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Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2006, Article ID 32097, Pages 1–11DOI 10.1155/WCN/2006/32097

Transient and Steady-State Analysis ofNonlinear RF and Microwave Circuits

Lei (Lana) Zhu and Carlos E. Christoffersen

Department of Electrical Engineering, Lakehead University, Thunder Bay, ON, Canada P7B 5E1

Received 1 November 2005; Accepted 28 February 2006

This paper offers a review of simulation methods currently available for the transient and steady-state analysis of nonlinear RFand microwave circuits. The most general method continues to be the time-marching approach used in Spice, but more recentmethods based on multiple time dimensions are particularly effective for RF and microwave circuits. We derive nodal formu-lations for the most widely used multiple time dimension methods. We put special emphasis on methods for the analysis ofoscillators based in the warped multitime partial differential equations (WaMPDE) approach. Case studies of a Colpitts oscil-lator and a voltage controlled Clapp-Gouriet oscillator are presented and discussed. The accuracy of the amplitude and phaseof these methods is investigated. It is shown that the exploitation of frequency-domain latency reduces the computational ef-fort.

Copyright © 2006 L. Zhu and C. E. Christoffersen. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

1. INTRODUCTION

The most common computer-aided circuit analysis tech-nique is the time-domain transient analysis using the time-marching approach to solve the system of ordinary differ-ential equations (ODEs) that represent the circuit. This ap-proach is used in Spice and many other circuit simulators. Al-though still very useful, this technique presents a number ofshortcomings for the analysis of RF and microwave circuits.Amongst these shortcomings is the huge amount of mem-ory and computation time required for the analysis of cir-cuits with widely separated time scales and/or excitations [1].This is often the case for mixers, power amplifiers, and oscil-lators. Several techniques have been developed to overcomethis problem. Some of them are available in modern circuitsimulators and are widely used. In this paper we present anoverview of these methods with special attention to the anal-ysis of oscillators.

We consider transient analysis methods first. In Section 3we focus on methods used to directly calculate the steady-state response of circuits. In Section 4 oscillator analysisis discussed with emphasis on methods based on multipletime dimensions. Case studies of a Colpitts oscillator anda voltage controlled Clapp-Gouriet oscillator are presentedand discussed in Section 5.

2. TRANSIENT ANALYSIS

A generic circuit can be described by the following system ofdifferential-algebraic equations:

Gu(t) + Cdu(t)dt

+dQ(u(t)

)

dt+ I(u(t)

) = S(t), (1)

here u(t) is the vector of state variables (nodal voltages andselected branch currents), G is a matrix of conductances, C isthe matrix representing the linear charge terms, Q(u(t)) andI(u(t)) are vector functions corresponding to the nonlineardevices, and S(t) is a vector that represents the sources.

The traditional time-marching technique consists in re-placing the derivatives of the state variables for an approx-imation using a numerical integration rule. For example, ifthe backward Euler rule is used, we have

du(tn+1

)

dt≈ u

(tn+1

)− u(tn)

tn+1 − tn. (2)

The same procedure is applied to the Q vector and the result-ing nonlinear algebraic system of equations is solved usingthe Newton method for each time step:

2 EURASIP Journal on Wireless Communications and Networking

Gun+1 + Cun+1 − un

h+Q(un+1

)−Q(un)

h+ I(un+1

) = Sn+1,

(3)

here un+1 = u(tn+1), un = u(tn), Sn+1 = S(tn+1), and h =tn+1 − tn for simplicity. This analysis is the most generalnonlinear circuit analysis. It is applicable to autonomous ornonautonomous circuits with any kind of excitation.

If a circuit presents rapid and slow variations simultane-ously, this analysis becomes inefficient because a small time-step is required to follow the fast variations with a long totalsimulation time. Consider, for example, a simulation to es-timate the spectral regrowth in an amplifier driven by a dig-itally modulated carrier [2]. If the rapid variations are os-cillatory, the circuit can be analysed using a more efficientapproach. Different implementations of this approach exist(e.g., see [3–5]), but the underlying theory is common to allof them and is presented in [1]. Consider a voltage describedby the following function:

v1(t) =[

1− exp(− t

τa

)][1 + sin

(2πτb

t)]

