THE harmonic balance (HB) method is probably the

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1 ECE 5 MODELING OF MULTIPHYSICS SYSTEMS The Haronic Balance Method Hans-Dieter Lang, Xingqi Zhang (ECE 5 project report, Spring ) Abstract The haronic balance ethod is used in ost current coercial RF siulation tools. This is due to the fact that it has certain advantages over other coon ethods used in SPICE, naely (odified) nodal analysis (MNA), which ake it better suited to siulate stiff probles and circuits containing transission lines, nonities and dispersive effects. In this project, the haronic balance algorith is derived and explained; a basic haronic balance solver is developed and ipleented in Matlab and applied to soe basic and non circuits. Index Ters haronic balance, non circuit, rf circuit I. INTRODUCTION THE haronic balance (HB) ethod is probably the ost coon technique to siulate icrowave and RF circuits, since it can efficiently account for transission line and non effects, operating in a periodic or quasi-periodic steady-state regie. It is known to be fast, effcient and versatile. With it being a (hybrid) frequency-doain ethod, dispersive effects can also be included and it coonly the ethod of choice for the optiization of circuits and circuit paraeters [5], []. The HB ethod is used in the industry s leading icrowave and RF siulation tools, such as Agilent s ADS/Genesys, AWR Microwave Office, Ansoft s Designer/Nexxi and Cadence Virtuoso Spectre. Nowadays, when people talk about haronic balance ethods, what is eant is what was originally known as piecewise haronic balance ethod [], thus in this project only this version will be considered. A. Tie-Doain Analysis The odified nodal analysis (MNA) ethod [], ost coonly used by SPICE-like circuit siulators, is a wellknown and powerful technique to for a atrix equation syste of the for Gx + Cẋ = Bu () fro a given circuit in the tie-doain. The ter odified refers to the fact that x does not contain only node voltages but also certain currents, to overcoe the shortcoings of the standard nodal analysis ethod and be able to siulate ideal voltage sources and inductors. Thus, MNA and tie-doain solutions in general are able to siulate circuits containing all kinds of luped circuit eleents but it has soe shortcoings: [] [5] Hans-Dieter Lang (ID 8599), and Xingqi Zhang, (ID ) are PhD and aster students, respectively, with the electroagnetics group, part of the ECE departeent of the University of Toronto, Ontario, Canada. Contact: {hd.lang,xingqi.zhang}@ail.utoronto.ca extending transient analysis out to steady-state is coputationally inefficient issues with stiff probles (poor convergence, long siulation tie, etc.) dispersion is hard to ipleent (since frequency-doain described odels cannot be ipleented) not effective for optiization and statistical analysis. These weaknesses only appear in special cases, including icrowave circuits (especially those including transission line effects) resulting in stiff probles (where usually several periods, e.g. to [] periods are neccessary for steadystate approxiations; for high-q problees even any ore), switched capacitor and crystal/cavity filters and ultitone excitation. B. Frequency-Doain Analysis Frequency-doain ethods usually solve for steady-state solutions by inverting a atrix equation syste siilar to [ G + jωc ] x(ω) = Bu(ω) () where each the solutions are obtained for each point in the frequency directly. For a sall nuber of frequencies this can therefore be considerably ore efficient than using tiedoain ethods, since the dynaic operations such as differentiation and integration becoe siple algebraic operations [5]. However, with the standard principle only solutions of tie-invariant circuits can be obtained. C. Haronic Balance A Hybrid Method The haronic balance ethod is a hybrid tie- and frequency-doain approach, which allows all the advantages of non tie-doain device odelling, cobined with the strength (naely efficiency) of the steady-state frequencydoain technique. The nae stes fro the idea of balancing the currents obtained fro the non subcircuits (via tie-doain) and the currents fro the subcircuits at the subcircuit interfaces, in the frequency-doain, as illustrated in Fig.. As discussed in the lectures, powerful odel-order reduction (MOR) techniques were developed to overcoe these probles by approxiating only parts of interest. However, in this project only full-order probles (not approxiated) are considered. The balancing can also be done in the tie doain, which is usually called tie-doain haronic balancing or wavefor balancing [5], which however is not considered any further, here.

