of current in a set of edges joining two subcircuits. Duality also applies to the technique, i.e., it can match the voltage on either side of a set of nodes, although only the former case will be considered in this example. The edges are chosen in such a way that nonlinear elements are partitioned into one subcircuit, and linear elements into the other. The edges at the linear/ nonlinear interface connect the two circuits and define corresponding nodes; current flowing out of one circuit must equal that flowing into the other. Matching the frequency components in each edge satisfies the continuity equation for current. The current at each edge is obtained by a process of iteration so that dependencies are satisfied for both the linear and nonlinear sides of the circuit.

Suppose the nonlinear circuit is represented by a nonlinear set of equations

where g is an arbitrary nonlinear function (and can include differentiation and integration), and iJ and uJ are the J th edge current and voltage, respectively. The dependent variables iJ are nonlinear functions of the independent variables uJ at some point in time Ts. Periodic, steady-state operation is assumed so that intergrals and derivatives at Ts may be determined.

The linear circuit may be represented by an N x ( N + M ) matrix, obtained by standard linear circuit analysis programs, e.g., ref. [46]. The M additional variables are the additional external nodes (or edges) at which applied voltages (or currents) are present. The linear circuit matrix is calculated at each frequency component present

in the circuit. In the case of an applied input signaI which contains harmonically related components at w, 2w, . . . , q w , there will be (q + 1) matrices relating the independent variables at each edge to the dependent variables:

X

fork = 0,1, . . . , q where the H,,(kw) are imped- ance or transfer ratios depending on which of the variables are voltages and which are currents. The purpose of the harmonic balance procedure is to find a simultaneous solution to eqs. (10) and (11) for u I , u2, . . . , u N , so that i,, i2, . . . , iN may be determined. Figure 1 illustrates the application of the technique to a three-terminal device such as a FET. Two edges constitute the FET gate input and the FET drain output, separating the nonlinear FET elements into one subcircuit and the parasitics, matching, and output networks into another (linear) subcircuit. The third edge is the source of the FET and is chosen as the reference, so that N = 2. Here u , and u2 are the independent variables; i , and i2 are the dependent variables. Additional applied inputs are the external voltages V , and V,. The desired output variables such as the current and voltage in the load can be found once il and i2 are determined.

Eq. (10) is stated in the time domain and eq. (11) in the frequency domain but simultaneous solution requires that they be expressed in the same domain, usually the frequency domain. Time- to-frequency conversion is most conveniently achieved using the discrete Fourier transform (DFT) which involves matrix multiplication and

28 Gilrnore and Steer

involves O(N?) operations where N is the number of frequencies. However. if the number of frequency variables is an integer power of two, the numerically efficient fast Fourier transform (FFT) , an O(N log N ) algorithm, may be used. The FFT algorithm is so efficient that the number of dependent variables is often extended so that the FFT rather than the DFT algorithm can be used. If estimates of u,(t) for .I = 1, . . . , N at some time TS are substituted into eq. (lo), i, can be found at time Tr. If this is done at time instants Ts . 2Ts. . . . . LT,y an L-point sequence of time samples of i, results. The Nyquist sampling theorem states that if a sequence of points is obtained by sampling a waveform at a rate that is at least twice the highest frequency component contained, the original waveform can be reconstructed. If the waveform contains only discrete fequencies which are spaced by integral multiples of w , up to qw, one can set Ts = ( 2 ~ ) / [ ( 2 q f l ) w ] with L = (2q + 1) to satisfy the Nyquist criterion. and can extract the desired frequency components at w from the L-point sequence by using the discrete Fourier transform.

An initial estimate must be made for i, and u, because they are not known a priori. Iteration between eqs. (10) and (11) is performed using the DFT or FFT to obtain the frequency components from the time samples obtained from eq. (10) until a setf-consistent set of variables (i.e.. those which satisfy the current continuity equations) is at- tained. The algorithm used in the analysis is given below:

1. Initial guesses are established for the current phasors c ( k w ) at the interface edges at the DC, fundamental, and harmonic frequencies ( k = 0, 1, . . . , q ) . The overbar refers to the current flowing in the linear “side” of the interface edges.