, (4)

where τa and τb are time constants. A graphical representa-tion of this function with τa = 5 seconds and τb = 20 ms isshown in Figure 1. Many sample points are required to rep-resent this function. For example, 5000 samples were neededin Figure 1 for a total time of 5 seconds. Envelope-followingmethods [4, 5] take advantage on the fact that the oscilla-tions do not change much in adjacent periods. They followthe shape of the envelope of signals using a time-step muchgreater than one period of the rapid oscillation and calcu-late the full response of the circuit with a small time-steponly once in a while. Envelope-following methods have beenrecognised [6] as the most promising methods for the anal-ysis of radio components such as mixers. It was proved in[1] that the idea in envelope-following methods can be seenas a particular case of a more general approach called multi-partial differential equations (MPDE). The main idea in thetime domain envelope-following (TD-ENV) method usingthe MPDE approach is to represent signals in more than onetime dimension according to the scale of variation. The sig-nals must be periodic in at least one of the dimensions andthe period must be constant and known. Note that this con-dition excludes autonomous circuits. We consider oscillatorsin Section 4. For example, v1(t) is replaced by the followingbidimensional function:

v1(t1, t2

) =[

1− exp(− t1

τa

)][1 + sin

(2πτb

t2

)]. (5)

This function is plotted in Figure 2, for the same values of τaand τb used before. Only 100 sample points were necessary torepresent the waveform in the same time interval. The orig-inal function can be easily recovered by setting t1 = t andt2 = t. Equation (1) must be modified as follows:

Gu + C(∂u

∂t1+

∂u

∂t2

)+∂Q(u)∂t1

+∂Q(u)∂t2

+ I(u) = S(t1, t2

),

(6)

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

00 1 2 3 4 5

Time (s)

v 1(V

)

Figure 1: The voltage v1 as a function of time.

where u and S represent the bivariate versions of the statevariables and the source vector, respectively. The time planewith the boundaries is shown in Figure 3. In the figure, T isthe period along the t2 axis. It was demonstrated in [1] thata solution of (6) with t1 set equal to t2 (the diagonal linesin Figure 3) is also a solution to (1). To obtain the transientresponse of the circuit, a steady-state problem in the t2 di-rection must be solved for every time-step along t1. The ef-ficiency of the method is better when the size of the time-step along t1 is many times greater than T . Unfortunatelythis is not always possible because fast variations along t1occur frequently. This problem can be alleviated using anadaptive time-step in the direction of t1. A time-step controlalgorithm for this purpose is presented in [7]. Another fac-tor that has a great effect in the rate of variations along t1 isthe choice of boundary conditions. The boundary conditionsof the MPDE are determined by the initial conditions of theoriginal ODE only at t1 = 0 and t2 = 0. Work [8] presents amethod to choose the boundary conditions in the rest of thet1 = 0 line to avoid fast variations along the t1 dimension.The solution of the steady-state problem along the t2 dimen-sion can be accomplished using several methods describedin Section 3. One popular choice is to use harmonic balance.Recently an approach using wavelets in the t2 dimension waspresented [9].

The MPDE method was presented here for two time di-mensions, but if the problem presents more than two rates,more time dimensions can be used as necessary. Work [10]considers a phase-locked loop (PLL) simulation using threetime scales.

3. STEADY-STATE ANALYSIS

Steady-state methods are of great interest for the analysis ofRF and microwave circuits. The most widely used steady-state analysis method is harmonic balance (HB). Some of theadvantages of this method are that the solution directly givesthe harmonic content of the circuit response and that it is

L. Zhu and C. E. Christoffersen 3

1.41.2

10.80.60.40.2

0

0.020.015

0.010.005 1

23

45

v 1(V

)

t2 (s)

t1 (s)

Figure 2: The bidimensional voltage v1(t1, t2).

t2

T

0T 2T 3T

t1

Figure 3: The time plane used for the MPDE method.

easier to include RF circuit components that are better mod-elled in frequency domain such as transmission lines, filters,and others.

