A SOLID STATE CONTROLLER FOR INDUCTION LOADS
by
George Havas
Submitted in Partial Fulfillment of the Requirements
for the Degree of
Master of Science in Engineering
in the
Electrical Engineering
Program
Adviser
De n of the Gra uate School
YOUNGSTOWN STATE UNIVERSITY
August, 1972
ABSTRACT
A SOLID STATE CONTROLLER FOR INDUCTION LOADS
George Havas
Master of Science in Engineering
Youngstown State University, 1972
This thesis presents the results of an investigation
concerning a thyristor controller with an induction heating load.
Approximate and exact methods of analysis are utilized to predict
system behavior. Unstable regions and an interesting jump phenomenon
are predicted and experimentally verified. Extensive use has been
made of a digital computer in predicting system performance.
ii
ACKN~DGEMENTS
The author wishes to express his gratitude to Professor
M. Siman of the Department of Electrical Engineering, Youngstown
State University, for his continuous guidance and encouragement in
the preparation of this thesis.
The financial assistance of Ajax Magnethermic Corporation
is also gratefully acknowledged.
iii
iv
TABLE OF CONTENTS
PAGE
ABSTRACT ? ? . ?
ACKNOWLEDGEMENTS
TABLE OF CONTENTS
LIST OF FIGURES
CHAPTER
ii
iii
iv
vi
Mode B Operation.
A PRACTICAL STEADY STATE ANALYSIS
Discussion .
Assumptions
INTRODUCTION ? ? . ? .
DEFINITION OF TERMS
1
3
3
4
5
6
7
9
10
11
11
13
18
19
19
20
21
23
29
? ? ? ? ? " 0
Equivalent Circuit Models
Analysis Using Circle Diagrams ?
Basic Controller
Control Scheme . .
Basic Operation
Representation of Load .
Definition of Terms
Conclusions . . ? . .
Discussion .
Methods of Analysis
Mathematical Derivations .
Mode A Operation.
EXACT ANALYSIS .
I.
II.
IV.
III.
v
PAGE
CHAPTER
IV. EXACT ANALYSIS (Continued)
Voltage and Current Expressions in the Circuit ? 31
Determination of Rms Circuit Quantities 31
Power and Power Factor ? 32
Harmonic Analysis 32
Prediction of System Performance ? 33
Conduction Angle vs. Firing Angle 34
. . . . 46
63
72
78
83
84
85
Line Current Harmonics ?
Power, Input Power Factor, Rms Line Current and Load
Voltage. 0 ?????
Experimental Verification
Comparison of Approximate and Exact Theoretical
Results ? ? ? ?
Conclusions
V. CONCLUSIONS
REFERENCES ? ? . ? .
LIST OF FIGURES
vi
FIGURE PAGE
1. Basic Controller Schematic ? ? ? ? ? ? ? ? 3
2. Modes of Operation and Symmetrical Phase Control 4
3. Basic Model of Parallel Compensated Induction Load 6
4. Controller Circuit Diagram ? 10
5. Harmonic Model of Controller 12
6. Fundamental Model of Controller 12
7. Circle Diagram, XL = .2, PF = 1.0 15
8. Circle Diagram, XL = .2, PF = .9 16
9. Circle Diagram, XL = .2, PF = -.9 17
10. Complete Controller Circuit 22
11. Equivalent Circuit for Mode A Operation 23
12. Equivalent Circuit for Mode B Operation 29
13. Conduction Angle vs . Firing Angle for Q = 1.0 and
XL = . 0714 . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 35
14. Conduction Angle vs. Firing Angle for Q = 1.0 and
XL = .1 . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 36
15. Conduction Angle VS. Firing Angle for Q = 1.0 and
XL = .2 . . ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 37
16. Conduction Angle VS. Firing Angle for Q = 1.0 and
XL = .286 ? ? ? ? ? ? ? ? ? ? ? ? ? 38
17. Conduction Angle vs. Firing Angle for Q = 2.0 and
XL = .0714 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 39
18. Conduction Angle VS. Firing Angle for Q = 2.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? ? 40
19. Conduction Angle vs. Firing Angle for Q = 2.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? 41
vii
FIGURE PAGE
20. Conduction Angle vs. Firing Angle for Q = 5.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? 42
2l. Conduction Angle vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? 43
22. Conduction Angle vs. Firing Angle for Q = 10.0 and
XL = .1 ? ? ? ? . ? ? ? ? 44
23. Conduction Angle vs. Firing Angle for Q = 10.0 and
XL = .2 ? ? ? ? ? ? ? ? 45
24. Rms Line Current vs. Firing Angle for Q = 5.0 and XL=?l 47
25. Average Output Power vs. Firing Angle for Q = 5.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? ? 48
26. Rms Output Voltage vs. Firing Angle for Q = 5.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? ? 49
27. Input Power Factor vs. Firing Angle for Q = 5.0 and
XL = .1 ? ? ? ? 0 ? ? ? ? ? ? ? . ? 50
28. Rms Line Current vs. Firing Angle for Q=5.0 and XL=?2 . 51
29. Average Output Power vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? . ? . 52
30. Rms Output Voltage vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? 53
3l. Input Power Factor vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? 54
32. Rms Line Current vs. Firing Angle for Q= 10.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? ? ? 55
33. Average Output Power vs. Firing Angle for Q = 10.0 and
XL = .1 ? ? ? ? ? ? ? ? ? ? . . 0 ? ? ? . . 56
34. Rms Output Voltage vs. Firing Angle for Q = 10.0 and
XL = .1 ? ? ? ? ? 0 ? ? ? ? ? ? ? ? . 57
35. Input Power Factor vs. Firing Angle for Q = 10.0 and
XL = .1 ? " ? ? ? ? ? ? ? ? ? ? ? ? ? ? 58
36. Rms Line Current vs. Firing Angle for Q = 10.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? . . ? ? ? ? 59
viii
FIGURE PAGE
37. Average Output Power vs. Firing Angle for Q = 10.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 60
38. Rms Output Voltage vs. Firing Angle for Q = 10.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 61
39. Input Power Factor vs. Firing Angle for Q = 10.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? ? ? 62
40. Line Current Harmonics vS.Firing Angle for Q = 1.0,
XL = .1 and PF = .9 ? ? ? 64
41. Line Current Harmonics vs. Firing Angle for Q = 1.0,
XL = .2 and PF = 1.0 ? ? ? ? ? 65
42. Line Current Harmonics vs. Firing Angle for Q = 1.0,
XL = .2 and PF = .9 ? ? ? ? ? ? ? ? ? 66
43. Line Current Harmonics vs .Firing Angle for Q = 2.0 ,
XL = .1 and PF = 1.0 ? ? ? ? ? ? ? ? ? ? 67
44. Line Current Harmonics vs. Firing Angle for Q = 2.0,
XL - .1 and PF = .9 ? ? ? ? ? ? ? ? ? 68
45. Line Current Harmonics vs. Firing Angle for Q = 2.0,
XL = .2 and PF = -.9 ? ? ? 69
46. Line Current Harmonics vs. Firing Angle for Q = 2.0,
XL = .2 and PF = 1.0 ? ? ? ? ? ? ? ? ? 70
47. Line Current Harmonics vs. Firing Angle for Q = 2.0,
XL = .2 and PF .9 ? ? ? ? ? ? ? ? ? ? 71
48. Rms Line Current vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? 73
49. Average Output Power vs. Firing Angle for Q = 5.0 and
XL = .2 ? ? ? ? ? ? ? ? ? ? ? 74
50. Rms Output Voltage vs . Firing Angle for Q = 5.0 and
XL = . 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? 75
51. Conduction Angle vs. Firing Angle for Q = 1.0, XL = .1
and PF = 1.0 ? ? ? ? ? ? ? ? ? ? 76
52. Conduction Angle vs. Firing Angle for Q = 1.0, XL = .2
and PF = -.9 ? ? ? ? . ? ? ? 77
53. Output Power vs. Firing Angle for Q = 1.0, XL = .2 and
PF = 1.0 ? ? ? ? ? ? . . . ? ? ? ? ? ? ? 79
FIGURE
54.