2. The hybrid matrix for the linear circuit H(kw) is calculated at DC, the driving frequency - w , and each harmonic. This is used with i ,(kw) and the applied external voltages in eq. ( l l ) , to calculate the unknown phasor components of voltage at each of the N edges.

3. Using an expression u,(t) = Real[E;=, u,(kw)&”] to derive the time value of the edge voltages at times t = Ts. 2T,, . . . , LT,. and a similar expression for derivatives, the time samples of voltage and its derivatives may be calculated at each of the N edges.

4.

5 .

6.

7.

Values of i,(t) in the nonlinear “side” of the interface edges may be obtained at corresponding time instants by substitution of the time samples of voltage u,(t) and its derivatives into eq. (10). Using the DFT or FFT, the harmonic phasor components i ,(kw) may be extracted from the L-point sequence of i,(t), if the sequence consists of samples obtained at the Nyquist rate. An error function is formed to compare the “nonlinear” current estimates iJ with the “linear” estimates so that

4

= c (li,(kw) + <(kw) /* (12)

+ . . . + liN(kW) + &(kw)l2)

k = O

The continuity equation for current states that the “nonlinear” currents must equal the “linear” currents. This corresponds to zero error function as a solution. The error function is minimized by forming new initial guesses for the current phasors i ( k w ) from the old estimates, and repeating steps of eqs. (2)-(7) until the error function lies below some threshold. At this point, the linear and nonlinear partitions give self-consistent results, since the currents on each “side” of the interface edges are equal. The quantities i, and u, are thus determined, and the voltage (or current) can be found at any desired node in the circuit (e.g., at the load) by linear analysis.

The fixed-point relaxation method (also known as the “p” factor method) of Hicks and Khan [38] can be used in this example to achieve convergence and force the error function to zero, by allowing the phasor currents to more closely approximate their true values on successive iterations. After the jth iteration of the loop, consider the “nonlinear” current in the Jth edge i f”( t ) = Eki$(kw)elkw‘, with the corresponding “linear” current Gj( t ) = EkGJ(kw)eJko‘. The next iteration - is then carried out with G(’+l)(kw) formed by i , IJ+’) (kw) = p ( - i$(kw)) + (1 - p)GJ(kw) where p is determined by convergence considerations and 0 < p 5 1. Hicks and Khan and other authors [37, 41 1 have investigated various criteria for conver-


gence. Kerr [36] used a multiple-reflection method which was shown by Hicks and Khan to be a spe- cial case of the p factor method. Camacho-Pen- alosa [41] developed an algorithm for determining the optimum p factor. Although fixed point iteration is very simple and quite efficient, it has poor convergence properties dependent on the imped- ance ratios (at each harmonic) between the linear and nonlinear circuit partitions. Hicks and Khan proposed the use of identity networks at the interface to modify this ratio and speed convergence. An identity network is a shunt connection of two parallel impedances of equal and opposite sign (or of two series impedances for a series connection), one element of which is lumped with the linear network (the Y matrix), and the other with the nonlinear network (with I ( V ) ) . The method also requires that each node be connected to at least one linear element, for the presence of a floating node would prevent the voltage at that node from being iterated as part of the linear circuit [i.e., in the left-hand side of eq. (19)]. Con- vergence properties are particularly poor with multifrequency excitation.

The example above illustrates the partitioning between the linear and nonlinear sides of the circuit. The linear side is analyzed efficiently in the frequency-domain, and the nonlinear side in the time-domain. The discrete Fourier transforms or fast Fourier transforms were used to “mesh” the two domains, although we will see shortly that other methods can be used where noncommen- surate sinusoidal signals (those with no integral multiple of a common period) are present in the system. A hybrid “H” matrix allows either voltage or current to be the independent variable, and allows nonlinearities which are either voltage-controlled or current-controlled to be incorporated.