In HB each element of the u(t) vector in (1) is representedby a set of phasors,

un(t) = �{ K∑

k=0

Ukne

jkωt

}

, (7)

where n is the state variable number, k is the harmonic num-ber, and ω = 2π/T . Equation (1) can now be expressed foreach angular frequency (kω) as follows:

(G + CΩk

)Uk + Ik + ΩkQk = Sk. (8)

In this equation Ωk is a diagonal matrix with nonzero ele-ments equal to jkω. The Sk vector represents the sources atangular frequency kω. The nonlinear devices in the circuitare normally modelled in time domain and the Ik and Qk

vectors are formed taking the kth harmonic component ofthe Fourier transform of the respective time-domain vectors.The nonlinear devices could be modelled directly in the fre-quency domain using Volterra series [11]. For weekly non-linear circuits Volterra series result in more accurate modelsand are also useful for analytical calculations for design [6].

The unknowns in (8) are the Fourier coefficients (Uk).The system of algebraic equations formed by applying (8) forall harmonics is normally solved using the Newton method.The number of simultaneous unknowns in this equationis equal to (2K + 1)n. In many cases this number can be

very large in the order of several tens of thousands. In thesecases the solution can still be found with a reasonable CPUtime using inexact Newton methods [12] and Krylov sub-space methods [2] (note that for smaller problems the regu-lar Newton method is more efficient).

Another relevant development is the exploitation offrequency-domain latency in HB [13–15]. These techniquestake advantage of the fact that in most circuits the numberof harmonics necessary to represent each variable (voltage,current) is not the same. By using a different number of har-monics in each variable, a significant reduction in the com-putational effort is achieved.

There are many problems of interest where the signals arenot strictly periodic but quasiperiodic. For example, considerthe following voltage waveform:

v2(t) = V2 cos(ω1t)

cos(ω2t). (9)

No period can be defined for this signal if ω1/ω2 is not a ra-tional number. Even in the case that the signal is strictly peri-odic, it is frequently more convenient to treat it as quasiperi-odic. Two approaches exist to treat this problem in HB. Thefirst of them is called artificial frequency mapping and thesecond is to use a multidimensional Fourier transform [16].

It is often acknowledged [17–19] that for strongly non-linear circuits the HB method may not be the best becausea large number of harmonics is necessary to represent thesignals. The most common alternative to HB is the shootingmethod [20]. The shooting method works by solving (3) forone period of the excitation and finding what initial condi-tions result in the same state of the circuit at the end of theperiod. Other approaches using wavelets [17] and adaptivebasis functions [18] are still in the research stage and are notwidely used yet. The state variables in these methods are rep-resented as a linear combination of a set of basis functions.Equation (1) can then be transformed into a nonlinear alge-braic equation.

3.1. Steady-state analysis and the MPDE

The MPDE approach is also useful for steady-state analysis ofquasiperiodic excitations by considering each period in a dif-ferent time dimension. Equation (6) is applicable with peri-odic boundary conditions in both t1 and t2 dimensions. TheMPDE approach is thus not only useful to analyse circuitswith widely separated excitation frequencies but also for cir-cuits with closely spaced excitation frequencies [19]. The HBwith multidimensional Fourier transform approach can beseen as a particular case of the MPDE when both time di-mensions are considered in the frequency domain.

4. ANALYSIS OF OSCILLATORS

Oscillator analysis is a difficult task [20–26]. Except for regu-lar time-marching transient analysis (3), all the methods thatwere reviewed so far must be modified to analyse oscillators.

For transient analysis, an alternative to the traditionaltime-marching approach called warped multitime partialdifferential equation (WaMPDE) was presented in [22, 27,


28]. This approach deals with the fact that the period (orequivalently, the local frequency) in one of the time dimen-sions is no longer constant. In the WaMPDE this is solvedby warping one of the time scales in MPDE to have a con-stant normalised period. As a result the local frequency isnormalised to a constant value and the warped time becomesa function of time. The t1 time axis in the MPDE is now re-named τ2 (they are otherwise equivalent) and the warpedtime scale is named τ1. The relation between τ1 and τ2 isgiven by

τ1 =∫ t

0ω(τ2)dτ2, (10)

where ω(τ2) is the unknown local frequency. Substituting t1and t2 in (6) we obtain the WaMPDE nodal equation:

Gu + C(ω(τ2) ∂u

∂τ1+

∂u

∂τ2

)

+ ω(τ2)∂Q(u)

∂τ1+∂Q(u)∂τ2

+ I(u) = S(τ1, τ2

).