55.
56.
Output Power vs. Firing Angle for Q = 10.0, XL = .2
and PF = 1.0 ..??..
Real vs. Reactive Fundamental Input Power for Q = 1.0,
XL = .2 and PF = 1.0 . . . .
Real vs. Reactive Fundamental Input Power for Q = 10.0,
XL = .2 and PF = 1.0 . . . .
ix
PAGE
80
81
82
1
C~PITR I
INTRODUCTION
The phenomenal development in the past few years of the four 
layer semiconductor device, the thyristor, more commonly known as an
SCR (silicon-controlled rectifier), opened a brand-new dimension in
efficient and economical switching and controlling of large blocks of
electrical power. SCR's have been used for controlling rotating equip 
ment, as switching devices in frequency converters and in many other
industrial and commercial applications.
In this thesis, the use of SCR's in switching and controlling
power supplied to large induction heating or melting loads is in 
vestigated.
The utilization of SCR's has the following inherent advantages
over conventional electromechanical or magnetic means of power control:
1. Higher Efficiency.
2. No moving parts.
3. No component deterioration.
4. Small physical size.
5. Transient-free starting.
6. Half-cycle fault protection.
7. Extremely fast response time.
However, the use of SCR's requires a deeper and better under 
standing of the complete system so that its performance under various
load conditions can be predicted. The prediction of system performance is
2
not straight-forward due to the non-sinusoidal nature of the SCR
currents during the controlled mode of operation.
In the following chapters, the definition of basic terms,
a method of approximate analysis, the derivation of equations necessary
for exact analysis, digital simulation of the system and the comparison
of predicted performance versus experimental results is given.
In choosing the method of analysis for the digital simulations,
the following factors were of primary concern:
1. Accuracy of calculations.
2. Limited computer working core capacity (8K).
3. Limited computer time available.
In other words, the highest degree of accuracy was desired
with short computational time using a computer (IBM 1130) with
limited working core capacity.
An attempt has been made throughout this study to consider
cases of practical importance and to link the theoretical results
meaningfully to practical considerations.
3
CHAPTER II
DEFINITION OF TERMS
This chapter introduces the basic controller configuration,
its control scheme and operation and gives a definition of the basic
terms used throughout the following chapters, as well as a short dis-
cussion of the general nature and range of the load.
Basic Controller
Th.l
L I --------I
iL .I I
Th.2 Vc Lo I
v C I
1.0
Ro I
L- ------- _-.J
Load
Fig.l Basic controller schematic
Fig.l is a schematic diagram of the controller. It consists of
a series inductance L, antiparallel connected thyristor switches Th.l,
Th.2, and a parallel-tuned tank circuit which is a simplified lumped
parameter representation of an induction load with its power factor
correcting capacitances.
4
Control Scheme
Stepless, continuous control of the power delivered to the
load can be obtained by the use of a sYmmetrical phase control.
Th.l in Th.2 in Th.l in Th.2 in
I? cond., I I- condo a I I?condo I?condo I I
I I I ( I
I I I , II
I I I I~\T+e:p I
0 I" I , "51\ I
1
'ModeI kde IGPI Mode Mode Mode Mode
A B A- A A-
I I
I I
I I
I UJtI
I ,
GP~l I I0 D
?vst.
Fig. 2 Modes of operation and sYmmetrical phase control
Fig.2 illustrates the principle of sYmmetrical phase control.
The input sinusoid v represents the time reference source. Gating
pulses GPI and GP2 are derived and controlled in such a manner that
for any integral cycle they have a fixed interval between the zero
?
5
crossing of the reference supply and the start of the gate pulses. The
interval between the start of each gate pulse and the previous zero
crossing of the reference supply is defined as the firing angle ~ ?
Symmetrical control of the load voltage, current or power is achieved
by continuously varying the firing angle on an integral cycle basis.
The symmetrical operation of the controller is extremely important
in eliminating direct current and even harmonics from the supply
lines.
Basic Operation
The steady state operation of the controller can be described
as a repetition of two transient modes. Fig.2 illustrates these modes.
Let us assume that Th.l is connected so that it conducts
current during the positive half cycle. Mode A starts when Th.l is
rendered conductive at firing angle ~ by gate pulse GPI. During this
mode, current flows from the supply to the load. When the current
through Th.l goes through zero, Th.l opens. This is when Mode B begins.
During this mode, both Th.l and Th.2 are open, that is, the load is
disconnected from the supply. The load now proceeds to resonate due
to the charge on the capacitor and current in the load inductance
from the previous mode. Mode B continues until UJ1:.=1T-t-ep, when Th.2
is rendered conductive, repeating Mode A for the negative half-cycle.
The steady state operation of the system then consists of the repeti 
tion of Modes A and B in the positive and negative half-cycles.
6
Representation of Load
The true reflected impedance of an induction heating load is a
complicated function dependent on the coupling between load and work
coil and on the resistivity, permeability and temperature of the load.
The frequency of operation and the geometry of the load also contribute
to the reflected impedance.
However, if one considers the total net effect of all the
factors mentioned above, the induction heating load can be represented
by a series RL network where R is the reflected resistance and L is
the reflected inductance of the load. Since the typical induction load
has a low power factor typical values are .1 .3, with occasional
values up to .7 parallel power factor correcting capacitors are
used for better supply current versus power utilization. Normally, an
induction heating load is corrected to nearly unity power factor. The
basic model of the load is shown in Fig.3?
? c
Fig. 3 Basic model of parallel compensated induction load
7
Definition of'Terms
characterized by:
power factor.
The parallel load as shown in Fig.3 consists of the basic
(1)
( 2)
fundamental supply frequency
Lo
Ro C
::=
Q
Req
and
The quality factor Q is defined by the equation
uJLo::=
"R;""
f is the
The equivalent impedance of the load at unity power factor is
Req - the impedance of the load to fundamental frequency at unity
parameters Ro, Lo and C. In the following chapters, the load will be
P.E the load power factor.