2 ECE 5 MODELING OF MULTIPHYSICS SYSTEMS frequency ω s is the lowest frequency in the syste (apart fro DC) and thus, usually ω = ω s is used. However, ω could also be chosen as / or /ω s, etc. In this project report the following notation will be used for the discrete Fourier transfor (DFT, FFT): i subcircuit ^i non subcircuit Fig. : Illustration of the haronic balance principle: the algorith tries to balance the currents (haronics) fro the and non subcircuits in the frequency doain by iteratively coparing the and applying adequate changes. D. Alternatives There are only a few alternatives to tie-doain transient ethods (possibly using a prohibitively sall tie step) and the haronic balance ethod, for exaple: Volterra series [6], which decopose the nonity into a weighted power series. This is only suitable for weakly non probles, where only a few higher-order ters are sufficient to discribe all effects. Shooting ethods [], where the transient response is bypassed by iteratively choosing the initial conditions to achieve steady state. Due to efficiency considerations, this is only suitable for periodic probles with a sall period. E. Notation Throughout this docuent italic letters refer to scalar variables, sall bold letters to vectors and capital bold letters to atrices, everywhere. For the voltages and currents, the sae sybols are used for tie and frequency doain, i.e. v, i (and x as in Eqns. () and ()); the distinction will either becoe apparent fro the context or the respective doain is specified in brackets, whenever necessary. The quantities resulting fro nonities are set with a hat, e.g. ^i, to separate the fro the quantities (appearing at the sae node). A. Preliinaries II. BALANCING THE HARMONICS HB uses the assuption that the total voltage can be (approxiately, but accurately enough) coposed of the finite Fourier series K = v(k) e jkωt K n, n N () k= K+ with the phasors v(k) C. Thus, the signal is periodic v(t + T ) = () with period T = π/ω. In the single-tone cases with siple nonities it is usually assued that the excitation There are also forulations for alost periodic functions, see [9], []. v(ω) = F (5) where using the twiddle factor w = e jπ/k the DFT atrix of diension K K is w w w k w K w F = w w k w (K ) K..... w k w k w k w k(k )..... w K w (K ) w k(k ) w (K ) and siilarly = F v(ω). Further, using the fact that F = KF * (6) F has never actually to be inverted. Note that only for this derivation the DFT is stated in atrix for; in the later ipleentation the FFT algorith is used. Assuing that the output signal is real, we know that its spectru is Heritian; thus we only have to consider K + frequencies (K haronics and DC), where for axiu efficency of the FFT algorith K should be chosen as n with n N. B. Non Systes The ost general for of equation to odel non circuits is [] d q(v) + i(v) + u = (7) dt where the fact that v = and u = u is oitted to keep the equation less crowded. Note that both q( ) and i( ) can be non functions; the first aps the vector v consisting of ostly voltage potentials to a vector whose entries are ostly sus of capacitive charges or inductive fluxes and the latter ostly accounts for conductive and non source effects. If q( ) is (i.e. there are no non capacitive or inductive effects ) and i(v) = Gv + can be split in a and a non part this results in the non equivalent of Eqn. (): C + G + g() + u = (8) Within this project only this type of non systes will be considered. Non inductive effects are rarely used in icrowave circuits [] but non capacitors are coonly found, ostly in for of varactors.