(11)

An additional equation is required to balance the introduc-tion of the unknown local frequency. This equation is formedby imposing a smooth phase variation along τ2 [22].

Often the HB method is used in the τ1 dimension. Theneach element of the u(t) vector in (1) is represented by

un(τ1, τ2

) = �{ K∑

k=0

Ukn

(τ2)e jkτ1

}

, (12)

where again k is the harmonic number and the period in thewarped time scale (τ1) is normalised to 2π (i.e., ω = 1). Wecan reformulate now (11) for each harmonic (k):

(G + CΩk

)Uk + C

∂Uk

∂τ2+ ΩkQk +

∂Qk

∂τ2+ Ik − Sk = 0.

(13)

As stated before, the phase of one of the variables must thenbe fixed to restore the number of unknowns to be equal tothe number of equations. That can be achieved by setting theimaginary part of one of the variables to be zero,

�(U1n

) = 0. (14)

Equation (13) is discretised in the τ2 direction using thebackward Euler (BE) rule, trapezoidal rule, or any other nu-merical integration method. The resulting algebraic nonlin-ear system is then solved with the Newton-Raphson methodfor each value of τ2. This technique is referred in this work astime-frequency envelope transient (TFET).

4.1. Transient

Given initial conditions for (1), the corresponding bound-ary conditions in the TFET analysis can be obtained from ashort section of the transient response of the oscillator ob-tained from a time-marching simulation [29]. The choice of

accurate boundary conditions is important if a good agree-ment between time-marching and TFET analyses is desired.

To improve the efficiency of the simulation, the num-ber of harmonics for each variable un can be adaptively con-trolled [15]. This is sometimes referred as frequency-domainlatency exploitation. This is achieved as follows. At the end ofthe calculation for each step of τ2, the magnitudes of the lasttwo harmonics are considered. If they are greater than somethreshold value, then the number of harmonics for that vari-able (l) is increased by one. If they are smaller than anotherthreshold, then l is decreased by one. Otherwise it is left un-changed. One advantage of this approach is that the numberof harmonics is increased or reduced as needed. Each rowof (13) (nodal equation at one frequency) is considered at anumber of frequencies equal to the number of harmonics ofthe corresponding nodal voltage. In this way the number ofequations is always kept equal to the number of unknowns.

An adaptive time-step control algorithm is used in orderto minimise the number of time steps [29]. The time-stepalong τ2 is adaptively changed according to a local truncationerror estimation,

hnew = hold

(εmax

ε

)1/m

, (15)

where εmax is the maximum acceptable truncation error, εis the current truncation error, and m is a number that de-pends on the integration method being used (m = 3 fortrapezoidal integration). This substantially reduces the com-putational cost and improves the accuracy of the TFET.

4.2. Steady-state

For the steady-state analysis of oscillators, harmonic bal-ance (HB) has been the dominant approach in recent years.Though HB is one of the most important frequency-domaintechniques, it still has some limitations. In particular a goodinitial guess is needed to make HB converge to the desired os-cillatory solution. It is especially difficult to get a good initialguess of the oscillator frequency. Several methods have beenproposed to improve this limitation. For example, in [25]the HB equations are modified by including the Kurokawacondition to eliminate the DC solution. In [26] a voltagesource probe at the fundamental frequency that is an opencircuit at all other frequencies is inserted to avoid the DC so-lution. By means of an iterative process the amplitude andfrequency of the probe are adjusted until there is no currentthrough the probe. At this point the autonomous solution isfound. A similar probe concept with the addition of a con-tinuation method has been proposed more recently [23] toimprove convergence. Accelerated transients have been used[15, 20, 30] to find the steady-state regime. In order to ac-celerate the finding of the steady-state regime, the transientbehaviour of the circuit is artificially reduced in [30]. In [20]an envelope-transient analysis is used to improve the conver-gence of the shooting method.