Q - the quality factor of the load.
given by the equation
In general, Req will be normalized to unity.
where YJ=2.:ttf in rad/sec
in Hertz.
In a practical circuit, the load power factor is held close to
unity by periodic adjustments of the parallel capacitors. However, due
to the rapid changes of the load, the power factor is held within a
fixed lead-lag band. To simulate these conditions, the value of the
parallel capacitance C is varied,with Ro and Lo held constant
until the desired load power factor is obtained. The normalized
magnitude of the load impedance at any operating power factor is then
given by the equation
1
nl..U C
0--------1 In
12
Fig. 5
where n = 3, 5, 700.
Harmonic model of controller
Fig. 6 Fundamental model of controller
13
A brief numerical example using the harmonic representation of
the compensated load illustrates well the shunting effect of the load
power factor correcting capacitor. For a load whose Q =: 3, it can be
shown that at the fundamental frequency,
Ro(l) 1 =: 1 =: .1- Q2+1 10
XLo(l) =: Q Ro =: 3 =: 03 and10
Xc(l) =: 1 =: 1 . 033Q 3
At the third harmonic frequency,
Ro(3) =: 1 =: .110
XLo(3) =: 9 =: .9 and10
Xc(3) =: 1 =: .119 .
The above simple calculations show that at the third harmonic
frequency the capacitance has about one ninth the impedance of the in-
ductive load. It is not difficult to see that for higher harmonics the
capacitance acts nearly as a short circuit compared to the inductive
load.
Analysis Using Circle Diagrams
Making use of the fundamental model, the controller's behavior
under various load conditions can be analyzed with the help of circle
diagrams. The following figures show the controller behavior from full
conduction to <p =: 1800 0
In all three figures,
In Figo 7, Req = 1.0 Ohm, PF = 1.0
In Fig.B, PF = 09 and in Fig.9, PF = - .9.
= .2 Ohm.
14
the constant supply voltage V is assumed as a reference. As
the variable reactance uJLv in the equivalent circuit of Fig. 6 is
increased, the locus of the line current phasor IL describes the
semi-circle ao The projection of IL on the real axis gives the
average power. The locus of the output voltage phasor Vc describes
the semi-circle bOo Note that phasor Vc is in phase, leads or lags
phasor IL' depending on the power factor of the load. The cosine
of the angle between the reference phasor V and the phasor IL is
the input power factor of the system. The magnitude and phase of the
voltage across the line reactances ~L and uuLv can be readily
obtained by making use of the fact that the voltage drops across the
load and the line reactances must always add up to the supply voltage.
The behavior of line current, output voltage and power, as well
as the fundamental power factor of the system, can be obtained readily
throughout the whole control range. A verification of the accuracy of
this approximate method of analysis will be presented in Chapter IV
where experimental results will be compared to predicted results, using
the approximate as well as the exact computer method of analysis.
Real P,IL
15
Lead
-.4 -.2
.8
.6
.2
.2
.4
.6
.8
Req
PFin Circle
Reactive P,IL
in Ohms (p. u .)
Fig.7
Impedance
in Ohms(p.u.)
Circle Diagram, XL ~ .2, PF 1.0
Lead Real P,IL
.4
.6
.4
Impedance
in Ohms(p.u.)
of lL
.8in Ohms(p.u.)
16
Fig.8 Circle Diagram, XL = .2, PF = .9
Lead
Real P,IL
.6
.4
.6
.8
Impedance
in Ohms(p.u.)
Angle of PF
Reactance in Ohms(pouo)
17
Fig. 9 Circle Diagram, XL = .2~ PF -.9
Conclusions
In this chapter, a steady state analysis of the controller
has been developed. Simplifying assumptions were made which allowed
the use of circle diagrams. An approximate system performance for
load Q's ~ 3 was predicted. Based on this analysis, a Q-independent
nature of the system can be predicted for load Q's ~ 3 ?
18
19
CHAPTER IV
EXACT ANALYSIS
An exact analysis of the controller system is presented in
this chapter. The analysis which is developed has the advantage that
transient as well as steady state results can be obtained without the
use of an excessive amount of computer time. Furthermore, the following
is presented in this chapter:
(a) Mathematical derivations.
(b) The development of equations for the circuit voltages and
currents.
(c) Prediction of system performance.
(d) Experimental verification of theoretical results.
(e) Comparison of exact and approximate theoretical results.
Discussion
In order to predict the exact steady state and transient per 
formance of the system, to study system behavior under widely varying
load conditions, to observe the relationships between circuit compo 
nents and system performance and to find the criteria for the most
economical and best performing system, an exact analysis must be
undertaken.
When a system's performance must be evaluated for widely vary 
ing conditions, experimental methods of analysis must be ruled out
since they are time consuming and costly. A simulation method employing
20
a digital computer lends itself particularly well to handling large
amounts of data. If a large number of variables are involved, a com 
puter is a necessity.
While the experimental work can not be compl~tely replaced by
computer simulation, this simulation, if used judiciously, can be an
effective and efficient aid in design work.
Methods of Analysis
There are many mathematical methods available for analyzing
dynamic systems. The most commonly used techniques are the numerical
and the operational methods. In recent years, the method of state
variables, in conjunction with the Laplace transform and its inverse,
has gained wide usage.
The state variable method is based on the fact that the output
of a system depends entirely on its condition or state in the past
(initial conditions) and on its present input (forcing function). The
state variable method has a number of advantages over the conventional
loop and/or node analysis techniques. One of these is that the state
variable method decomposes a complex system into a set of simpler inter 
acting systems which then can be solved one at a time quite readily.
Another distinct advantage of the state variable method is that direct
information about the past state of the system is immediately obtain 
able from the state variables.
Each of the above methods requires the development of the
system equations. In applying numerical techniques, less analytical
work is required but computer time may be excessive. The selection of
21
appropriate step sizes can have a great bearing on obtaining stable
and accurate solutions. The operational methods require a great deal
more analytical work but much less computer time is required. Step
sizes are of no great significance and a relatively clear picture can
be obtained between the relationship of the circuit parameters and the
circuit functions.
In this chapter, the state variable method is used in conjunc 
tion with Laplace and inverse transformations. This method was chosen
because of the following considerations:
(a) Suitable for computer-aided analysis.
(b) Short computer time required.
(c) Solutions as a function of time are exact and are independent of
step size.
(d) Vital circuit voltages and currents can be obtained directly.
Mathematical Derivations
When using the state variables method, the system state equa 
tions must be developed and written in matrix form. Next, the Laplace
transform of the matrices is taken and the system is solved for the
state variables with all possible initial conditions. Inverse transform 
ation of the system then yields the state variables in the time domain.
When the time functions of the state variables (currents, voltages) are
obtained, their simulation on a digital computer can proceed directly.
