IEEE Transactions on Power Delivery, Vol. 9, No.

3, July 1994 1609

Dynamic Response of a Thyristor Controlled Switched Capacitor

Sasan G. Jalali Robert H. Lasseter Ian Dobson
Student Member Fellow Member
Electrical and Computer Engineering Department
University of Wisconsin, Madison, WI 53706

Abstract: This paper computes the small signal dy- cal degrees. The natural frequencies of oscillation range
namic response of a thyristor controlled series capaci- from 8.9 Hz to 1.9 Hz.
tor system for use in control design. The computation The control design for Kayenta was achieved by us
includes the effects of synchronization and the nonlin- ing these transfer functions in a closed loop model al-
earity due to thyristor switchings. Eigenvalues of the lowing the use of standard tools to optimize the control
small signal dynamic response are computed and used response. This method is effective but provides little
to study the dynamic response of the Kayenta system insight into the behavior of the system and requires d e
using different methods of synchronization and a closed tailed EMTP simulations to find the necessary transfer
loop control. functions. This paper presents a first principles method
for computing eigenvalues of the small signal response
Keywords: FACTS, TCSC, Kayenta, TCR.
of TCSC systems. The method takes account of the
nonlinearities of the thyristor switchings and the syn-
Introduction chronization and controller dynamics. Related work in-
The optimization of controllers for a Thyristor Con- cludes computations of the closed loop dynamic response
trolled Series Capacitor (TCSC) is critical for its use in of controlled AC/DC or resonant convertors [3,4,5]and
power systems. This is a difficult task due to the non- analysis of instabilities, damping, and resonance in an
linearities introduced by the switching of the thyristors. open loop static Var control system [6,7,8].
In particular, the dynamic response of a TCSC changes
as a function of its operating point. This dependence is Kayenta System
discussed in papers describing the design of the Kayenta The 230 kV, 330 Mvar TCSC system shown in
system [1,2]. Figure 1 was installed in northeastern Arizona at the
Kayenta Substation. There are two conventional 165
Table 1. Complex poles for Kayenta System [l] Mvar series capacitor banks with a nominal reactance
of 55 R each. One of these segments is divided into 40 R
(T=200 s = -14.5 f j56.2 and 15 R. The 15 R unit is configured as a TCSC so
U = 40° s = -13.5 f j48.1 that 15 R of capacitance is in parallel with a Thyristor
u=500 s = -10.6 f j30.2 Controlled Reactor (TCR) of 2.56 R.

jim
(T=6O0 s = -13.5 f j11.9

The design of the Kayenta controller was achieved

by first finding the open loop response of line current to I

changes in the firing point using a detailed EMTP model.

I
The resulting envelope of the line current response en-
abled control engineers to find a transfer function which
approximates the dynamics of the system around an o p
erating point. The resulting fourth order transfer func-
tion has two poles on the real axis and a complex pair of
poles. The dependence of the complex poles on the oper- Figure 1. Kayenta System
ating point is shown in Table 1. The operating points are
specified by o,the thyristor conduction time in electri-
The fundamental impedance of the TCSC is similar
to that of a parallel LC circuit with a variable inductance.
94 WM 065-3 PWRD A paper recommended and approved With the thyristors off and conduction time o = Oo, the
by the IEEE Transmission and Distribution Committee
of the IEEE Power Engineering Society for presentat- impedance per phase is 15 R. As U is increased, the ca-
ion at the IEEE/PES 1994 Winter Meeting, New York, pacitance increases to a large value at the fundamental
New York, January 30 - February 3, 1994. Manuscript resonance point o = 74O [l]. It is assumed that control
submitted August 2 1993 ; made available for printing
December 6, 1993. limits prevent even temporary operation near the res*
nance. It is also assumed that operation will be limited to
the capacitive region except for full bypass at G = 180O.