A. Posing the Circuit Analysis Problem Although there is no standard nomenclature to describe the harmonic balance formulation, that in ref. 1471 will be used here. This formulation is sometimes known as a nodal formulation because the partitioning of the circuit into linear and nonlinear subcircuits is not necessary. Every node in the nonlinear subcircuit is, therefore, considered to be connected to the linear subcircuit.

If the total circuit has N nodes, and if u is the vector of node voltage waveforms, then applying Kirchoffs current law (KCL) to each node yields

a system of equations

where we have chosen the nonlinear circuit to contain only voltage-controlled resistors and capacitors for representational ease. The quantities i and q are the sum of the currents and charges entering the nodes from the nonlinearities, y is the matrix impulse response of the linear circuit with all the nonlinear devices removed, and is are the external source currents.

In the frequency-domain, the convolution in- tergral maps into Y V , where V contains the Four- ier coefficients of the voltage at each node and at each harmonic, and Y is a block node admittance matrix for the linear portion of the circuit. The system of eq. (13) then becomes, on transforming into the frequency-domain

(14) F ( V ) = I ( V ) + RQ(V)

+ Y V + f , = O

where is a matrix with frequency coefficients (terms such as j k o ) representing the differentiation step. The notation here uses small letters to represent the time-domain waveforms and capital letters the frequency-domain spectra. This equation is, then, just KCL in the frequency-domain for a nonlinear circuit. We have implicitly used Parseval’s theorem in implying that the total en- ergy of the system is equivalent when calculated in either the time- or frequency-domain. Har- monic balance seeks a solution to eq. (14) by matching harmonic quantities at the linear-nonlinear interface. The first two terms are spectra of waveforms calculated in the time-domain via the nonlinear model, i.e.,

F( V ) = !fi(!f-’V) + fln:fq(:f-lV) (15) + YV + z,

where Y is the Fourier transform and Y - ’ is the inverse Fourier transform.

6. Formulation of the Harmonic Balance Equations The harmonic balance approach requires the nonlinear devices to be algebraic, i.e., without mem-

30 Gilmore arid Steer

ory. This is a requirement to ensure that the response of the nonlinear devices is quasi-periodic. This is not a restriction on nonlinear capacitances and inductances because they may always be mapped into a combination of a purely linear re- active component and a nonlinear resistance. This can be simply seen because if i = C(i , u ) d v / d f , the derivative is an algebraic operation in the frequency-domain (the jo term simply becomes a multiplicative coefficient of the original waveform). and hence, the current phasors can be determined given all of the voltage phasors and their frequencies.

The mathematical approach described in the section above is a nodal form of the harmonic balance equations, in which the circuit unknowns are the voltages at every node in the circuit. This method has advantages when the number of nonlinearities in the circuit is large. or similar in size to the number of linear elements. An alternative approach [45.49,50] is the piecewise harmonic balance method, which is based upon segmenting the circuit into a purely linear part, connected at a series of ports through which the remaining nonlinear components are interconnected to the linear subcircuit. This approach is conceptually simpler because the linear circuit can be collapsed and represented by a reduced Y matrix in which only the ports connected to the nonlinearities appear as external ports. The circuit unknowns then become the port voltages and currents at the linear- to-nonlinear interface.

Both methods would appear to require that the currents entering any node from a nonlinear element can be explicitly expressed in terms of the nodal voltages. However. the piecewise harmonic balance approach can be formulated to remove this restriction by defining state-variables rather than nodal voltages as the independent variables [49]. Modified nodal analysis also removes this constraint. In any case, however. the system of nonlinear equations that must be solved is of the form

where X is a vector of unknowns. typically the voltages at each frequency component and node considered in the system. A solution to a set of equations of this form can be found in several different ways. Each equation in the system of equations has the dimensions of current, and rep- resents either the current flowing into a node or

the current flowing in a branch joining a linear subnetwork with a nonlinear subnetwork.