In the simulation process, the end conditions of a circuit operating
mode are the initial conditions of the next mode of circuit operation.
When the initial conditions for the periodically repeated modes attain a
22
steady state value the circuit is considered in steady state operation.
The complete circuit is shown in Fig.10. In developing the
system equations, the following simplifying assumptions are made:
(a) Thyristor switches used in the circuit are ideal switches,
that is, they turn on and off instantaneously, have zero imped-
ance when conducting current and infinite impedance when block-
ing the flow of current.
(b) Losses in circuit elements are negligible compared to the power
in the load.
Th.l
.. L
ol.L
.1
h.2 V c
v C
~o Ro
Fig. 10 Complete controller circuit
The complete analysis requires the analysis of two transient
modes of circuit operation.
Mode A Operation
Mode A operation is shown in Fig.ll.
L
23
v
+
c
Fig. 11 Equivalent circuit for Mode A operation
The thyristor switch is closed and current flows from the
supply to the load. Using Fig.11, the state variable equations are~
diL = Vc + v ( 6)
dt L L
dvc
= iL i o ( 7)
dt C C
d' Vc ioRo1.0 (8)
dt = Lo ~
Written in matrix form, Equations (6) , (7) and "(8) are:
24
diL 0 0 1
dt L
dio 0 - Ro 1=
Lo Lodt
dvc 1 - 1 0
<it c c
+
1
L
o
o (9)
Equation (9) can be written in general matrix form as
?~=Ax+Br (10)
The general form of the sinusoidal forcing function v can be written
as
v = A sin (UJ t +4 )
Expanding equation (11) yields
v = A (sinw t cos c:I> + cos u..J t sin eP )
and the Laplace transform of v is:
(11)
(12)
1- (v) = A (t.u cos ef:> + s sin.q,)s2 +L.U 2 (13)
Taking the Laplace transform of equation (10) and solving for
x(s) with all possible initial conditions, we have
!(s) == [s,!, - ~J -1 ~(O+) + [s,!, - ~] -1 ~(s) ~(s) (14)
Written in terms of the circuit componenmof Fig.ll, equation
(14) is:
25
l L(s) s2 + Ro + 1 1 s(-1) Ro iL(O+)s- -- ---Lo LoC LC L LLo
10 (s) = 1 s2 + 1 s ....L io(O+)LoC LC Lo
Vc(s) s 1 Ro 1 s2 Ro Vc(O~C LoC C Lo
det (sI - !]
s2 + s Ro 1 1 s(-l) _ Ro-+-L
o LoC LC L Lto
+ 1 s2 + 1 1LoC LC Lo
s l Ro 1 s2 + Ro+-- s - s -C LoC C L
o
1
L
o
o
x [-s""'2-+;;';;~'--""'2-
det [SI - AJ
(\.0 cos cj> + s sin <p ~ (15)
given by
where det [sI - !] is the determinant of the matrix [sI - ~ and is
det r s_1 __ = s3 + s2 Ro + s (1 + I) + ~ (16)L Lo LC LoC LLoC
The roots of equation (16) are the roots of the characteristic
equation of the system.' They may be either all real or one may be real
while the other two are complex conjugate pairs. In what follows, the
latter case is assumed since in all practical induction loads the load
circuit is of the underdamped oscillatory type.
The first part of matrix equation (15) is only a function of
the initial conditions and the second part is a function of the forcing
function or input only. This shows clearly how the outputs or state
variables of a given system are strictly a function of the state in the
past (initial conditions1or part one of equation (15), and of the present
26
input (forcing function), or part two of equation (15).
Once the roots of det [Sl - A] =0 are found, the exact ex 
pressions for the state variables as a function of time can be obtained
with the aid of the inverse Laplace transformations. The state variables
as a function of time can be written as:
Yt Al - A2X + A3XZ c. -Xt= AL E:. - sin (Zt +ClC,) + (Y _ X)2 + Z2 ~
+ Bl - BZX + B3X 2 - B4X3 E. -Xt
(X2 + W 2) [(Y-X)2 + zZJ
+ AB sin (UJ t + o(~) + B E:. -Yt sin (Zt +(3) (17)
where
ALlo{l = Al - AZ (Y - jZ) + A3 (Y - jZ)Z
Z (X - Y + jZ)
Bl + B2 (jw) + B3(jw) 2 + B4(j U) ) 3ABlot~ =
LUrrj U3 + Y) 2 + Z2] (J 0..> + X)
Bl~ = Bl - B2 (Y  
Z [(Y -
jZ) + B3 (Y  
jZ)Z +t..U 2J jZ) 2 - B4 (Y   jZ)3
-YtE.= CL ) + Cl - C2X + C3X2 -Xtsin (Zt +0' (Y _ X)2 +Z2 ?..
Dl - D2X tcE.. -X
+ (X2 +lJ.J Z) [(Y - X)Z + z2J
+ CB sineUJ t + o%,) + DeE -Yt sin(Zt +d) (18)
where
CL~ + Cl - C2 (Y - jZ) + C3 (Y - jZ) 2
Z (X - Y + jZ)
Dl + D2 (jW)CB~=
W [(j LU + Y) Z + Z2J (jW + X)
27
Did = Dl - D2 (Y - jZ) (X - Y +
jZ)
Vc Yt El - E2X + E3X2 t:. -XtEL c. - sin(Zt +t=") + (Y _ X)2 + Z2
+ Fl - F2X + F3X2 -Xt
(X2 +lJ..)2) [(Y - X) 2 ..(. Z2J E.
+ EB sin(LtJ t +f'L) + F e. -Yt sin (Zt + E5 ) ( 19)
where
El - E2 (Y - jZ) + E3 (Y - jZ) 2EL\P, =
Z (X - Y + jZ)
Fl - F2 (jUJ) + F3(jL.U)2
EBlf':z. = luw + y)2 + z2]
(jUJ + X)
Fl - F2 (Y - jZ) + F3 (Y - jZ) 2FlSL
=
Z [(Y - jZ) 2 +UJ 2] (X - Y + jZ)
In the above equations, X is the real root of
det [s.! - A] = 0 and Y and Z are the real and imaginary parts
of the complex conjugate roots of det [s.! - A] = 00 The subscripted
constants in terms of the circuit constants are:
Al
A2 = 1
L
A3 = iL(O+)
Bl A UJ cos pLLoC
B2 = A <.u cos 1> Ro + A sinpLL
o LLoC
B3 = Au..> cos ep + A sin 4> Ro
L LLo
B4 = A sin 4>L
28
Cl
C2
C3
=
=
1
LC
Dl
D2
= A u.J cos c:I>
LLoC
A to sin ep=
LLoC
El
E2
=
=
Ro
LoC
E3
Fl = A U) cos cI> Ro
LLoC
F2 = A w cos ep Ro + A sin et> Ro
LLoC LLoC
F3
where
= A sin <pLC
A = amplitude of supply voltage (Volts)
f = supply frequency (Hertz)
u..J = 2 -no f = angular frequency of supply voltage (rad/sec)
~ = phase angle of supply voltage (also firing angle) (degrees)
Equations (17), (18) and (19) completely define the system
response for Mode A. All other system voltages and currents can be
easily calculated from the above equations. Mode A hence is ready for
digital simulation.