0885-8977/94/$04.00 Q 1994 IEEE

 1610

Basic Concepts of TCSC Dynamics to small disturbances. In particular, the steady state is
The dynamics of any system which is periodic with exponentially stable if the eigenvalues of the Jacobian
period T can be studied by sampling the system states evaluated at z o lie strictly inside the unit circle. The
once per period. If the resulting values of the state vari- eigenvalues or poles lie in the z-plane familiar from sam-
ables are equal to the previous sampled states, the sys- pled data systems. The expected relationship between
tem is in steady state. The system dynamics can be s-plane and z-plane poles applies so that an s-plane pole
described as the change in the sampled states from one s = --a f j w maps to the z-plane as z = e-aTe*jwT.
sampled point to the next. This concept is formalized as The magnitude 1. = e--oT defines the damping so that
the Poincarb mapping F from dynamical systems theory a pole inside the unit circle has positive damping and a
[9] which maps the system states forward in time from pole outside the unit circle implies instability. The pcr
+ +
t o to t o T.That is, F[z(to)]= z(t0 T)where z r e p lar angle, wT defines the frequency of oscillation about
resents the system states. If F[z(to)]= z(to),the map the sampling fiequency. As polar angle traverses from 00
has a fixed point z(t0)and the system is in steady state. to 3600 the modulation1frequency increases from zero to
Figure 2 describes the system dynamics as the the sampling frequency of 60 Hz. Care must be taken to
TCSC states evolve over a period T. The conduction correctly interpret the angle in the presence of aliasing.
periods starting at times 40, 4112, 4 1 are defined by the All the computed examples of this paper are half
control system and the method used to synchronize the wave symmetric. Half wave symmetry means that the
thyristor firing times to the system states. The turn off system states are equal in magnitude and opposite in sign
times TO, 7112, 7 1 are defined as the times at which the to the system states half period or T/2 later. Moreover,
thyristor current becomes zero. The dependence of these the firing times 40 and 4112 and the thyristor switch off
switching times on the system states causes the system times TO and r l p differ by exactly half a period.
nonlinearity.
At the beginning of the period to, the initial states Open loop response
are the vector z o and a thyristor is on. This conduc- This section compares the effects of synchronization
tion mode ends when the thyristor current goes through with respect to zeros of TCSC voltage and line current
zero at TO. The off or non-conducting mode starts at by computing the dynamic response and eigenvalues as
0 varies [8].
TO and continues until the next firing pulse is applied at
951/2, where the subscript 1/2 refers to the half period.
The system progresses through a thyristor firing at time - simulation
4112, a thyristor turn off at 7112 and a thyristor firing
1.5 0 Jacobian
at 4 1 until arriving at the end of the period at t o T +
' with states F(z0). The Poincarb map F ( z 0 ) takes into
account the dependence of the switching times on the
system states and can be computed by integrating over
the period the appropriate state equations as determined
by the switching times.

to ti =to +T
on , , on , off , on
40 i
I

TO
Off
I

4ip
I

71/2
I

41 ;
I
71
1.5 1
1

20 :F&O)
Figure 2. System dynamics over one period.
The small signal stability of the TCSC can now be
I o IBasecwent(A)
L

computed from the Jacobian of the Poincarb map evalu- -0.5

ated at a sampled steady state 20.
(b)
Jacobian = DF(z0) 0.1 0.2 0.3
T i e (seconds)
The detailed derivation of the Poincarb map and its Jaco- Figure 3. Response of line current to 4" step change
bian is given in Appendices B and C. The eigenvalues of in U with synchronization on (a) voltage (b) line current.
the Jacobian DF provide information on the small signal
dynamics of the system. More precisely, the eigenvalues
are the poles of the sampled data transfer function which The Kayenta system is used for all examples. The
is the best linear approximation to the system response dynamic open loop response for both voltage and current
 1611

synchronization are shown in Figure 3. The solid lines while at other values they lie on the real axis. This is
represent the solution of the state equations for succes- best seen in figure 4(a) and (b). The letters A,B,C in-
sive sample points. A detailed EMTP simulation was dicate the values of U used in Figure 3. The dynamics
done to confirm these results. The dots are derived by calculated in [l]by fitting a fourth order transfer function
using the Jacobian to compute the dynamic response of are also shown as rectangles in Figure 4, The differences
the transfer function. The Jacobian computation closely may be due to the fitting methods used or the inclusion
approximates the simulated dynamic response. Figure 3 of filters or other dynamical components in the modeling.
clearly shows the dependence of the dynamics on U . The Note that one of the eigenvalues leaves the unit
data using voltage synchronization agree with other p u b circle for values of U greater than 1400 which implies
lished results [l]. that the halfwave symmetric solution becomes unstable.
EMTP simulations verify this fact. (More detailed anal-
ysis shows that the stability is lost by an eigenvalue of
the Jacobian of the halfwave map leaving the unit circle
at -1 in a period doubling bifurcation.)