It is worthwhile remembering that independent of formulation, the harmonic balance equations discussed so far equate the sum of all the currents flowing into some set of nodes. It is a nodal formulation based on Kirchoffs current law. Just as for linear simulators, a mesh formulation based solely on Kirchoffs voltage law is also possible but has not been implemented in any programs of which the authors are aware.

Rizzoli et al. [49-531 use a state-variable approach to overcome the problem of requiring the nonlinearities to be expressed explicitly as voltage-controlled current sources. Although the voltages and currents at the nonlinear ports must ul- timately be evaluated to use in the harmonic balance equations, they need not be explicitly dependent on each other, but may be related through an alternate set of independent control variables, referred to as the state-variables of the circuit. The state variables can then be embedded inside the nonlinearity rather than at the nonlinear ports. The program then iterates on the state-variables (rather than interface currents, as described in the initial example), and calls the nonlinear models with time-samples of the state-variables. The model then returns the interface (terminal) voltage and current at the same instants of time, which it cal- culates from the values of the state-variables and the components. These interface quantities are used within the linear subcircuit, but it is the values of the state-variables which are adjusted to satisfy eq. (16).

For example, consider the nonlinear model of a diode in Figure 2, which has a nonlinear capacitor and nonlinear resistor in series, and which may be embedded within a larger circuit. Choos- ing the junction voltage u, as the state-variable allows us to write a very simple user-defined model as:

du dt

il = C-’ + iF u , = ui + Rsil

Figure 2. Simple linear model for a microwave Schottky diode. i F is the ideal diode current.


with Rs = Ro - 7/C(ui) and where il and u1 are the interface current and voltages calculated within the model to use within the harmonic balance algorithm. This model is called repeatedly from within the harmonic balance program with u, as input, and il and u1 as output. The state- variable approach has allowed two nonlinearities to be connected in series, which would not be possible if the method required il to be calculated as a function of u i . This situation can arise even in very simple device models, such as a Schottky diode which has a nonlinear space-charge capac- itance in series with a nonlinear spreading resistance.

Proper choice of state variables removes the restrictions that would otherwise arise in the piecewise harmonic balance method. With an im- proper choice every nonlinear port current is required to be an explicit function of the port voltage which would then have to be an independent variable. The ability to use implicit equations to describe the nonlinear equations is a major advantage of the piecewise approach and allows much greater generality in modeling devices, because either voltage or current may then be the con- trolling parameter.

An alternative way of representing the harmonic balance equations is in the time-domain [54-561. (This approach has been referred to as time-domain harmonic balance [%I.) Rather than represent the desired solution by the coefficients of the imposed linear combination of sinusoids, the response can be represented in sampled data form. Thus, the time-domain determining equations are satisfied for discrete time samples. Start- ing from eq. (14), consider the nth node. Then, conversion to the time-domain is achieved by mu- tliplying through by the matrix which performs the inverse Fourier transformation r-l:

(17) + r-I C Y,,,v, + r-izs.fl = o N

m = l

Recognizing the inverse transformation in these terms results in

m = l

This is a time-domain form of the harmonic

balance equations which can be solved using a finite difference approach. The coefficients which multiply qn and v, are constants and can be computed just once. This approach is sometimes known as the waveform balance method [56], in which the unknown quantities are the samples of the waveforms themselves. However, the approach is mathematically identical to achieving balance in the frequency-domain, although for a nodal approach the Jacobian matrix needed for solution will be dense, because unlike the harmonic components at purely linear nodes, the waveform samples are not orthogonal and sensitivity between adjacent samples is therefore nonzero. Conse- quently, the waveform balance method is numerically inefficient and only practical when the waveform is periodic. For the periodic case good results were reported by Hwang et al. for a monolithically integrated amplifier [56]. It can be expected to be impractical with multitone excitation as then the Jacobian will be very large as well as being dense, so that the amount of linear algebra will be pro- hibitive.