Mode B Operation
Mode B operation is shown in Fig.12.
L
29
v
----..... i o
C
+
Vc
Ro
Fig. 12 Equivalent Circuit for Mode B Operation
The thyristor switch is open. No current flows in the line in-
ductor. The load circuit continues to oscillate due to the initial
voltage on the load capacitor and the initial current in the load in-
ductance.
Writing the circuit equations
't1 5
iodt dio + i o Ro (20)Vc(O+) = C + L o .......-.-
0 . dt
assuming that the current is a decaying sinusoid,
= E. -at (Kl cos bt + K2 sin bt) (21)
where a =~ 2 Lo and b =w; then applying initial
30
conditions at t = 0 which are the final values of Mode A, gives:
i o = io(O~
Kl = io(O+)
dio = Vc(o+) io(O+) Ro
dt Lo (22)
K2 Vc(O+)
Lo b
io(~ a
b
( 23)
Substituting Kl and K2 into equation (21), gives:
i o = ?.. -at [ io(O+) cos bt + (vc(o+) io(O+) a) sin b~
Lo b b
and
Vc = Vc(O+) L dio i o Ro0--dt
(24)
( 25)
where dio is given by the equationat
dio = -a :io + E.. -at
[- bio(o+) sin bt +( Vc(O+)dt Lo io(O+) a :J
( 26)
This concludes the development of the equations for Mode B.
The system equations developed for Modes A and B describe
the system response as instantaneous, functions of time, Utilizing these
equations, a complete digital simulation of the system has been under-
taken,
31
Voltage and Current Expressions in the Circuit
Equations (17), (18) and (19) yield instantaneous values for
iL, i o and Vc' Utilizing these equations, the remaining circuit
voltages and currents can be found readily.
The current ic in the load capacitor is:
Mode A ic = iL i o
Mode B i C = - i o
The voltage VI.. across the series inductor is:
Mode A VI.. == v Vc
Mode B vL = 0
The voltage vTh across the thyristors is:
Mode A VTh = 0
Mode B vTh = v Vc
( 27)
(28)
(29)
(30)
(31)
(32)
Determination of Rms Circuit Quantities
Once steady state operating conditions are reached in the
circuit, the rms circuit voltages and currents are of primary interest.
The rms quantities were arrived at in the following manner:
n=mL..
'1'\: t
(33)
where = instantaneous value of current, t incremental time and
32
T = rnA t _is the period.
Power and Power Factor
The average power transferred to the load is given by:
P = (34)
where 10 is the rms value of i o .
The input power factor is given by:
PFin P= Vh
Harmonic Analysis
(35)
In order to obtain the harmonic content of the line current,
Fourier analysis is utilized.
00
iL = ao + L.. (an cos nUJt + bn sin nuJt)
r.=l
or
00
iL = L Cn sin (nUJ t + en )
"..,:\
T
where ao = 1 J iL(t) dtT
T
an = ~S iL(t) cos nwt dt
?T
bn = 2 S iL(t) sin nwt dtT
0
Cn = Van2 + b 2n
en = tan-1 (an/bn)
(36)
(37)
33
The magnitude of the fundamental component of iL and its phase
angle with respect to the line voltage v is given by :
C1 =ya12 + b1 2
Qph = e:p - tan-1 (a1/bl)
(38)
(39)
by:
The fundamental input power factor of the system then is given
= cos Qph (40)
and the fundamental input power by:
= (41)
Prediction of System Performance
The choice of system parameters for the results presented in the
following sections were based on:
1. Economical considerations for a practical system,
2. Considerations involving a practical load,
3. Practical and theoretical considerations involving system stability.
The following system parameters are used throughout the fo11ow-
ing sections:
XL = .0714, .1, .2, .286
Req = 1.0
PF = -.9, 1.0, .9
Q = 1, 2, 5, 10
34
Conduction Angle Versus Firing Angle
The plots of conduction angle versus firing angle for load
power factors of -.9 (lagging), 1.0 and .9 (leading), with various
values of line inductances are shown in Figs. 13 through 23. The plots
are shown in the order of increasing Q's. Examination of the curves
permits the following observations to be made:
1. The conduction angle is a non-linear function of the firing angle.
2. For the same firing angle the conduction angle decreases with XL.
3. For Q ~ 2 and XL ~ .2, the system is stable for all firing
angles.
40 For Q ~ 5, these characteristic curves are nearly independent of
the load Q.
5. Stability increases with load Q.
6. The system is more stable for leading power factor loads.
7. Stability increases with XL ?
8. Instabilities occur at low firing angles.
9. For certain conditions, a jump phenomenon can be observed.
10. Under certain conditions, the system does not become unstable but
two jumps occur as the firing angle is increased or decreased
(see arrows on Figs. 15 and 16).
The system is considered unstable if unsymmetrical currents
flow in the supply line despite the symmetrical gating of the thyristors.
This causes d.c. and subharmonics to flow in the line, causing the satu 
ration of the supply transformer.
35
180
---- Instability
100 120 140 160
PF = 1.0
PF = -.9
60
----""
o
30
60
\
120 \
\
\
\
\
90 \
-20
Firing angle in degrees
Fig. 13 Conduction angle vs. firing angle
for Q =1.0 and XL = .0714
PF = -.9
\
\ Instability
\ ----
\ PF 1.0
\ =
00 ~
OJ
OJ \l-I
bO 120 \OJ
't:l \
p \
'.-1
\OJ
\
bO PF .9p
t1l 90
p
0?.-1
ol-l
U::l
't:lP
0 60u
36
-20 o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 14 Conduction angle vs. firing angle
for Q = 1.0 and XL = .1
37
180
180
PF = -.9
100
PF = 1.0
80
30
UJ PF = .9
<lJ
<lJH
tlll
<lJ 120'0
s::?.-1
<lJ
.-l
tllls::
~ 90
s::0
-.-I
.j.J
CJ;::l
'0s::
0
Co) 60
-20
Firing angle in degrees
Fig. 15 Conduction angle vs. firing angle
for Q = 1.0 and XL = .2
-20
180
o 20 40 60 80 100 120 140 160 180
38
Firing angle in degrees
Fig. 16 Conduction angle vs. firing angle
for Q = 1.0 and XL = .286
C/)
Q)
Q)
l-IbO
Q)
"'C
{:lo
.r-!