I 0 Data from Table 1 I

0 (degrees)

Figure 4. Eigenvalues for voltage synchronization.

(a) locus of eigenvalues as U varies (b) Attenuation; real
pole=---- , complex pole- (c) Modulation frequency
0 (degrees)

Figure 4 shows the details of the calculated eigenval- Figure 5. Eigenvalues for current synchronization.
ues when the firing is synchronized to zeros of the voltage (a) locus of eigenvalues as U varies (b) Attenuation; real
across the TCSC. The system has four eigenvalues. For pole=---- ,complex pole- (c) Modulation frequency
some values of U these eigenvalues form complex pairs
1612
Figure 5 shows the details of the calculated eigen- The dynamic response to step changes in a is shown
values when the firing is synchronized to zeros of the line in Figure 7. The step changes in a are the same as for the
current. This synchronization was used in the final con- open loop case. The dynamics are greatly improved when
trols for Kayenta. In general the line current is very sts- compared to the open loop response shown in Figure 3b.
ble and less susceptible to ambient harmonics. The half The dynamics are almost uniform across the operating
wave symmetric steady state is unstable for a between range of ~7and overshoots and undershoots are greatly
69O and 76O and is shown as a gray area in Figure 5. The reduced. The transients settle in half the time required
differences in the eigenvalues of Figures 4 and 5 show the in the open loop cases.
significant differences in dynamic response between volt-
age and current synchronizations.
Closed loop control
A basic control issue for TCSC is the dependence of
the dynamic response on the operating point of a;see
Figure 3. To reduce this variance, a feedback control on
a is proposed. In this controller the error function is
the difference between a requested and a measured
value a,,,.The measured value a, is updated twice per
period while the requested value can change continuously
depending on the action of higher level controllers. The
controller is a PI controller as shown in Figure 6. The
controller gains Kp = -1OExp [-d(65 - 0re,)/2] and
Ki = -23-24 Exp [(a,,,- 65)/2] were chosen to develop
a uniform response over the TCSC impedance ranging
from 1.0 to 3.0 p.u. capacitive, or a less than 6 4 O (the
normal capacitive range of operation).

Figure 6. a Controller.
35 t
The formulation of the Poincar6 map of the closed
loop system and the computation of its Jacobian are de-
scribed in Appendix D. For this example, line current
synchronization was assumed and two extra states were
added to represent the control.

- Simulation
Jacobian
I 8 16 24
(3
32 40
(degrees)
48 56

Figure 8. Eigendues of the closed loop system.

(a) locus of eigenvalues as a varies (b) Attenuation; real
pole=---- ,complex pole- (c) Modulation frequency.

The closed loop eigenvalues are shown in Figure 8.

In this example, the operation of TCSC was limited to
Time (seconds)
the capacitive region, or a less than 64O. Compared to
the open loop response in Figure 5 there are important
Figure 7. Closed loop system response. differences. Comparing the dotted curves in Figure 5 (b)
and Figure 8 (b) shows a marked difference in the damp
 1613