Another new approach improving the convergence ofthe HB analysis of oscillators was presented in [15]. Thisapproach is based on an accelerated TFET method and is


summarised here. A TFET simulation is started with theboundary conditions set to the DC bias point values. An ex-citation current,

iS(τ1, τ2

) =⎧⎨

⎩I0 cos

(τ1)

if τ2 ≤ ta,

0 otherwise,(16)

is injected from the ground node into one of the nodes whereoscillations are expected. Here, I0 is a small real number (nor-mally a few μA) and ta is set equal to the initial time-step sizealong τ2 (h). The purpose of iS is to start oscillations by mov-ing the system away from the equilibrium point. The systemwill then naturally tend to reach the desired oscillatory steadystate. The key of this work is to accelerate the TFET simula-tion to reach a point in τ2 close to the steady state in the mini-mum possible number of Newton iterations and then use thestate at that point as the initial guess of a regular autonomousHB analysis. Since the focus is on the steady state, it is notnecessary to calculate the transient evolution with great pre-cision as long as it converges close to the actual steady stateof the circuit. Thus the time-step along τ2 and other parame-ters in the simulation are controlled to minimise the numberof Newton iterations [15]. When the local frequency func-tion becomes constant and the difference between two peri-odic solutions along τ2 becomes small, the accelerated TFETis stopped and a regular oscillator HB analysis is started us-ing the last solution along τ2 as the initial guess. Note thatthe regular oscillator HB analysis is obtained by setting allderivatives with respect to τ2 to be zero in (13).

5. CASE STUDIES AND DISCUSSION

In this section we perform a transient and steady-state anal-ysis of two oscillators: a Colpitts oscillator and voltage con-trolled oscillator (VCO). In each oscillator, the regular timedomain simulation is provided followed by a WaMPDE-based simulation, and then both results are compared.

5.1. Colpitts oscillator

The Colpitts oscillator taken from [31] uses a capacitive volt-age divider in the LC tank circuit. In the circuit shown inFigure 4, the component values are: C1 = C2 = 2 pF, Cc =400 pF, Ce = 100 pF, L1 = 1 μH, R1 = 8 kΩ, R2 = 2 kΩ,Rc = 2.4 kΩ, Re = 1.3 kΩ, Vcc = 11 V, BF = 100, andBR = 1.

5.1.1. Transient analysis

This oscillator exhibits an extremely long initial transientcompared with the oscillation period. Part of the transientsimulation is presented in Figure 5. The CPU time in a1.5 GHz computer using Matlab was 57 seconds for a simula-tion stop time of 10 μs. Clearly time-marching simulation isvery time-consuming for the analysis of this circuit. In or-der to obtain an accurate result, the acceptable truncationerror in this simulation was reduced until no phase differ-ence could be observed with further reductions. In this way

Vcc

Rc

R1

Cc

Q1

R2 RE CE

C1 C2

L1Vout

Figure 4: Schematic of a Colpitts oscillator.

10

9

8

7

6

5

4

3

2

10 0.2 0.4 0.6 0.8 1

×10−4Time (s)

Ou

tpu

tvo

ltag

e(V

)

Figure 5: Transient response of the Colpitts oscillator.

we can take the result of this time-marching simulation as areference for both magnitude and phase information.

We now discuss the simulation results using the TFETapproach. The TFET simulation starts from the specified ini-tial conditions and the stop time along τ2 is set to 0.07 sec-onds. The initial number of harmonics is 11 and the adaptiveharmonic balance automatically adjusts this number as re-quired. The adaptive time-step algorithm increases the stepsize according to the local truncation error. These two pro-visions largely speed up the simulation. Total CPU time is199 seconds. A time-marching simulation would require ap-proximately 15 hours to produce the same result. Figure 6shows the multitime expression of the output voltage.

Very good agreement between TFET simulation and thetime-marching simulation has been achieved as shown in thetop of Figure 7. There is a phase error accumulation along τ2

shown at the bottom of Figure 7. The relative amplitude errorand absolute phase error of the first harmonic in TFET andODE simulations are presented in Figure 8. The amplitudeerror is very small, but the phase error may be important in


15

10

5

04

32

10−15

−10−5

0

Ou

tpu

tvo

ltag

e(V

)

The warped timeLogarithm of the real time

×10−8

Figure 6: Bidimensional representation of output voltage.

