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Physics Education AprilJune 2006 31

Numerical Evaluation of DifferentMethods

Used for Charging Capacitor Banksin Pulsed Power Applications

C.P.NAVATHE AND S.NIGAM

Laser Plasma Division, Raja Ramanna Centre for

Advanced Technology, Indore, India, 452013

[email protected]

ABSTRACT

In pulsed power applications like solid-state pulsed lasers, the

energy is first stored in large capacitor banks that are charged using

different types of power supplies. It is necessary to know various

parameters like charging time and efficiency of such schemes for the

design and selection of components. Although it is possible to write

analytical equations for these circuits, the solutions are too

cumbersome due to the presence of rectifying diodes, which controlthe direction and flow of current depending on their bias. They can

be solved more easily using numerical methods and with the use of

digital computers. In this paper, we have considered charging of

capacitor banks by limiting the primary current with resistor,

capacitor and inductor, charging by voltage doubler and by constant

current technique. In each case, the equation governing the charging

process is derived, algorithm suitable for computer solution is given

and numerical data is demonstrated for various values of

components. This analysis will be extremely useful for getting

understanding of charging process involved in various

configurations of power supplies during the design of capacitor

banks for energy storage.

PACS: 84.30.Bv, 84.30.Jc

Keywords: Capacitors, charging circuit, constant current power supply

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32 Physics Education AprilJune 2006

Introduction

The concepts of conversion of AC waveform

into DC using various types of rectifier

circuits and analysis of such circuits are

presented in detail in textbooks on power

electronics.1

However, such circuits are

modified when the capacitor filter takes the

form of an energy storage device, as in the

case of solid-state lasers. Here, the capacitor

bank is charged using either an AC or a DC

source for a time as required for maintaining

desired repetition rate of the laser, and

discharged in flashlamps for achieving the

lasing action. The charging process iscontrolled by means of a current limiting

device in the primary or secondary circuit. A

mention of such circuits is also found inwell-known books on solid-state lasers.2 A

detailed analysis of DC and AC charging

techniques has been given by Glasoe,3 but it

mainly covers cases, which involve half and

full cycle charging. No detailed analysis of

circuits used for charging capacitor banks

over several cycles is seen in published

literature on either lasers or power

electronics. Here, an attempt is made to give

an understanding of the behaviour of basic

capacitor charging circuits used for laser

power supplies, with slow charging of the

capacitor banks, and numerical evaluation of

their performance.

Different Methods of Charging

-

+

Z

V

C

1P

line

Figure 1. A simple circuit for charging capacitor bank.

In rectifier circuits, an AC source is applied

to a transformer primary and secondary is

connected through a rectifier circuit to the

capacitor. This can be generalized in a form

shown in Figure 1. Here impedance Z1P can

either be resistive, inductive or capacitive or

combination of more than one type.

This circuit is redrawn in Figure 2 with

primary impedance and voltage source,

which are connected to the primary side, are

transferred to the secondary side of the

transformer. If we take transformer turns

ratio as 1:n then the primary impedance

transferred to secondary Z1 = n2Z1P and

voltage transferred to Vac becomes Vac =

nVline. Here we have assumed ideal

transformer, neglecting its losses and

magnetizing reactance.

-

+

Z

V

C

1

ac

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Figure 2. Circuit in Figure 1 redrawn without transformer.

The charging current is limited due to theimpedance in the primary, which charges the

capacitor bank in time varying up to several

seconds. Solution of such a circuit by

differential equations is possible but

extremely cumbersome as the equation has to

be solved for each cycle and initial

conditions vary every time. Hence numerical

solution is better option for solving such

problem. Alternately, it can be solved using

PSPICE, which can simulate all the elements

in this circuit.

Performance Criteria

There are standard criteria for evaluating

performance of DC power supplies, such as

regulation, ripple, accuracy etc. We have

considered following parameters for

measuring their performance.

1. Charging time: This is the time taken tocharge the capacitor bank to a fixed

value (90%, chosen arbitrarily) of themaximum voltage. This time is

necessary to ensure a required repetition

rate of firing the laser.

2. Efficiency: This is the ratio ofEc andEp,

where Ec is the energy stored in

capacitor bank andEp is energy suppliedby the source.