3 LANG / ZHANG: THE HARMONIC BALANCE METHOD C. Derivation The following brief derivation focuses on the circuit shown in Fig. but can easily be generalized. The derivation is necessary since the forulas in [6] are incoplete, see to contain soe errors and did not lead to a useful algorith. The underlying idea is siilar, but follows ore the one in [], with a different attept at the Jacobian. excitation R i s i v = i ^i v non Fig. : Siple circuit consisting of three parts. The currents fro the non and subcircuits (sides) at node have to be balanced, eaning the sae (but of opposite sign), in for all haronics in the frequency doain k [,, K]: i (kω ) + ^i (kω ) = k i (ω) +^i (ω) = (9) ) Linear subcircuit: The current is given by i (ω) = Y v i (ω) +Y v = Y s +Yv. () i s(ω) and since the node nuber is the only reaining node, this specification will be oitted to siplify notation. Thus stands i(ω) = Y s + Yv = i s (ω) + i lin (ω). () ) Non subcircuit: The non current ^i(ω) is obtained by taking the inverse Fourier transfor of the (presuably not yet correct) node voltage in the frequency doain, applying the nonity in the tie doain, and transforing back to the frequency doain. Note that for this procedure, the voltage vector is used in full (not just K entries) using the conjugate-coplex values at the right spots and thus K tie saples are obtained: v(ω) v K * (ω) R K = F. v * (ω) v(ω) C K+ (5) ) The algorith: The HB algorith works as illustrated in Fig., naely: ) Initial guess v (could also be in frequency doain). Most coonly a zero initial value is used, however also daped versions of the excitation and ipulses (or noise, with broad spectral content) can be good start values 5. ) Apply nonity in the tie doain, either via non conductivity ^i = or directly as voltagedependent current as for the diode current in this case: ^i(v) = ^i d () = I s ( e v/v T ) (6) in the actual vector for ^i = ^i (v ) (7a) = ^i d (v ) = ^i d ( F v (ω) ) (7b) ) Go back to frequency doain ^i (ω) = F ^i = F ^id ( F v (ω) ) (8) The source coponents are Y () Y (ω )... i s (ω) = Y s = Y(kω)... Y(Kω). ) Check if the haronics are balanced using the cost function in the frequency doain f(v ) = i (ω) +^i (ω) (9a) = Y s + Yv + F ^i d ( F v ) (9b) 5 Especially when considering oscillators (without excitations), as shown later, zero initial guess leads to wrong (or rather the trivial zero) solutions = Diag[y (ω)] () and the coponents due to the interface voltage v siilarly, are, i lin (ω) = Yv = Diag[y (ω)] v () where Y and Y are the respective (generally frequencydependened) entries of the adittance atrix of the subcircuit (in the frequency doain, thus a vector of K + entries). Diag[ ] is an operator putting the input vector in diagonal atrix for. And the vector of unknowns in the frequency doain is v = [ v(), v(ω ), v(ω ),, v(kω ),, v(kω ) ] T () Initial guess v Nonity ^i = v Tie doain v ^i F F Frequency doain v (ω) i (ω) ^i (ω) Update v + = v J f(v) f(v) < ε? Fig. : Algorith working principle. converged v(ω)

4 ECE 5 MODELING OF MULTIPHYSICS SYSTEMS ) If the convergence criteria were not satisfied, updated the voltages using the cost function and its Jacobian (still in the frequency doain): v + = v J f(v ) () 5) Start the process over, by going to the tie doain, to apply the nonity once ore. The ain proble is finding the Jacobian: J = df(v) dv v=v (ω) J ij = f i(v) Following the previous notation, the Jacobian can be found colun-wise according to (for the jth colun) J j = f(v) = Y j + ^i(ω). () The derivative of the non current can be found using in the definition fro Eqn. (7b) [ F v ^i ] d (F v(ω)) j = F ^id (F v(ω)) () where the place holder τ = F v(ω) is introduced to solve the partial derivative of the non diode current according to (F v(ω)) = ^i d (τ) τ (a) ^id τ τ=f v = ^i d(f v(ω)) * F j (b) where F j is the jth colun of the IDFT atrix and the asterisk denotes a point-wise ultiplication (not a convolution). The derivative of the current can be found analytically in this case, since the diode current is known in for of a siple exponential function; thus ^i d can reain in the result. A better notation is foring a diagonal atrix, to ultiply the vector with: d (F ^i v(ω)) = Diag(^i d(f v(ω)))f j. () After this, coon atrix ultiplication can be applied which leads to the sae result. Thus, one-by-one (for each colun j independently) the coluns of the Jacobian are obtained, leading to the total atrix J = Y + F Diag(^i d(f v(ω)))f (5) where we only the j specifying the respective row of the IDFT atrix had to be reoved. Note that this result derivation differs 6 fro the one in [6]. As stated before, we do not have to actually invert the DFT atrix but can use the conjugate-coplex instead. Also, since the diension of the tie-doain values is R K but the one of the frequency-doain values is v(ω) C K+, the DFT atrix is of (rectangular) size K + K and the inverse turns out to be not just the conjugate-coplex but the Heritian. Thus, the ipleented Jacobian is finally [ J = Y + KF Diag(i d(kf H v(ω)))f H] (6) and is of size C K+ K+. 