9876543210.5 10.55 10.6 10.65 10.7 10.75 10.8 10.85 10.9

Time (μs)

Ou

tpu

tvo

ltag

e

Proposed WaMPDE solution compared to time-marching simulation

Time marchingTFET

(a)

109876543

99.5 99.55 99.6 99.65 99.7 99.75 99.8 99.85

Time (μs)

Ou

tpu

tvo

ltag

e

Time marchingTFET

Proposed WaMPDE solution compared to time-marching simulation

(b)

Figure 7: TFET solution compared to time-marching solution.

some cases. It should be noted that a comparable phase errorwould be obtained with a time-marching simulation if theacceptable truncation error is not set to a very small num-ber. The phase error in the TFET simulation can be reducedif the acceptable local truncation error in the TFET simula-tion is reduced as the results of Figure 9 indicate. However,the reduction in the acceptable local truncation error in-creases the number of time steps along τ2 and this results in a

0.8

0.6

0.4

0.2

00 20 40 60 80 100 120

Time (μs)

Th

eam

plit

ude

erro

r(%

)

(a)

15

10

5

00 20 40 60 80 100 120

Time (μs)

Th

eph

ase

erro

r(d

egre

e)

(b)

Figure 8: Relative error in the magnitude of the first harmonic.

10

9

8

7

6

5

4

3

9.97 9.975 9.98 9.985

×10−5Time (s)

Ou

tpu

tvo

ltag

e(V

)

Time marchingTFET with large LTE (0.01)TFET with small LTE (0.0001)

Figure 9: Comparison of time-marching and TFET with a smallermaximum local truncation error.

significantly longer simulation time. Finally, Figure 10 showsthe size of the Jacobian matrix and the size of the time-stepalong τ2 for each step in τ2. The size of the Jacobian matrixis proportional to the total number of harmonics consideredin the simulation. As the variations in τ2 become smooth,the time-step is increased and the size of the Jacobian matrixis increased as more harmonics are generated in the nodalvoltages.


190

180

170

160

150

140

130

120

110

100

0

−1

−2

−3

−4

−5

−6

−7

−8

−90 10 20 30 40 50 60 70 80

Sample number along τ2

Jaco

bian

size

Loga

rith

mof

tim

est

ep

Jacobian sizeTime step

Figure 10: Size of the Jacobian matrix and the size of the time-stepalong τ2 for each step in τ2.

10

8

6

4

2

02

1.51

0.50 −8

−6−4

−20

Ou

tpu

tvo

ltag

e(V

)

The warped timeLogarithm of the real time

×10−7

Figure 11: Bidimensional representation of output voltage.

5.1.2. Steady-state analysis

The excitation current (iS) was applied to the base node withI0 set to 10 μA. The bidimensional plot of the output voltageas a function of τ1 and τ2 is shown in Figure 11. The CPUtime of the accelerated TFET and HB analyses combined is52 seconds.

The oscillator frequency determined by the warped func-tion ω(τ2) is 5.04 MHz as shown in Figure 12. This figure il-lustrates the robustness of the proposed method. Differentinitial frequencies converge to the correct value.

In Figure 13 the steady state is compared with the fi-nal line of the multitime simulation. This shows how closethe result from the accelerated TFET to the steady state is.Figure 14 shows the Jacobian matrix size and the number ofNewton iterations at each time-step (sample number).The

×107

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.20 1 2 3 4 5 6 7

×10−7Time (s)

Loca

lfre

quen

cy(H

z)

IC:0.5fIC:fIC:2f

IC:3fIC:4f

Figure 12: Fundamental frequency as a function of τ2.

9

8

7

6

5

4

30 0.5 1 1.5 2

×10−7Time (s)

Ou

tpu

tvo

ltag

e(V

)

Steady-state solution

Final multitime solution

Figure 13: Steady-state solution compared to final line in multitimesolution.

adaptive control of the time-step along τ2 keeps the numberof Newton iterations small for each value of τ2. The Jacobianmatrix size increases as the number of harmonics increases.

5.2. Clapp-Gouriet oscillator

In this section, we present the analysis of a VCO circuitbased on the Clapp-Gouriet configuration [32]. Figure 15shows the electrical schematic of the VCO. In this circuit


140

120

100

80

60

40

20

0

28

24

20

16

12

8

4

00 10 20 30 40 50 60 70


Jaco

bian

size

New

ton

iter

atio

ns

Jacobian sizeNewton iterations

Figure 14: Jacobian size and number of Newton iterations.

Vcc

RcR1

L1

Q1

C1

C2

C3

D1

C4 R3Vout

RL

RE

Vdc

R2

Figure 15: A VCO using Clapp-Gouriet configuration.