3. Peak current: This is the peak value ofsource current, which will decide the

specification of diodes.

In this paper, we describe various

methods used for charging capacitors and

explain them using mathematical equations.

This is followed by numerical examples, in

which we have primarily made use of

PSPICE, followed by MS Excel and C to

calculate required results. It must be noted

that as our aim is mainly to calculate the

parameters mentioned above, transienteffects arising from diodes are ignored. Also,

results calculated in various examples are

only representative as they will depend on

characteristics of diodes and resolution used

in simulation.

Case 1: Charging with Resistor

Let us refer to Figure 3. IfZ1 is resistive, the

capacitor C gets charged when the source

voltage (Vac) exceeds the sum of the voltage

on the capacitor (Vc) and forward voltage

drop across two diodes (Vf). The equation

governing the charging process can bewritten as follows.

-

+

R

I

+

-

CVac

Vc

D D

D D

1 2

3 4

Figure 3. Charging a capacitor with current limiting resistor.

When Vac > (Vc + 2Vf) in positive cycle or

Vac > (Vc + 2Vf) in negative cycle,

we have

Vac = iR + 2Vf+ 2iRf+ Vc +1

Cidt (1)

where Vc = voltage already present on

capacitorVf= forward drop of diodeRf= forward resistance of diode

Also,

Vac = Vmsin(t) (2)

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where Vm = peak AC voltage

= 2f

The current flows into capacitor C in

same direction in both positive and negative

half cycles of sine wave. The combined

expression can be written in discrete form as

follows.When |Vac(n)| > (Vc(n) + 2Vf),

|Vac(n)| =I(n)(R + 2Rf) + 2Vf+

Vc(n 1) +1

CI n T ( ) (3)

For n = 0, 1, 2, 3 .

where T is sampling time, n denotes thesample number and the absolute value isused as sine wave is rectified. The voltage

across capacitor is represented by summation

of previous voltage and charge transferred in

current sample period. It can be noted again

that equation in this form gives direct insightof the behaviour of the circuit. The error in

calculation is small because T is muchsmaller than the shortest time constant

involved in the circuit.

Solving forI(n), we get

I nV n V n V

R R

T

C

ac c f

f

( )| ( )| ( ( ) )

( )

= +

+ +

1 2

2

(4)

and

I(n) = 0, if |Vac (n)| (Vc(n) + 2Vf) (5)

and for Vc(n)

Vc(n) =1

CI(n)T+ Vc(n 1) (6)

Expressions 4, 5 and 6 can be used for

calculating the building up of charge oncapacitor with initial values at t= 0, i.e. For

n = 0, Vc(0) = 0 andI(0) = 0.

Further, the energy stored in the capacitorafter nth sample is

Ec =1

2C[Vc(n)]

2 (7)

and the energy dissipated by the source is

Ep = V n I n T acn

n

( ) ( )=

0

(8)

The charging efficiency can be defined as

= E

E

c

p

100% (9)

For a DC source, the charging of the

capacitor is exponential. For this, the

charging efficiency is found to be4

=1

2

1 100%( )

et

RC (10)

Figure 4. Charging capacitor with current limiting resistor with AC and DC source, R=1000

ohm, C=100 F, AC voltage source= 1000 sin(100t), DC voltage source= 1000 V.

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In this case, it is possible to compare

charging pattern with the exponentialcharging observed with a DC source. For this

purpose, we have taken DC source voltage

equal to the AC peak voltage. The charging

pattern of a typical capacitor and resistor

combination with AC as well as DC source is

shown in a graph in Figure 4. The frequency

of AC source is taken as line frequency, i.e.

50 Hz and the peak amplitude is 1000 volt,

i.e. the source can be represented as 1000

sin(100t). (Same value of AC source is used

in all examples in this paper.) The currentwaveform for the same configuration and

few cycles is shown in Figure 5. It can also

be verified that for the same time constant,with different values ofR and C, the values

of charging time and charging efficiency

remain same and only the peak value of

charging current increases with decrease in

resistor value. Results for 3 cases with same

time constant are shown in Table 1.

Figure 5. Current waveform for charging capacitor with current limiting resistor withAC source, R=1000 ohm, C=100 F, AC voltage source= 1000 sin (100t).