ing near 64O. In the open loop case, the damping has an Institute for Math and its Applications workshop
oscillatory behavior showing very low attenuation close paper, March 1993, to appear in Systems and Con-
to 64O whereas in the closed loop case, the attenuation trol Theory for Power Systems (eds. J. Chow, P.
is much more uniform. The modulation frequencies also Kokotovic, R.J. Thomas), IMA volumes in mathe-
show more uniform behavior for the closed loop case. matics and its applications, Springer Verlag.
[8] S.G. Jalali, “Harmonics and Instabilities in Thyris-
Conclusion tor Based Switching Circuits”, Ph.D. Thesis, Uni-
The effects of synchronization and thyristor switch- versity of Wisconsin at Madison, 1993.
ing on system dynamics have traditionally been difficult [9] J.M.T. Thompson, H.B. Stewart, Nonlinear Dy-
to include in models for control design. Detailed and namics and chaos: geometrical methods for scien-
time consuming simulations have been used to approxi- tists and engineers, John Wiley, London, 1987.
mate transfer functions for the control design at a selec- [lo] I. Dobson, S.G. Jalali, “Surprising simplification of
tion of operating points. This paper derives formulas for the Jacobian of a diode switching circuit”, IEEE
the linearized dynamics of a TCSC system with synchro- Intl. Symp. on Circuits and Systems, Chicago, IL,
nization and a feedback control. Evaluating the formulas May 1993, pp. 2652-2655.
yields the system Jacobian and eigenvalues as a function [ll] R. Rajaraman, I. Dobson, S.G. Jalali, “Nonlin-
of firing angle U which can then be used in controller de- ear dynamics and switching time bifurcations of a
sign. It is more insightful and much quicker to evaluate thyristor controlled reactor” , IEEE Intl. Symp. on
formulas for a general operating point than to numer- Circuits and Systems, Chicago, IL, May 1993, pp.
ically approximate transfer functions with simulations. 2180-2183.
Currently the authors are investigating the applications
of these methods to SSR damping using TCSC, the de- Appendix A. System Modeling
sign of SVC compensators and HVDC controllers. This section describes the system modeling with
thyristor firing synchronized with voltage or current ZB
Acknowledgment ros. During the thyristor conduction time, the system
The authors gratefully acknowledge funding in part I , ( t ) ,VS(t))’,where I,
state vector is z ( t ) = ( I r ( t ) Vr(t),
,
from EPRI under contracts RP 4000-29 and RP 8010-30 is the thyristor current, V, is the thyristor controlled ca-
and from NSF PYI grant ECS-9157192. pacitor voltage, I, is the line current and V, is the fixed
References capacitor voltage. The system dynamics are described
[l]N. Christl, R.Hedin, K. Sadek et al., “Advanced se- by the linear differential equations:
ries compensation (ASC) with thyristor controlled X=h+Bu (All
impedance”, CIGRk 14/37/3&05, Aug. 1992.
[2] R. Johnson, P. Krause, A. Montoya, N. Christl, R. where

(
Hedin, “Power system studies and modeling for the 0 L;’ 0 0 0
c;l
(
Kayenta 230kV substation advanced series compen- -c;1 0
A= o L;’ -R,L;’ B= +1)
sation”, IEE Fifth Intl. Conf. on AC and DC Power
transmission, Sept. 1991, London UK. 0 0 cy1 0
[3] J.P. Louis, “Non-linear and linearized models for , L, = 406 mH and R, = 19.89 fl are the total line
control systems including static convertors”, Third impedance and resistance, C, = 27.9 pF is the fixed
Intl. Federation on Automatic Control Symp. on capacitor, L, = 6.8 mH and C, = 177 pF are the thyris-
Control in power electronics and electrical drives, tor controlled reactor and capacitor, and the net source
Lausanne, Switzerland, Sept. 1983 pp 9-16. voltage u ( t ) = 67sinwt kV. During the off time of each
[4] M. Grotzbach, R. von Lutz, “Unified modeling thyristor, I, is identically zero and the system state vec-
of rectifier-controlled DC-power supplies”, IEEE tor is y ( t ) = (Vc(t),13(t),V3(t))t
and the system dynam-
Trans. on Power Electronics, Vol. 1, No. 2, April ics are
1986, pp. 90-100. y = PAPty PBu + (A21

(:: : :)
[5] G.C. Verghese, M.E. Elbuluk, J.G. Kassakian, “A
general approach to sampled-data modeling for
power electronic circuits”, IEEE Trans. on Power
where P is the projection matrix P = 0 0 1 0 .
Electronics, Vol. 1, No. 2, April 1986, pp. 76-89.
The circuit state at the turn on time 40 is denoted
[6] S.G. Jalali, I. Dobson, R.H.Lasseter, “Instabilities
either by ~ ( 4 0 or
) by ~ ( 4 0 and
) these are related by
due to bifurcation of switching times in a thyris-
tor controlled reactor” , Power Electronics Special- 440) = P”(40) (A31
ists Conf., Toledo, Spain, July 1992, pp. 546-552.
The state at the turn off time TO is denoted either by
[7] I. Dobson, S.G. Jalali, R. Rajaraman, “Damping
~ ( 7 0 or
) TO) and these are related by
and Resonance in a High Power Switching Circuit”,
Y(T0) = P4.0) . (A41
1614

The thyristor turn on times at 41/2and 41 depend By the chain rule, the Jacobian of the half period map
on the firing scheme: is:
(a) Synchronizingthe firing with respect to the zeros
of the voltage V, is given by