+Jg
'8o 60
C,)
30
-20 0
Firing
---- Instability
PF = - .9
:::: .9
in degrees
39
Fig. 17 Conduction angle vs. firing angle
for Q = 2.0 and XL = .0714
40
Instability
180
= .9
= 1.0
180
PF = - .9
\
\
\
\
\
\
\
\
80
30
(l)
.--Ico
s::
Cll 90
s::o
.~
.j.Ju
;:l'"8
o
C,) 60
-20
Firing angle in degrees
Fig. 18 Conduction angle vs. firing angle
for Q = 200 and XL = .1
41
180
18010080604020o
til
Q)
Q)
HbO
Q)
'"0
-20
Firing angle in degrees
Fig. 19 Conduction angle vs. firing angle
for Q = 2.0 and XL = .2
42
180o-20
180
= - .9
PF = 1.0
Ul
Q) = .9
Q)
l-lbO
Q)
'0
l=:-r-!
Q)
.-I
bOl=:
ell
l=:0
.r-!.u
CJ::l
'0l=:
0 6
to)
Firing angle in degrees
Fig. 20 Conduction angle vs. firing angle
for Q = 5.0 and XL = .1
180
43
-20 o 20 40 60 80 100 120 140 160 180
Fig. 21
Firing angle in degrees
Conduction angle vs. firing angle
for Q = 5.0 and XL = .2
180
Ul
Q)
Q)
HbO
Q)
"0
l::-.-I
90
Q)
...-l
bOl::
til
s::0
-.-I
ol-J 60
()
;:1]
0u
30
-20 0
PF = - .9
PF = 1.0
PF = .9
Firing angle in degrees
Fig. 22 Conduction angle vs. firing angle
for Q = 10.0 and XL = .1
44
45
180
~PF = - .9
PF = 1.0
= .9
40
00 120
Q)
Q)
l-l00
Q)
'tl
l::-r-!
90
Q)
,.....I
00l::
Cd
l::0
-r-!
+J 60
()::l
'tll::
0
U
Firing angle in degrees
Fig. 23 Conduction angle vs. firing angle
for Q = 10.0 and XL = .2
Power, Input Power Factor,
Rms Line Current and Load Voltage
46
Figs. 24 through 31 show curves of the average output power P,
input power factor PFin, rms line current IL and rms output voltage Vc
versus the firing angle ep for a load Q = 5.0. Fi.gs. 32 through 39
show the same for a load Q = 10.0.
Examination of the plots permits the following observations to
be made:
1. All curves are a non-linear function of the firing angle.
2. For the same firing angle, all quantities decrease with XL.
3. For inductive loads, all values decrease uniformly as the firing
angle is increased.
4. For unity power factor loads, all values decrease in general
after a slight increase as the firing angle is increased.
5. For leading power factor loads, all values decrease after a
marked increase as the firing angle is increased.
6. The curves are independent of the value of Q.
PF == .9
== 1.0
----,-
== -.9La
".......
;:l
0p..
'-"
til
OJH
OJ .8
~<Xl
l:::-..-I
+Jl:::
OJ .6H
H;:l
()
OJl:::
-..-I
..-l
til .4
~
47
-20 a 20 40 60 80 100 120 140 160 180
Fig.24
Firing angle in degrees
Rms line current vs. firing angle
for Q == 5.0 and XL == .1
1.4
.,-...d
1.0
0
0..-
Ul
-l-J-l-J
till3
I:: .8.r-!
H<ll
~0
0..
-l-J .6::3
0..-l-J
::30
<ll00
tilH
<ll .4
~
.2
-20 o 20 40 60 80 100 120 140 160 180
48
Fig. 25
Firing angle in degrees
Average output power vs. firing angle
for Q = 5.0 and XL = .1
-l.J
;::l
0..-l.J
;::lo
Ule
p:l'
-20 o 20 40 60 80 100 120 140 160 180
49
Fig. 26
Firing angle in degrees
Rms output voltage vs. firing angle
for Q = 5.0 and XL = .1
1.2
50
-20 o 20 40 60 80 100 120 140 160 180
Fig. 27
Firing angle in degrees
Input power factor vs. firing angle
for Q = 5.0 and XL = .1
51
1.4
180
PF = 1.0
PF = -.9
1204020o
LO
1.2 = .9
- 0
,,-....
::l.
p..
'-"
rt.l .8
Q)
H
Q)
~<
J::.,-l
.6
.j.l
J::
Q)
HH
::l
()
Q)
J:: .4.,-l
,..-l
rt.lS
P:::
.2
Firing angle in degrees
Fig. 28 Rms line current vs. firing angle
for Q = 5.0 and XL = .2
1.4
52
-20 o 20 40 60 80 100 120 140 160 180
Fig. 29
Firing angle in degrees
Average output power vs. firing angle
for Q ~ 5.0 and XL ~ .2
Q)
bOtil
-IJ
.-Io
:>
-20
1.4
o 20 40 60 80 100 120 140 160 180
53
Fig. 30
Firing angle in degrees
RIDs output voltage vs. firing angle
for Q = 5.0 and XL = .2
PF .9
1.2
H0
.j.J
CJtil
4-l
H .6(l)
~0
0.
.j.J
;:j
0.r::
H .4
-20 o 20 40 60 80 100 120 140 160 180
54
Fig. 31
Firing angle in degrees
Input power factor vs. firing angle
for Q = 5.0 and XL = .2
PF = .9
PF 1.0
PF = -.9
"""'?
::l.
0.
'-" .8
(I)
QJ
l-l
QJ
~<
t::..-I
.6
-I-Jt::
QJ
l-ll-l
::l0
QJt::
.,-1
r-l
(I)e
p::
.2
55
-20 o 20 40 60 80 100 120 140 160 180
Fig. 32
Firing angle in degrees
Rms line current vs. firing angle
for Q = 10.0 and XL = .1
1.
PF = .9
PF = 1.01.0
r-..
::l? PF
= -.9p..
'-"
UJ
.j.J
.j.J .13
ttl
~
t::
'r-!
~OJ
~ .6p..
.j.J
::lp..
.j.J
::l0
OJ .4
bOttl
~OJ
:><
56
-20 o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 33 Average output power VB. firing angle
for Q = 10.0 and XL = .1
1.2
1.0
,.......
::l .8.
0.
'-'
rJloIJ
~ 0
>
Q
-r-! .6
Q)
00til
oIJ
~ 0
:>
oIJ .4::l
0.oIJ
::l0
rJl
~
.2
57
-20 o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 34 Rms output voltage vs. firing angle
for Q = 10.0 and XL = .1
1.2
58
-20 o 20 40 60 80 100 120 140 160 180
Fig. 35
Firing angle in degrees
Input power factor vs. firing angle
for Q = 10.0 and XL = .1
104
CIJ
(])
l-l
(])t
-20 o 20 40 60 80 100 120 140 160 180
59
Fig. 36
Firing angle in degrees
Rms line current vs. firing angle
for Q = 10.0 and XL = .2
60
1.4
PF .9
1.0
PF = 1.0
,.......
::l.
p.. PF = -.9
"-/
til
+J
+Jtll
~
s::OM
H
Q) .6
~p..
+J::l
p..
+J::l
0 .4
Q)
00tll
H
Q)
~
.2
-20 o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 37 Average output power vs. firing angle
for Q = 10.0 and XL = .2
1.2
........