Table 1. Charging circuit with current limiting resistor and same time constant, C=100 F, voltage

source: 1000 sin (100t).

S.No. R

(ohm)

C

(F)

Time

constant

Tdc

charging

time with

DC source(second)

Tac

charging time

with AC

source(second)

Energy

lost by

source

(joule)

Peak

current

(ampere)

Charging

efficiency

%

Tac/Tdc

1 10000 10 0.1 0.23 0.715 7.97 0.097 50.84 3.11

2 1000 100 0.1 0.23 0.715 79.67 0.967 50.84 3.11

3 100 1000 0.1 0.23 0.716 796.62 9.666 50.84 3.11

Next in Table 2, a comparison is made ofvarious parameters for the same circuit by

changing the value of resistance to change

the time constant, while keeping value of

capacitor constant. When the time constant is

varied, it is seen that the ratio of charging

time with AC source with respect to DC

source is very high for low time constant (

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current source. In a constant current source,

the voltage at the output is decided by loadcurrent and load resistance, so the voltage

drop across load increases linearly. The

sinusoidal waveform being roughly similar to

linear one for a short duration, it helps in

improving efficiency. With increasing , theefficiency steadily decreases, but charging

time remains nearly 3 times .

Table 2. Charging circuit with current limiting resistor and increasing time constant, C=100 F, voltage


S.No. R(ohm)

C

(F)

Time

constantTdc

charging

time with

DC source

(second)

Tac

charging

time with

AC source

(second)

Energy

lost by

source

(joule)

Peak

current

(ampere)

Charging

efficiency %Tac/Tdc

1 1 100 0.0001 0.00023 0.004 43.18 30.90 93.79 16.05

2 10 100 0.001 0.0023 0.005 59.02 24.53 68.62 2.13

3 20 100 0.002 0.0046 0.015 67.91 19.66 59.64 3.17

4 50 100 0.005 0.0115 0.035 76.16 12.32 53.18 3.05

5 100 100 0.01 0.023 0.074 78.61 7.60 51.52 3.23

6 200 100 0.02 0.046 0.145 79.40 4.31 51.01 3.15

7 500 100 0.05 0.115 0.356 79.63 1.88 50.86 3.09

8 1000 100 0.1 0.23 0.715 79.66 0.97 50.84 3.11

9 2000 100 0.2 0.46 1.435 79.66 0.49 50.84 3.12

10 10000 100 1 2.3 7.505 79.62 0.10 50.87 3.26

11 20000 100 2 4.6 14.994 79.72 0.05 50.80 3.26

12 50000 100 5 11.5 37.215 79.78 0.02 50.77 3.24

Table 3 describes the effect of change of

frequency. If we vary the frequency of thesource, it is observed that the fastest charging

occurs for a frequencyf approximately given

by the relationship

=fRC

1

2

1

2

Table 3. Charging circuit with current limiting resistor and increasing frequency, C=100 F, voltage

source: 1000 sin (2ft).

S.No. R(ohm)

C

(F)

Freq(Hz)

Tac

charging time

with AC source

(second)

Energy lostby power

supply

(joule)

Peakcurrent

(ampere)

Chargingefficiency

%

1 1000 100 0.1 1.895 45.21 0.06 89.58

2 1000 100 0.5 0.490 59.01 0.25 68.64

3 1000 100 0.7 0.404 63.43 0.31 63.854 1000 100 1 0.730 67.97 0.39 59.58

5 1000 100 10 0.724 79.46 0.86 50.97

6 1000 100 100 0.763 76.83 0.96 52.72

7 1000 100 1000 0.754 74.79 0.99 54.16

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At frequencies lower than this value, the

charging is slower as the sine wave takeslonger time to reach the maximum. On the

other hand, for higher frequencies, there is no

sufficient time for the capacitor to charge in

one cycle, because of which charging

extends over number of cycles.

As frequency increases, the impedance of

capacitor reduces and it causes increase in

peak current. It is also observed that the

charging efficiency decreases with increase

in frequency. Performance of a typical

combination withR = 100 ohm and C= 100

F is shown in Table 3.