C. Solution Strategies for Solving the Equations The solution of the set of eqs. (14) can be obtained by several methods. One method, known as relaxation, uses no derivative information and is relatively simple and fast, but is not robust. Alter- natively, gradient methods can be used to solve either a system of equations (e.g., using Newton- Raphson) or to minimize an objective function using a quasi-Newton or search method.

Relaxation Methods This method was the one used in the example. Relaxation methods use fixed point iteration to solve eq. (14). Rewriting eq. (14) yields

where real and imaginary components are now being separated to eliminate complex quantities.

The fixed point relaxation method was first used by Kerr [37] and later by Hicks and Khan [38] in determining the local oscillator waveforms at mixer diodes. It has also been used in the harmonic balance procedure to analyze MESFET amplifiers [43,57,58]. The relaxation method, when successful, can result in much faster computation


times than gradient methods. This is partially due to decoupling of the frequency components, as their values can be computed separately in a nonlinear function, and they are held constant when applied to the linear network. The interaction of the harmonics is associated only with the calculation in the nonlinear network, because the linear network is independent at each harmonic. Its main advantage is that a Jacobian is not required with consequent reduction in memory and CPU usage.

Optimization and Minimization Methods This method uses an optimizer to minimize F(V)F*(V) . i.e., it seeks a least-squares minimum of an error function representing the residual currents flowing into each node. The optimizer will typically be a global optimizer that uses a quasi-Newton method. The method tends to be inefficient because information about each of the contributors to the error function is lost in the calculation of the sum of the squares. However. the advantage of minimizing an objective function using an optimizer is that any arbitrary circuit parameters (such as power or frequency) can be added into the objective function and so be optimized simultaneously [49].

This method is particularly cumbersome for circuits with a large number of nodes and harmonics. As with all the harmonic balance methods, the number of nodes used in eq. (16) can be reduced by "burying" internal nodes within the linear network, which then becomes a single multiterminal subcircuit as far as harmonic balance is concerned. The system of equations in (16) is then reduced accordingly. Once the "interfacial" node voltages are known. any internal node voltage can be found by using simple linear analysis and the full Y matrix for the linear circuit.

Newton Methods This method has the best convergence properties of all the methods and, provided that the initial solution guess is close. the rate of convergence is quadratic (i.e.. at each subsequent iteration the error is the square of the previous error).

If j denotes the j th Newton-Raphson iteration, and F(V(1)) = 0 is a solution to eq. (14), then expanding F as a Taylor's series gives

or

or

where J ( V"!) is the Jacobian. As before, the node voltage spectra are trans-

formed into the time-domain, applied to the nonlinearities, and the resulting current waveforms converted back into the frequency-domain. JF( V ) is the Jacobian

dQ J A V ) = - d F = - + 0 - + Y . (23) l3V av av Now I and V are separated into real and imaginary parts as before and so Jd V ) is organized as a block matrix. Each block is itself a 2 x 2 matrix, representing the partial derivative of the real and imaginary parts of F at one node with respect to the real and imaginary parts of the voltage at another node, over all harmonics of the voltages. J F is of order 2HN by 2HN, where H is the number of harmonics and N is the number of nodes. The calculation of J F is quite complex if it is performed numerically (using numerical differences) as in refs. [48,53,59-621. Forming the Jacobian numerically can consume the majority of the computation time. However, it can be calculated analytically as in refs. [63-671 and as proposed in ref. [68].