C1 = 82 pF, C2 = 220 pF, C3 = 47 pF, C4 = 330 pF,L1 = 102.55 μH, R1 = 220 kΩ, R2 = 22 kΩ, R3 = 47Ω,Rc = 2.2 kΩ, Re = 220Ω, Rl = 100Ω, Vcc = 12 V, BF = 70,and BR = 5. The oscillator frequency is tuned by a diode con-nected to a control voltage, Vdc. The circuit is analysed witha sinusoidal control voltage,

Vdc = 3 + sin(2π104t

)V. (17)

The initial transient is shown in Figure 16. In TFET anal-ysis, the initial number of harmonics is set to 8 and the adap-tive HB algorithm automatically adjusts this number for eachnode every time-step along τ2. The bidimensional plot of theoutput voltage is shown in Figure 17.

Good agreement between time-marching and TFETanalyses is achieved as shown in Figure 18. A plot showingthe first harmonic magnitude and phase difference between

1

0.5

0

−0.5

−1

−1.5

−2

−2.50 10 20 30 40 50

Time (ms)

Ou

tpu

tvo

ltag

e(V

)

Figure 16: Transient response of the VCO.

1

0

−1

−2

−38

64

20 0

2

4

Ou

tpu

tvo

ltag

e(V

)

The warped time The real time (s)×10−4

Figure 17: Bidimensional representation of output voltage of VCO.

the time-marching and TFET analyses is shown in Figure 19.As with the previous case study, there is some phase errorthat can be reduced by reducing the tolerance of the localtruncation error. Finally, Figure 20 shows the size of the Ja-cobian matrix and the size of the time-step along τ2 for eachstep in τ2. It can be observed that both quantities follow thevariations of the transient and the control voltage.

6. CONCLUSIONS

We have presented a review of simulation methods cur-rently available for the transient and steady-state analysis ofnonlinear RF and microwave circuits. Although the time-marching approach used in Spice will continue to be widelyused, envelope-following methods are particularly effectivefor RF and microwave circuits and they are becoming apopular choice. Recent improvements to oscillator analysiswere summarised and case studies of a Colpitts oscillator


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Figure 18: Time-marching and TFET results compared.

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ase

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egre

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Figure 19: Comparison of ODE and warped MPDE in first har-monic.

and a voltage controlled Clapp-Gouriet oscillator were pre-sented. In both cases there was good agreement betweentime-marching and TFET analyses, but it was noted thatsome phase error may occur if the local truncation error tol-erance is not kept small enough. This may be an importantconsideration for the analysis of PLLs and should be fur-ther investigated. The simulations indicated that the adaptiveHB technique significantly reduces the computational effort

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Figure 20: Size of the Jacobian matrix and the size of the time-stepalong τ2 for each step in τ2 for the VCO.

by reducing the size of the Jacobian matrix in the Newtonmethod. It was shown that an accelerated TFET analysis canbe effectively used to improve the convergence of the HB os-cillator analysis.

ACKNOWLEDGMENT

This work was supported by Natural Sciences and Engineer-ing Research Council of Canada (NSERC).

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Lei (Lana) Zhu received her Bachelor de-gree of Electrical Engineering from South-east University in China in August 1998.From 1994 to 2001, she was a Junior De-sign Engineer and later a Design Engineer inElectrical Engineering Design Departmentof Nanjing Power Supply Bureau, China.She received her M.S. degree from LakeheadUniversity, Canada, in November 2005. Herresearch interest includes circuits analysisand simulation in multiple time axes, oscillator design, phase-locked loops, and general communication circuits. Currently sheworks at Manitoba Hydro, Canada.


Carlos E. Christoffersen received the Elec-tronic Engineer degree at the National Uni-versity of Rosario, Argentina, in 1993. From1993 to 1995, he was a Research Fellow ofthe National Research Council of Argentina(CONICET). He received an M.S. degreeand a Ph.D. degree in electrical engineer-ing in 1998 and 2000, respectively, fromNorth Carolina State University (NCSU).Currently he is an Assistant Professor inthe Department of Electrical Engineering at Lakehead University,Thunder Bay, Canada. He is a Member of the IEEE. His currentresearch interests include analogue and RF circuit computer-aideddesign including electromagnetic and thermal interactions.

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