Case 2: Charging with Capacitor

The capacitor bank can be charged bylimiting the current using another smaller

capacitor, C(see Figure 6). TheR represents

unavoidable circuit resistance. Now the AC

supply charges both Cand Cin first quarter

cycle. As the Vac starts decreasing, C

discharges and the current flows in opposite

direction which further charges C. Similar

process occurs in the negative cycle, in

which C is charged and discharged in

opposite polarity. The equations governing

this process can be written by modifying

earlier ones, as follows.

Vac(n) =I(n)(R + 2Rf) + 2Vf+ Vc(n 1) +

Vc (n 1) +

1 1

+

CI n T

CI n T ( ) | ( )| (11)

The sign ofI(n) has to be chosen properly

as per the quarter cycle. Also, it is necessary

to compare Vac with Vc and Vc and check if

current can flow in one of the two alternate

paths. After some manipulations, it can be

seen that if

|Vac(n) Vc (n 1)| > Vc (n 1) + 2 Vf,

'C

T

C

T)R2R(

V2)1n(V)1n(V)n(V)n(I

f

fc'cac

+

++

=

(12)

I(n) = 0, if |Vac(n) Vc(n 1)|

Vc(n 1) + 2 Vf (13)

The sign ofI(n) is positive when voltage

is rising (i.e. during 270-0-90) andnegative when voltage is falling (i.e. during

90-180-270).

V nI n T

CV nc c =

+ ( )

( )( )

1 (14)

V nI n T

CV nc c( )

| ( )|( )= +

1 (15)

The resistance mentioned in this equation

mainly represents resistance of wires and

contacts. If it is minimized, dissipation

occurs only in diodes. As a result, the

efficiency of charging approaches 99%.

-

+

R

I

C'V

+ -

+

-

C

Vac

Vc

c'

2D

D1

4DD

3

Figure 6. Charging a capacitor with current limiting capacitor.

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However, the current is not in phase with

the voltage. The simulated current waveformin PSPICE shows some spikes, probably due

to the discharge of junction capacitance of

reverse biased diodes. A typical charging

waveform with charging current is shown in

Figure 7 and results for typical values of

charging capacitors are shown in Table 4.

Again, if we see performance for various

frequencies, it is obvious that with increasingfrequency, impedance offered by both

capacitors goes down, causing higher current

and therefore, faster charging. There is no

effect on charging efficiency.

Figure 7. Current waveform for charging capacitor with current limiting capacitor with ACsource, C=10 F, R=0 ohm, C=100 F, AC voltage source= 1000 sin (100t).

Table 4. Charging circuit with current limiting capacitor,R=0, C=100 F, voltage source:

1000 sin (100).

S.No. C

(F)

C

(F)

Tac

charging time

with AC

source(second)

Ep

energy lost

by the AC

source(joule)

Ec

energy

stored by

capacitor(joule)

Chargingefficiency

%

Peak current(ampere)

1 1 100 1.163 40.72 40.5 99.46 0.60

2 5 100 0.234 40.68 40.5 99.55 2.14

3 10 100 0.123 40.73 40.5 99.44 3.90

4 20 100 0.063 40.74 40.5 99.41 5.95

5 50 100 0.024 40.79 40.5 99.29 10.47

6 100 100 0.013 41.15 40.5 98.41 16.28

Case 3: Charging with Inductor

Figure 8 shows the circuit for charging with

an inductor. In this case, a voltage equal to

the difference of input ac voltage and voltage

across capacitor and resistor is impressed on

the inductor. This induces current in theinductor, which is integrated. This current in

turn charges the capacitor. The equations

governing this process are modified as

follows. If

|Vac(n)| Vc(n 1) + 2 Vf andI(n 1) 0

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then

Vac(n) Vl(n) = Vc(n 1) +

2Vf+RI(n 1) (16)

where Vl indicates voltage across inductor.

-

+

V

I

R

L

-

V

+

Cac

c

3D

1 2D

4

D

D

Figure 8. Charging a capacitor with current limiting inductor.

The current in inductor (and hence in the

circuit) is then

I(n) =I(n 1) +V n T

I

l ( ) (17)

Thus the capacitor voltage is found to be

Vc(n) = Vc(n 1) +| ( )|I n T

C

(18)

In this case, the impedance is decided by

the series combination of inductance and

capacitance. If there is no resistance in the

circuit, there is no power loss and efficiency

is high. A typical charging waveform with an

inductor is shown in Figure 9.