The Newton-Raphson method is the most com- monly used approach in harmonic balance simulators, and generally the best one to achieve convergence. A major drawback can be the amount of memory required to store and invert the Ja- cobian matrix, and the speed with which it can be inverted. The methods fail if the Jacobian is sin- gular. However, the Jacobian itself is very useful as its properties enable it to be used for other purposes as well. In particular, the Jacobian can be used in the determination of stability, in calculating the conversion matrix for mixers, and in the adjoint technique for calculating the sensitivity of nonlinear circuits.

It should be noted that the system of equations must be square in order to invert the Jacobian. Otherwise, the system must be solved using a nonlinear least-squares approach such as the Gauss- Newton method:


The Jacobian will not be square when additional constraints, such as optimization criteria, are lumped into the harmonic balance system of equations, as in [68]. In such a case, a nonlinear least-squares approach such as that above, or a quasi-Newton or direct search approach must be used. The former approach is not preferred because the Jacobian is dense and possibly quite large.

The Jacobian can be formulated either analytically or numerically. Numerical calculation of the Jacobian can be effective when the number of nonlinearities is small, but for large circuits, small sensitivities between harmonic components of state variables at different ports can result in small numbers (comparable with the machine precision). This causes ill conditioning for matrix inversion, resulting in inaccurate updates of the unknowns and a large number of iterations and possibly noncon- vergence. Analytical calculation of the Jacobian involves considerable calculation and transformation, but is the preferred method and still requires less computation than numerical evaluation of the Jacobian. The Jacobian matrix is

dF J = - av and consists of terms Jmn( V ) such as

where m and n are node numbers, and

Now, dZm/dV, is the derivative of the harmonic component of current at the mth node with respect to the harmonic component of voltage at the nth node. The derivative can only be formulated analytically in the time-domain using the chain rule and

where r is again the matrix multiplier representing the DFT. So

Now i (u ) is just the known nonlinear time domain function for the model, and if it is algebraic, its partial derivative can be found analytically. Fi- nally, using the fact that r-’V,, = u,, we have

This is computed as follows [69]. First form the vector

d = [(di/du),] = ( d i ( ~ , ) / d ~ ( ~ , ) , . . . , (31) di(T,) / d U ( T , ) } .

If we define r = [ x i ] (i.e., a matrix with elements y l i ) and r-’ = [rim] we form the vector d = [(di/ ~ u ) ~ Y , ~ ] with m held constant. Now

T a = {dZ,/dV,, . . . , dIN/dV,} (32)

that is, we have a vector of the derivatives of the current phasors at all frequencies with respect to the voltage at the fth frequency.

To accelerate convergence, the Jacobian can be reused in a “chord” iteration. The combination of a Newton ieration with a number of chord iterations is known as the Samanskii method. In fact, for the nodal harmonic balance method, the Jacobian is sparse, as there is no harmonic coupling between components for those nodes to which no nonlinearity is attached, i.e., terms such as aim/du, are zero for m # n for a purely linear equation. The sparsity of the Jacobian can be uti- lized to simplify the inversion process and storage requirements. For the piecewise harmonic balance method, however, every port is connected to a nonlinear device, so that the Jacobian matrix is dense. However, with this approach, the number of circuit variables is correspondingly reduced, because the linear network has been collapsed into a port representation. Filicori et al. [70] compare the computational differences between the nodal method and the circuit partitioning approach of the piecewise method, and conclude that the adoption of sparse techniques with the nodal method of harmonic balance for calculation of the Jacobian is justified only when the number of nonlinear elements is relatively large (greater than approximately 10) and when the ratio between the numbers of the linear and nonlinear branches is quite small.

Intermediate between the Newton and chord iteration methods is the block Newton method [64,71]. Here terms relating different frequency


components are deleted from the Jacobian. Thus, the Jacobian is block diagonal (i.e., it consists of submatrices arranged along the diagonal) and inversion of the Jacobian requires inversion of the much smaller submatrices. The inherent assumption in this process is that the various frequency components are weakly coupled. This approxi- mation is usually valid upto several decibels of gain compression. It is usually not adequate for such strongly nonlinear circuits as diode ring mixers. The block Newton iteration scheme can be combined with the chord method for a further reduction in computational complexity.