6 There appear to be soe istakes in the derivation in appendix A.6. D. Ipleentation of the algorith The next few subsections express soe details of the ipleentation of the algorith in Matlab. ) Nonity stap: In this project the MNA code fro previous assignents was adapted to now also account for (single-port) nonities. The new coand is defined as Nvalue node+ nodewhere as usually value can by any value to distinguish the nonity and node+ and node- are the anode and cathode nodes, respectively. The MNA code then interprets this as a voltage source (thereby also adding the current through it to the vector of unknown variables) with unit voltage. Further, it produces the input atrix B R N Ns, where N is the total nuber of variables and N s is the nuber of sources and nonities, along with the source vector u C Ns, as shown in Eqns. () and (), whereas previously a source vector b = Bu R N was used, directly. ) Adittance atrices: With the input atrix B and the source vector u separate, the way to get the required source and nonity adittance atrices is straightforward: According to the definition, the (self-) adittance at (non) port and the adittance as seen fro the source are: Y = i Y s = i (7) v vn=, n vn=, n Since all v n are either source or non port voltages and thus accounted for in the source vector u, the atrix entries can be obtained according to, for exaple v = (G + jkω C) Bu (8) Y (kω ) = i NL u NL u NL= i NL (9) where u n refers to an excitation vector with only entry n (for port n) equal to, thus exciting the syste only at that one port (with unit aplitude) and i NL v is an entry in the vector of unknowns v (consisting of the voltages of the regular and currents of the odified MNA equations). The way the new NodalAnalysis() script is set up, the non currents are always at the end of the vector of unknowns v. While this way a very general code results, there are soe shortcoings: whenever the nonities of the circuit to be siulated lie (alost) in parallel to one another, by shorting one out, others get shorted out as well and the adittance atrix only has (alost) zero entries. Thus, these circuits, e.g. full-wave rectifier, can not be siulated with the current ipleentation. ) Source stepping: Source stepping proved to increase the rate of convergence of non probles for the MNA ethod, before. The sae is true for HB at an even greater extent. Soe probles, especially involving strong nonities (e.g. diodes) along with dynaic eleents (capacitors and/or inductors) do not converge without it, but instead the solution becoes unstable and blows up or leads to singular atrices. In this project the siplest way of source stepping was ipleented. Since the source is only contained in the cost

5 LANG / ZHANG: THE HARMONIC BALANCE METHOD 5 function, the source stepping variable λ is used there, leading to f(v) = λ Y s + Yv + F i d ( F v ) () where the initial value in the range of λ [.,.5] was ost coonly used and the update is λ + = λ + Δλ { Δλ Δλ + if v converged in M steps = Δλ if v did not converge in M steps () where the initial update Δλ was usually chosen in the range Δλ [.,.5]. Using source stepping proved ost effective with zero initial values; v =. For fair coparisons the sae was ipleented in the MNA ethod. ) Speed-up using FFT: As entioned during the derivation, ipleenting the algorith using the DFT in atrix for is not efficient, especially when a large nuber of haronics is looked for. The atrix ultiplications fro left can siply be replaced with the respective FFT/IFFT functions 7. The ultiplication fro right with the Heritian transpose F H fro right can be done using the Matlab coands KF [ Diag(i d())f H] = ifft(fft(diag(did)). ). () where. stands for the (noral) transpose and did is the differentiated diode current vector in the tie doain, ^i d, consisting of K saples. Note that this result (despite the fact that an IFFT was used last) is in the frequency doain and that only a (square) subatrix ranging over the entries fro indices to K + are used (since the negative side of the spectru is the coplex conjugate of the positive side). 5) Coplex variables: Norally, the variables have to be split into real and iaginary parts [], thereby alost doubling the size of the proble. However, since this algorith was ipleented in Matlab, which can work with coplex variables directly, this was not done here. Additionally, it is coon to zero out [] the iaginary part of the DC coponent in the spectru (since it is expected to be zero in the end anyway) or to not consider it at all, when the variables are split up. Personal experience shows that this in fact can help the rate of convergence for soe circuits, but also leads to ore severe probles for others. Thus, if convergence is not achieved or achieved to slowly, zeroing/not zeroing out the iaginary part of the DC could help. III. NONLINEAR CIRCUITS Essentially, with non circuits, the transfer characteristic is a function of the signal level, in addition to all the other paraeters for circuits. As Stephen Maas, well-known researcher, author and now director of technology at AWR, writes in [] "All electronic circuits are non: this is a fundaental truth of electronic engineering." 