;:l.
0..
'-"
til 08
+J
...-I0
:>
s::OM
Q)
CO 06t11
+J
...-I0
:>
+J
;:l
0..+J
;:l .4
0
til13
~
-20 o 20 40 60 80 100 120 140 160 180
61
Firing angle in degrees
Fig. 38 RIDS output voltage vs. firing angle
for Q = 10.0 and XL = .2
.j.J
::l?'
H
-20
1.2
o 20 40 60 80 100 120 140 160 180
62
Fig. 39
Firing angle in degrees
Input power factor vs. firing angle
for Q = 10.0 and XL = .2
63
Line Current Harmonics
Figs. 40 through 47 show the rms harmonic content of the line
current iL as a function of the firing angle. From the above mentioned
figures, the following observations can be made:
I. The fundamental component of iL for lagging and unity power
factor loads decreases as the firing angle is increased.
2. The fundamental component of iL for leading power factor loads
increases slightly before decreasing as the firing angle is in 
creased.
3. The maximum rms value of the higher harmonics decreases with XL.
4. The third harmonic component of iL is the dominant harmonic in
all cases.
5. The leading power factor loads have higher harmonic content.
6. The pulse nature of the line current at high firing angles is
implied by the nearly equal magnitudes of the harmonic components.
7. The third, fifth and seventh harmonic components have one, two
and three peaks, respectively.
8. The second peak of the fifth and the third peak of the seventh
harmonic are the largest for each harmonic.
1.2
1st
1.0
r..0
;:l0
0..
'-"
(J) .8
Cll
l-l
Cll
~<Xl
~
~
-I.l 06
~
Cll
l-ll-l
5th
CJ
Cll
~
'r-!
.--l
44
0
Cll
'"Cl;:l
-I.l
'r-!
~
bJ)
tlle
(J)e
~
64
-20 o 20 40 60 80
Firing angle
100 120
in degrees
140 160 180
Fig. 40 Line current harmonics vSo firing angle
for Q = 1.0, XL =.1 and PF = 09
1.2
ro-.
;j.
LO
'-"
CIl
(]j
l-<
(]j!
0 .8?.-1
-lJ0
(]j
l-<l-<
;j
()
(]j0
.6?.-1
.--l
4-1
0
(]j
"t:l;j
-lJ?.-1
0 .4co
tilS
CIlS
~
.2
o
65
o 20 40 60 80Firing angle 100 120in degrees 140 160 180
Fig. 41 Line current harmonics vs. firing angle
for Q = 1.0, XL =.2 and PF = 1.0
,-... 1st.
1.0~ .
p..
'-'
Q)
H
Q)
@"<
Q .8
?..-1
,j.J
Q 3rd
Q)
HH
~
()
Q) .6
Q
-..-I
..-I
'H
0
Q)
"Cl
~
,j.J
-..-I
Q
bOttl
e
00 5the
P:l
-20 0 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 42 Line current harmonics vs. firing angle
for Q = 1.0, XL = .2 and PF = .9
66
1.2
,,-....
::l
0
0.-
'-' LO
CIl
(lJ
$-I
(lJ
~<
!=i.,..j
.8
,j.I
!=i
(lJ
$-I
$-I::l
()
(lJ!=i
.,..j .6
...-l
4-10
(lJ
"0::l
,j.I
.,..j
!=i
bO .4
~
CIlS
ci:::.
.2
0
0 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig.43 Line current harmonics vs. firing angle
for Q = 2.0, XL = .1 and PF = 1.0
67
LO 1st,.......
::l0
0..
'-'
Ul
(lJ
l-I
(lJ
~ .8<
l:l..-/
-l-Jl:l
(lJ
l-Il-I
060
(lJ
l:l-.-/
.--I
4-l 5th
0
(lJ
"0::l
-l-J..-/
l:l
OJ)
~
Ul13
~
68
-20 o 20 40 60 80 100 120 140 160 180
Figo 44
Firing angle in degrees
Line current harmonics vSo firing angle
for Q = 200, XL =.1 and PF = 09
1.2
ro-.
::l0
1.0
'-'
fJl
QJ
l-l
QJft
<Xl
Q 08
.r-!
+JQ
OJl-l
l-l::l
()
QJ .6
Q.r-!
.--l
4-l0
QJ'"
::l
+J
'r-! 04Q
bOt1l
S
fJlS
~
02
o
69
o 20 40 60 80 100 140 10 160 180
Figo 45
Firing angle in degrees
Line current harmonics vS o firing angle
for Q = 2.0, XL = 02 and PF = -09
70
1.2
0
0 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 46 Line current harmonics vs. firing angle
for Q = 2.0, XL = .2 and PF = 1.0
r-. 1.0.
;:l.
0.
'-./
til
Q)
H
Q)
~ .8<
I::.,.l
+JI::
Q)
HH
.6;:l()
Q)
I::.,.l
~
4-l
0
Q) .4
'1:l;:l
+J
.,.l
I::co
CllS
til
~ .2
71
-20 o 20 40 60 80 100 120 140 160 180
Fig. 47
Firing angle in degrees
Line current harmonics vs. firing angle
for Q = 2.0, XL =.2 and PF = .9
72
Experimental Verification
The predicted results were verified for a number of load condi 
tions. Figs. 48, 49 and 50 show calculated and experimental results for
a stable load with Q = 5, XL =.2 and PF = -.9, 1.0, .9.
Fig.5l shows an unstable load condition and Fig.52 shows the
jump phenomenon. Note the very good agreement of predicted and experi 
mental values in Figs. 48 through 50. The inaccuracy in predicting the
start of instability, as indicated in Fig. 51, is due to the fact that
the value of the inductance in the experimental setup cannot be kept
constant, as assumed in the calculation. In Fig.52, the jump phenomenon
is verified. Very good accuracy was obtained on the return or increase
side of the characteristic curve but on the decrease side there is a
22.6% difference between predicted and experimental values. It has been
observed that in general better accuracies were obtained with higher
Q loads.
73
11 Experimental
180
- Calculated
120o
= .9
= 1.0
= -.9
1.0
"......
::J.
0..
'-" .8
CIl
Q)
H
Q)
~<:
l::l-r-!
.6
ol..ll::l
Q)
H
H::J
(J
Q)l::l
.4
.-I
CIl8
~
.2
-20
Firing angle in degrees
Fig. 48 Bros line current vs. firing angle
for Q = 5.0 and XL = .2
74
1.4
/!A Experimental
Calculated
1.2
PF = 09
PF = 1.01.0
PF = -09
r-..
::l0
0.......,
til
~
...-I0
>
~-,-/
H
Q)
~0
0.
~ ::l
0. .4
~ ::l
0
Q)
betll
H
Q):>
<11 02
-20 o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 49 Average output power VS o firing angle
for Q = 500 and XL = 02
j
j
A Experimental
Calculated
j
j
j
PF = .9
j
1.0
PF = LO
j
"""'
j
.
::l
PF -.9. -= j
0..