Like time-domain methods. convergence is dependent on the initial starting point used for the solution. Continuation methods [59,72,74], seek to approach the desired solution point by incre- mentally following a path from a known starting point. A continuation parameter +I is a single parameter that is varied in a continuous fashion between the two endpoints. Typically, a continuation parameter is chosen as some circuit excitation. such as the input power to an amplifier, although frequency or bias is also sometimes used [73]. A two-step procedure can be used, in which all DC and RF sources in the network are first set t o zero. The first step traces the solution from the known off state to one in a DC steady-state; the second step will trace the solution from this known state by increasing the RF sources to their final value. A t each point along the path, a solution is sought prior t o moving to the next point. This is known as a natural embedding of the continuation parameter, as each solution has a physical meaning. Solution at subsequent steps on the continuation curve is achieved simply by using the previous solution as the initial conditions for the next step. If a solution is not achieved at the next point, step reduction may be used, i .e. , the step size is cut in half repetitively until a solution is achieved. The step is again doubled in turn after a solution has been achieved.

Such a strategy has proved extremely useful in coaxing convergence from highly nonlinear circuits, such as power amplifiers driven well into saturation. In the examples section of this paper, sweeping the incident driving power up to the desired level enables a solution to be achieved even in saturation. If the initial conditions were otherwise unknown, convergence would have been most unlikely had no sweep been performed.

Continuation methods can be used to identify turning points and multiple solutions. For many classes of circuits, such as oscillators and fre-

quency dividers, the circuit response will show hysteresis with the tuning parameter. Several in- teresting examples a re given in refs. [68] and [74].

ACKNOWLEDGMENTS

This work was supported in part by a National Science Foundation Presidential Young Investi- gator Award Grant No. ECS-8657836 to M. B. Steer.

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BIOGRAPHY

Rowan Gilmore obtained the BE degree with First Class Honors from the Univer- sity of Queensland, Brisbane, Australia in 1976 and his DSc degree from Washington University in St. Louis in 1984. His graduate work involved GaAs FET modeling and the development of several original techniques used in the harmonic balance simulation of nonlinear circuits.

From 1976-1978, he was a project engineer with the Over- seas Telecommunications Commission (Australia). After ob- taining his graduate degree, he was a Senior Engineer with Central Microwave Co., developing microwave power amplifiers and oscillators. In 1985, he worked at Schlumberger Wire- line Services in Houston, Texas, where he was project manager responsible for the development of a wireline logging tool to determine the dielectric constant of downhole formations.

Since 1987, Dr. Gilmore has been with Compact Software, where he is presently Vice President of Engineering. He has been responsible for much of the development of their har- monicbalance-based software. He has authored over 20 papers on microwave circuit design. His research interests are in circuit simulation and device modeling

Michael Steer received his PhD in Elec- trical Engineering from the University of Queensland, Brisbane, Australia, in 1983 and is currently Assistant Professor of Electrical and Computer Engineering at North Carolina State University.

His research involves the simulation and computer-aided design of nonlinear microwave circuits and systems. He is currently

working on the simulation of microwave analog circuits, of high speed printed circuit boards and multichip modules, parameter extraction, computer-aided design of microwave circuits using simulated annealing, and millimeter-wave quasi- optical techniques. He presents a course on the National Tech- nological University entitled “Computer-Aided Circuit Anal- ysis” which is broadcast throughout the US. He has published more than 20 journal papers on nonlinear microwave circuit analysis, simulation of delta-sigma modulators, high frequency limits of transistors, microwave measurements, and equivalent circuits of diode mounts.

Dr. Steer is a senior member of the Institute of Electrical and Electronic Engineers and active in the Microwave Theory and Techniques Society. In 1987 he became a Presidential Young Investigator.

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