7 Note that the fft() and ifft() coands in Matlab use a different noralization than can be expected; thus the FFT result is divided by the length of the vector, whereas the IFFT is ultiplied by it. Still, it is not necessarily easy to find a good exaple for non and stiff circuits. The traditional way to show nonities is tho show the generation of new frequencies using a power series with ulti-ode excitation, copare spectra, etc. [5], [] In the following the use and perforance of the haronic balance algorith will be discussed for the following basic non circuits: Half-wave rectifier with filter capacitor Half-wave doubler (to show ultiple nonities) Mixer diode (to show effects of ultiple excitations) Van der Pol oscillator (to show a different kind of nonity) Clearly, this list could be extended alost indefinitely, but it is believed that the ost iportant characteristics of the haronic balance ethods can be seen fro these exaples and thus no further ciruits were considered. A. Half-wave rectifier with capacitor The first exaple, as shown in Fig., is basically the sae as used in the derivation of the haronic balance ethod (see Fig. ), only that in this case the diode is "floating" (i.e. has two non-zero nodes) and an additional dynaic eleent in for of the output RC filter has been added. excitation i s i R L R C i ^i v i ^i non Fig. : Half-wave rectifier with capacitor and load. Fig. 5 shows the expected result in the tie-doain: depending on the tie constant of the output RC circuit, it takes a large nuber of periods for the output voltage (v, blue) to reach steady-state (approxiatively, e.g. with an overall L error < 6 ). Voltage (V), Current (A) 5 5 Tie (s) v v i d x Fig. 5: Tie doain (MNA) result of the filtered half-wave rectifier circuit.

6 6 ECE 5 MODELING OF MULTIPHYSICS SYSTEMS While the errors are generally coparable (see Fig. 6), the coparison of the coputational efficiency leads to the expected conclusion: the haronic balance ethod is uch ore efficient in calculating the steady-state of this circuit. While the CPU tie of the MNA solution increases with B. Half-wave doubler rectifier The Delon bridge voltage doubler consists of two diodes and two capacitors, as shown in the scheatic in Fig. 8. To show that the siulator can handle two nonities totally independently fro one another and with different dynaics, the capacitor values were chosen differently: C = C = F, R L = kω and R = Ω, peak = V ( Hz). i ^i Absolute error MNA L HB L MNA L HB L MNA L HB L Nuber of haronics K+ i s excitation i R R L C C v d i ^i v d i ^i non Fig. 6: Coparison of the absolute errors of HB and MNA vs. nuber of haronics K +, for various tie constants τ = R LC. increasing tie constant τ (slower dynaic of the output, therefore longer tie until the steady-state is reached), it stays (alost) constant for HB. Thus, as long as the total nuber of haronics can be held sall (i.e. for exaple lower than 6), then HB is (uch) ore efficient. For large nubers of haronics, eventually the coputational deands of the HB increase faster than for the MNA. This akes sense, since for the tie-doain ethod to achieve ore haronics only soe ore saples in the tie-doain have to be calculated (and then Fourier transfored), whereas for the HB the ipedance atrices increase for each additional haronic. Thus, for large nubers of haronics (i.e. broadband siulations), the HB should probably not be the ethod of choice. Also, if the dynaics are very fast, the coputational deands are coparable. CPU tie (s) MNA = MNA = MNA = HB = HB = HB = Fig. 8: Delon bridge voltage doubler circuit. The tie- and frequency-doain plots in Fig. 9 show the expected behavoir: the dynaics in the upper and lower part of the rectifier differ, with the voltage at the upper capacitor C dropping ore than the one of capacitor C (since it is has a larger capacitance and thus a greater filter effect). A siilar result can be seen in the spectru: the DC-offset is larger at the second diode, while the first haronic is of a coparable level. Voltage Tie (a) Solution in tie doain Haronic k (b) Spectru of the solution v d v d v c v c V d V d Fig. 9: Solution of the Delon bridge voltage doubler. Nuber of haronics K+ Fig. 7: CPU tie consuption vs. nuber of haronics K +, for various RC values. Error C. Diode as ixer As coonly used in icrowave and RF circuits, due to its nonity, a diode can serve as frequency ixer eleents. Fig. 6 6 shows the basic diode ixer scheatic, where two Iteration excitations act on a single nonity.