'-"
til
j
.l-J .8
..-l0
>
j
c::
j
.r-!
Q)
bO
j
til .6
.l-J
..-l0
j
:>
.l-J
j
::l0..
.l-J::l
j
0 .4
til
j
E
~.
j
.2
j
j
j
j
-20 0 20 40 60 80 100 120 140 160 180
j
Firing angle in degrees j
Fig. 50 Rms output voltage
vs. firing angle j
for Q = 5.0 and XL = .2 j
j
j
j
j
j
--0-- Experimental
Calculated
----- Instability
,
~ ,
15 ,"\
\ \\ \
UJ
<IJ
<IJ 12
\-IbO
<IJ ,\'"Cl
r:::
.~ \
\
<IJ \\
.--l
bO 90r:::
tll
\r::: \
0
.~
-I.J
0::l ....
'"Cl 6 ::::----r:::
0 ~
t.,) ...
76
30
o
o 20 40 60 80 100 120 140 160 180
Fig. 51
Firing angle in degrees
Conduction angle vs. firing angle
for Q = 1.0, XL =.1 and PF = 1.0
77
30
o
o 20 40 60 80 100 120 140 160 180
Firing angle in degrees
Fig. 52 Conduction angle vSo firing angle
for Q = 1.0, XL = .2 and PF = -.9
77
180
Calculated
140
-'0'-- Experimental
60
o
30
180
150
"&[/) "-
Q)
Q)
l-l 120 1be
Q)
"0
P-r-! I
Q)
.-f 90 ~be
Pell
P I0
-r-!
.j.J
0::s
"0p 60
0
c:..:>
Firing angle in degrees
Fig. 52 Conduction angle vs. firing angle
for Q = 1.0, XL = .2 and PF = -.9
78
Comparison of Approximate
and Exact Theoretical Results
In developing the approximate analysis presented in Chapter III,
two fundamental assumptions were made:
(a) The fundamental component of the line current iL is the
dominant component in predicting system performance.
(b) The fundamental component of the line current iL decreases and
becomes lagging with respect to the supply voltage as the firing
angle 4> is increased.
The accuracy of the system behavior predicted by the computer
simulations has been demonstrated by Figs. 48 through 50. In Figs. 53
and 54, the average power and its fundamental component as a function
of firing angle qp has been plotted for loads of Q = 1 and Q = 10,
with PF = 1.0 and XL = .2. Note that at Q = 10 the average power
consists almost entirely of the fundamental component. That is, con-
sidering only the fundamental component of the average power would lead
to negligible practical error.
Figs. 55 and 56 plot fundamental power as a function of firing
angle ~ as predicted by computer simulation. The close correspondence
in both these figures is self-evident. The fundamental input power
factor and the input KVA requirements can also be obtained easily from
these curves.
79
1.2
160 18020
o
02
100
....... Average.
;:j.
P.
'-/
til
+J .8 Fundamental
+Jm
~
I:::OM
HIII
;3 .60
p.
+J
;:j
P.+J
;:j
0
04
Firing angle in degrees
Fig. 53 Output power vs. firing angle
for Q = 1.0,XL = .2 and PF = 1.0
80
1.2
180
o
.2
1.
r-..
::l.
AveragePo.......,
ell .8
+J
+J Fundamentalttl
~
r::.r-!
I-l .6QJ
~ Po
+J::l
Po+J
::l .40
Firing angle in degrees
Fig. 54 Output power VSo firing angle
for Q = 10.0, XL = .2 and PF = 1.0
1.2
4> =11.3?
1.0 __ 20?300--.. 400 Approximated
r-...
;:l0
p..
"-'
til 08
+J Computer
+Jtil Predicted
::.:
s::-r-!
\-I .6 1.0
Q)
~p..
.--l
til
Q)
\-I
.--l 04 PFin Circle
til
+Js::
Q)S
til'0
s::;:l
"'"' 02
o
.2 .4 .6 .8
Fundamental reactive power in Watts (p.uo)
81
Fig.55 Real
for
vs. reactive fundamental input power
Q = 1.0, XL =.2 and PF = 1.0
102
ct> ::::110 3?
1.0 --
Approximated
".......
::l Computer.
Po. Predicted
'-"
rt.l .8
+J
+Jtll
~
l::.,-j
100H
.6(j)~
Po.
.--l
tll
(j)
H
.--l .4tll
+Jl::
(j)S
PFin Circletll
'0l::
::l
~
.2
o
o .2 .4 .6 .8
Fundamental reactive power in Watts (p.u.)
82
Fig. 56 Real vs. reactive fundamental input power
for Q = 10.0, XL ::::.2 and PF =1.0
83
Conclusions
In this chapter, an exact computer simulation of the controller
has been developed. Stable and unstable operating conditions have been
predicted and experimentally verified.
The following important observations were made:
1. The system is unstable for low Q's and low per-unit line react-
anceD
2. The system stability increases as the load Q and the per-unit
line reactance is increased.
3. Loads with Q ~ 2 and XL? .2 are stable within the investigat 
ed load power factor limits.
4. A jump phenomenon is present in this periodically interrupted
circuit.
A short verification of the approximate method of analysis
developed in Chapter III has also been given.
84
CHAPTER V
CONCLUSIONS
The primary purpose of this thesis was to study the feasibility
and general behavior of the controller with an induction load.
An approximate and an exact method of system simulation were
developed. The system's behavior under various practical load condi 
tions was predicted and verified. System instabilities at low load Q's
and low values of per-unit line reactances and an interesting jump
phenomenon have also been predicted and verified. This phenomenon is
similar to that occurring in classical non-linear circuits with energy
storage elements. In this periodically interrupted circuit the anti 
parallel connected thyristor switches are the non-linear circuit
elements. They act as a harmonic generator similar to the way a
non-linearity acts in classical non-linear circuits.
From the design point of view, the system is stable and readily
controllable outside the indicated unstable region. The line harmonics
in the controlled region, with the third harmonic dominant, may be
objectionable from the power company's point of view.
For future investigations, various input filter configurations
could be considered to eliminate the harmonic currents from the supply
line. Also, an investigation with continuous gating may yield inter 
esting results. Some limited experimental work indicates the possibili 
ty of completely eliminating the instabilities.
REFERENCES
Books
Bedford, B.D. and Hoff, R.G. Principle of Inverter Circuits.
New York: John Wiley and Sons,Inco, 1964.
Hoyt, W.H o, Jr. and Kemmerley, J.Eo Engineering Circuit Analysis.
New York: McGraw-Hill Book Co., 1962.
Kuo, B.C o Linear Networks and Systemso New York: McGraw-Hill Book
Co., 1967 0
Lago, GoV. and Waidelich, D.L. Transients in Electric Circuits.
New York: The Ronald Press Co., 1958.
Articles
Siman, Matthew. The Circle Diagram as a Teaching Tool. IEEE Trans 
actions on Education, vol. E-ll,No.l, March 1968,pp.50-56.
85