41/2 = TWO + 7~ - ureq/2 (A51 Now we compute the partial derivatives of Ho in (Cl).
where T ~ Osatisfies V,(T,O) = 0 and is the requested bHo - eA(tl/a-61 12 )ptePAPt(+i/2-~o)peA(To - t o ) . (c2)
value of U . T,O is assumed to occur when the thyristor is ax0
conducting. since the only term of Ho(x0,TO,4112) depending on xo is
(b) Synchronizingthe firing with respect to the zeros the right hand side of (C2) times 20. An important sim-
of the line current I$ is given by plification proved in (3,101 and also used in [6,7] asserts
that
41/2 = 7c0 + ( n - areq)/2 (A61 -aH0
=o (C3)
a70
where T,O satisfies 18(7&)= 0. T,O is assumed to occur
Ho(z0, TO, ~ ~ 5 ~ 1can
2 ) also be written as
when both the thyristor are off.
Appendix B. Poincar6 map
This section sketches the construction of a Poincarh
map as in [6,10,8]. Given a time interval [sl,s2], it is
and differentiating and using (Al) gives
convenient to write ~ ( Z , S ~ , S Zfor ) the map which ad-
vances the state z ( s 1 ) at 51 to the state Z(52) at 52. If
the thyristor is on during all of the time interval [SI, SZ],
we write f ( x , 51, 52) as fon(x,51, 52). Similarly, if the
thyristor is off during all of [ s l , s 2 ] , we write f(y, 51, 52) (The notation y(&12-) means the limit of y(t) as t a p
as foff(y,S I , s a ) . A half period map advancing the state proaches 4112 from below.) Since (A2), (A4) and (Al)
from t o to tl12 may be written in terms of fori and foff imply that 3i(41,2-) = Pi(41/2+)and the form of P
taking into account the coordinate changes (A3) or (A4) implies that PtP - I = -ctc, where c = (1, O,O, 0), (C4)
at the switching times: may be rewritten as

Note that ~ k ( 4 ~ / 2 +is) the gradient of the thyristor cur-

The Poincar6 map F may now be written by composing rent as it turns on at 4 1 / 2 .
two successive half period maps and then neglecting the Now we compute~theterm Dqh1/2 in (Cl). The row
details of the time arguments: vector D 4 1 p is the gradient of the turn on time 4x12
with respect to XO. D41/2depends on the firing scheme:
(a) For voltage synchronization, differentiation of
(A5) yields D & / 2 = DT,Oso that we need to compute
DT,~. The constraint determining T ~ Ois 0 = Vr(Tu0) =
~ x ( T , o )= mfon(20,t o , T,O) where m = (0,l , O , 0). Dif-
ferentiation with respect to $0 yields 0 = mDfon +
Appendix C. Computation of the Jacobian m~lTuOD= ~ umeA(Tuo-to)
O + m$(~,o)D~,,oand
This section derives the formulas to compute the
Jacobian D F . Since the thyristor turn off time and the Ddlp = DT,O = -meA(Tuo-to)/m~(~,O) (C6a)
Poincarh map are discontinuous at a switching time bi- Note that mk(~,o) is the gradient of V, as it passes
furcation, we assume that the system is not at a switch- through zero at 7,o.
ing time bifurcation [6,11,8]. The first step is to compute (b) For current synchronization, differentiation of
the Jacobian of the half period map f(zo,t o , t1/2). De- (A6) yields D 4 1 / 2 = D T , ~ . The constraint determining
fine HO(ZO, TO, 4112) to be the right hand side of (Bl). HO
T,O is 0 = I#(T&) = ny(T,o) where n = (0,1,O). Differen-
expresses z ( t l / 2 ) as a function of XO, the turn off time TO tiating the constraint as in case (a) yields
and the turn on time 41/2.TO is a function of zo which
is determined by the constraint of zero thyristor current D 4 1 / 2 = DTCO
= -nDf(zo, t o , Tco)/n@(Tco)
at time TO. The turn on time 4 1 / 2 depends on 20 via the - -n e ~ A ~ ' ( ~ ~ - ~ ~ ) p e A/ni(T,o)
( ~ = ~ - t o(C6b)
)
firing strategy. The half period map may be written as
where the final expression for Df(so,to,~,o)was ob-
tained using the simplification (C3).
 1615