7 LANG / ZHANG: THE HARMONIC BALANCE METHOD 7 i s i v = R Voltage Input : v Input : v At diode: v Output: v i s 5 i 5 R C i ^i v i ^i Tie step t 7 n DC. 8 At diode: v ( ) Output: v ( ) R L R 5 6 Frequency k excitation non Fig. : Diode as ixer: tie signals and spectra. Fig. : Two-tone diode ixer circuit. The Taylor expansion of the diode current function reveals the power series ( ) i d (v) = I s (e v/v T v ) = + v v T vt + v vt () Thus, when the voltage consists of two cosines of different frequencies v = cos ω + cos ω () then because of the non conductance behavoir of the diode, besides the excitation frequencies, other frequencies result. Fro the second- and third-order ters follow: v = + cos ω t + cos ω t + cos(ω t ± ω t) v = 9 (cos ω t + cos ω ) + (cos ω t + cos ω t) + ( ) cos(ω t ± ω t) + cos(ω t ± ω t) This is confired by the haronic balance siulation, where the excitation frequencies were ω = 7ω and ω = ω and the second-order products +7 = 8 and 7 = as well as = and 7 = can be seen right away. The third-order products are already uch sall in aplitude, but together with higher-order ters for a ore "noisy" spectru around the doinant odes. As can also be seen, the output high-pass filter does its job well: there is no DC in the output signal. Multi-tone siulations with nonities generally lead to any non-zero values spread over a wide band of the spectru, thus the nuber of haronics K has to be chosen high enough, in order to avoid aliasing effects of frequencies which cannot be resolved correctly. The priary proble is to find the first haronic ω. As in this case, it can be chosen siply as the coon factor of the source frequencies, which however in soe cases could require a very finely resolved spectru and lead to an inefficient solution. Luckily there are soe special techniques available, which do allow better and ore efficient solutions. [], [] As in previous cases, also in this case source stepping greatly iproved convergence. For ore coplex probles it could be helpful to source-step the excitations separately (for exaple if there are other nonities only acting on one of the excications), but this was not further investigated in this project. D. Oscillator As last exaple, the well-known Van der Pol oscillator is introduced. Instead of a diode it contains a nonity in for of a resistor with a voltage-dependend conductivity: = v (5) i(v) = v = v v (6) Thus, the resistor has a negative conductance around for sall v. The total oscillator is a parallel circuit of this resistor, a capacitor and an inductor, as shown in Fig.. i C + i L = i L C v ^i = ^i R non Fig. : Siple circuit consisting of three parts. According to Kirchhoff s current law follows i L + i C + ^i R = (7) with the current/voltage relations of the capacitor and inductor i C = C v (8) v = L i L. (9)

8 8 ECE 5 MODELING OF MULTIPHYSICS SYSTEMS Using the KCL also follows i L = (i C +^i R ) for the inductor current. Filling in the currents through the resistor and the capacitor and filling everything into Eqn. (9) follows the differential equation v = L d dt (i C + ^i R ) = LC v v v v () v where the rule fro calculus d = v () dt v t was used. By taking the derivative of the current function and rerranging the ters follow the so-called Van der Pol differential equation LC v + L(v ) v + v = v + ε(v ) v + v = with ε = /C = L for the later (ore atheatical, [9]). Note that for ε = (and non-zero initial conditions) this becoes the noral haronic oscillator equation with sinusoidal solutions. For ε, this leads to the well-known liit cycle attractor proble, as illustrated in Fig. (b). of a plane surface which rotates everything around the axis noral to this docuent. Thus, for exaple an initial value at will rotate around that axis in circular otion and eventually end up at the sae spot again. When the abscissa denotes the voltage, then the ordinate denotes the derivative thereof, leading to the coon cosine and sine functions over tie (as the tornado rotates). In the non case the tornado is still present, but over a hilly region. There is a hill in the iddle of the doain, a (non-circular but closed) valley around it and then hills on the other side, all around the center hill. A point soewhere on the hill will be rotated tue to the tornado, but will also go down the ountain (leading to a generally non-circular otion, copared to the origin). Eventually all