The corresponding equations for half period map g ( z o , t o , t l / 2 ) advancing the state zo = (ZO,UO,EO)‘ to
f(x1/2,t1/2,t1) can be obtained from the results above the state 2112 = ( z 1 / 2 , ~ 1 p , e 1 / 2 ) ’ are (D4), (D2) and
by changing all of the subscripts ‘1/2’ to ‘1’ and all of (Dl). Differentiating these equations, using the simpli-
the subscripts ‘0’to ‘1/2). Finally, differentiating (B2) fication (C3), and omitting terms which vanish because
and using the chain rule, the Jacobian of the Poincark eo = e l 1 2 = 0 when DzOgis evaluated at the fixed point
map is gives
DZog(ZOz0, t o , t1/2) =
DF = D f ( X 1 / 2 , t 1 / 2 , tl)Df(ZO, t o , t1/2) (C8)
If the steady state waveform is assumed to be half wave
symmetric, then (C8) simplifies to
.-
(D5)
where 2 and E are given by (C2) and (C5) and the
term D 4 1 j 2 evaluates to
or, in detail for the case of voltage synchronization,
04112 + + +
= DTCO ( K p K i ( ~ , o ao - T O ) ) (WO
- DTO)
(D6)
In (D6), DrCois given by (C6b) and it remains to com-
-
pute D ~ O070. The constraint equation determining
TO is 0 = &(TO) = =(TO) = cfon(zo,t~,to ro), where +
c = (1,0,0,0). Differentiation with respect to zoyields
0 = cDfon+ c* lTo DTO= ~ e ~ ( ~ 0 -+’ &(To-)DTO
0 ) and
DTO= - c e A ( T o - t o ) / & ( ~ O - ) . Note that &(ro-) is the
In the case of current synchronization, one eigenvalue of gradient of the thyristor current as it turns off at TO.
DF is identically zero. Similarly, ~ 4 =0-ceA(do--to) / ci(40+) and hence
Appendix D. Closed loop control
The controller equations can be written by inspect- ~h - 0 7 0 = -ceA(70-tO)/C3t(T0-)+ceA(do-to) /440+)

ing figs. 2 and 6. The error function e is updated at each

The equations for the half wave map 9 ( ~ 1 / 2 tli2,
, tl)
thyristor switch off. The error function at time tl12 is
and its Jacobian D21/2g(21/2, t l / 2 , t l ) can be transcribed
from the corresponding results above by changing all
e1/2 = are, - (70- 40) (D1) of the subscripts ‘1/2’ to ‘1’ and all of the s u b
where 40 can be computed from XO. The integrator out- scripts ‘0’to ‘1/2’. The Poincar6 map G(z0) =
put U at time t l p is g(g(zo,to, tip), t1/2, ti) and finally the Poincar6 map Ja-
cobian can be computed from
all2+ Ki€O(TO - t o ) +
= a0 Ki€1/2(t1/2 - TO)
-to) +
= uo + K~Q(TO + 40-Ki(0req ~ 0 ) ( t 1 / 2- 70)(D2)
We remark that one eigenvalue of D,,G is identically
The midperiod turn on time 4 1 / 2 depends on rCo and the
zero and that (D7) does not simplify under half wave
control output at time 4112:
symmetry.
41/2 = ~ c o + ~ p ~ l / 2 + ~ o + ~ i ~ o ~ 7 0 - ~ O ) + ~ i ~ 1 / 2 ( 4 1 / 2 - 7Sasan
0) Jalali (S’92) received the BS, MS and PhD in
Electrical Engineering in 1988, 1990 and 1993 from the
and solving for 4112 and substituting for ell2 from (Dl) University of Wisconsin-Madison. His interests include
gives application of power electronics to utility systems
41/2 = Robert H. Lasseter (F’92) received the PhD degree
in Physics at the University of Pennsylvania, Philadel-
phia in 1971. He was a Consultant Engineer at Gen-
eral Electric Company until he joined the University of
Wisconsin-Madison in 1980. His main interest is the a p
plication of power electronics to utility systems.
x1/2 = ~ o ( ~ o , ~ o ( z4 1o/ )2 (, 5 0 , ao, Eo)) 034)
Ian Dobson (M’89) received the PhD in Electrical En-
gineering from Cornel1 University and joined the Univer-
The circuit state x is augmented by the integrator sity of Wisconsin-Madison faculty in 1989. His current
output and the error to form a 6 dimensional state vec- interests are bifurcations, nonlinear dynamics, voltage
tor z = (z,a, e)’. The equations for the half wave map collapse and switching circuits.