points will end up in the valley and keep traveling in periodic but non-circular fashion. The abscissa gain denotes the voltage, whereas the ordinate is its derivative. The resulting voltage fors are illustrated in Fig.., t v (a) ε = (), t (a) ε = (): no attractors (b) ε = (soewhat non), t v (b) ε = (non): attracting liit cycle Fig. : Circular force field and attractors for coon haronic oscillator and Van der Pol oscillator A siple figurative interpretation is as follows: in the haronic case (Fig. (a)) there is a "tornado" at the origin (c) ε = (strongly non) Fig. : Voltages and derivatives of the Van der Pol oscillator, copared to the coon haronic oscillator (ε = ). When trying to siulate this oscillator, the ain proble is that the oscillating frequency cannot be calculated in closed for and the approxiations are only accurate enough if ε is sall (i.e. the circuit just weakly non). [] Thus, we can only take a guess (i.e. the frequency of the haronic oscillator) and use a K large enough. Fig. 5

9 LANG / ZHANG: THE HARMONIC BALANCE METHOD 9 shows the results obtained fro MNA and HB siulations. Both reveal soe probles: since the spectru should be discrete (since the function is periodic), the MNA result is soewhat inaccurate; ost likely because there were not enough periods siulated and the steady-state has not been reached. The HB ethod resulted in a very spiky spectru; the voltage appears to be soewhat too sooth whereas the derivative appears to be not sooth enough. Also, since the oscillation (if there is any) always occurs at the fundaental frequency ω, but it is not defined what this actually is, the frequency cannot be obtained this way. Also in this case there v dv/dt Noralized tie (period) 6 8 Frequency (a) MNA Noralized tie v dv/dt IV. SUMMARY & CONCLUSIONS When it coes down to the question: tie- or frequencydoain? The answer for icrowave and RF circuits is: both. The haronic balance ethod operates in both doains subcircuits frequency doain non subcircuits tie doain balance currents at interfaces (it can therefore can be considered a hybrid ethod) and is a powerful instruent to siulate circuits containing nonities as well as stiff dynaics. REFERENCES [] M. S. Nakhla, J. Vlach, A Piecewise Haronic Balance Technique for Deterination of Periodic Response of Non Systes, IEEE Transactions on Circuits and Systes, Vol., No., February 976 [] C.-W. Ho, A. E. Ruehli, P. A. Brennan, The Modified Nodal Approach to Network Analysis, IEEE Transactions on Circuits and Systes, Vol., No. 6, June 975 [] R. J. Gilore, M. B. Steer, Non Circuit Analysis Using the Method of Haronic Balance A Review of the Art. Part I. Introductory Concepts, International journal of Microwave and Millieter-Wave Coputer-Aided Engineering, Vol., No., pp. -7, 99 [] R. J. Gilore, M. B. Steer, Non Circuit Analysis Using the Method of Haronic Balance A Review of the Art. Part II. Advanced Concepts, International journal of Microwave and Millieter-Wave Coputer-Aided Engineering, Vol., No., pp. 59-8, 99 [5] Kenneth S. Kundert, Jacob K. White, Alberto Sangiovanni-Vincentelli Steady-state ethods for siulating analog and icrowave circuits, Kluwer Acadeic Publishers, 99 [6] Franco Giannini, Giorgio Leuzzi, Non Microwave Circuit Design, John Wiley & Sons, Ltd, 5 [7] H. G. Brachtendorf, G. Welsch, R. Laur Fast siulation of the steady-state of circuits by the haronic balance technique, 995 IEEE International Syposiu on Circuits and Systes, ISCAS 95., Vol., pp. 88-9, May 995 [8] George D. Vendelin, Anthony M. Pavio, Ulrich L. Rohde, Microwave Circuit Design Using Linear and Non Techniques, nd Edition, John Wiley & Sons, Ltd, 5 [9] Paulo J. C. Rodrigues Coputer-aided analysis of non icrowave circuits, Artech House, Inc, 998 [] Stephen A. Maas Non Microwave and RF Circuits, Artech House, Inc, [] O. Nastov, R. Telichevesky, K. Kundert, J. White, Fundaentals of Fast Siulation Algoriths for RF Circuits, Invited Paper, Proceedings of the IEEE, Vol. 95, No., March 7 [] D. W. Jordan, P. Sith Non Ordinary Differential Equations, Fourth Edition, Oxford University Press, Noralized frequency (b) HB Fig. 5: Oscillator siulations. are special techniques [], [9], [] to deal with all these probles occuring when dealing with autonoous probles. Essentially, the frequencies have to be added to the unknowns and additional equations have to be put in place to solve for the.