If the value of inductor is such that it

resonates with the capacitor or is close to that

value, it gets charged in less than half cycle.Also, in this case the voltage can reach

higher than the source voltage and it depends

on the Q of the circuit. Actually, as the

charging process is restricted to one half

cycle, this case can be solved by analytical

equations, and the details are available in

Ref. 3. However, the efficiency is less as the

source looks like a dc source, i.e. charging

current flows only in one direction. Results

for typical values of inductors are listed in

Table 5.

Figure 9. Current waveform for charging capacitor with current limiting inductor with AC source,

L=5 H, R=0 ohm, C=100 F, AC voltage source= 1000 sin (100t).

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Table 5. Charging circuit with current limiting inductor,R=0, C=100 F, voltage source:

1000 sin (100t).

S.No. L

(henry)

C

(F)

Tac

charging timewith AC

source

(second)

Ec

energy storedby capacitor

(joule)

Ep

cnergy lost bysource

(joule)

Peak current

(ampere)

Charging

efficiency%

1 0.1 100 0.006 40.5 123.68 28.66 32.75

2 0.2 100 0.009 40.5 73.84 19.32 54.85

3 0.5 100 0.075 40.5 40.73 9.97 99.45

4 1 100 0.185 40.5 40.69 5.56 99.53

5 2 100 0.438 40.5 40.67 2.93 99.57

6 5 100 1.135 40.5 40.68 1.22 99.56

7 10 100 2.276 40.5 40.52 0.62 99.96

If now frequency is changed, the

impedance of the inductor increases with

frequency, thus the peak current decreases

and charging time increases. For a given L-C

combination, the minimum time is obtained

at resonant frequency, but charging

efficiency is least. For lower as well as

higher frequency, the charging takes place in

a time longer than this time and charging

efficiency will be higher than the value at

resonance. Table 6 shows performance of a

typical combination of L-C at different

frequencies.

Table 6. Charging circuit with current limiting inductor with increasing frequency,R=0, C=100 F,

voltage source: 1000 sin (2ft).

S.No. L(henry)

C

(F)

Freq(Hz)

Tac

charging time

with AC source

(second)

Ep

energy lost

by source

(joule)


Chargingefficiency

%

1 2 100 5 0.038 65.79 4.95 61.56

2 2 100 7 0.033 75.08 5.82 53.95

3 2 100 10 0.030 82.1 6.38 49.33

4 2 100 11.25 0.029 82.53 6.42 49.07

5 2 100 50 0.435 40.66 2.94 99.61

6 2 100 100 0.793 40.57 1.56 99.82

Relative Performance of Three Cases

We can compare now the performance of the

three combinations for a specific case. Here,

we have considered charging of 100 F

capacitor with each type of impedance forZ1.The value ofZ1 is taken to be 1000 ohm at 50

Hz. The calculated values are shown in Table7 and the charging curves are shown in

Figure 10.

From this table, it can be observed that if

a resistor is used, the time is longest and

efficiency poorest. The inductor provides a

rapid slope initially, which tapers later on.

The capacitor provides better slope than

resistor and also charging up to 90 % of peak

value in least time. Again, from the figure, itis seen that the inductive circuit provides

faster charging initially, but slows down as

compared to capacitive charging above 50 %

of peak voltage.

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Figure 10. Relative performance of charging capacitor bank of 100 F with R= 1000 ohm,or L=3.18 H, or C=3.18 F, Voltage source: 1000 sin (100t).

It should also be emphasized here that in

all the three cases discussed till now, we

have included results for a wide range of

values. However, it is not practical to use all

such combinations. Thus very low value of

inductor or resistor, or very high value of

capacitor is not suitable, as it will cause very

high peak currents in the charging circuit. In

practice, one will have to restrict charging

current to 1-2 ampere, as the rectifiers are

easily available in that range.

Table 7. Relative performance of 3 cases for equivalent impedance of 1000 ohm C=100 F, voltage


S.No. Currentlimiter

Valueof limiter

C(F)

Taccharging time

for AC

source

(second)

Ecenergy

stored by

capacitor

(joule)

Epenergy lost

by AC

source

(joule)

Efficiency%


1 R 1000 100 0.72 40.50 79.66 50.84 0.97

2 C 3.18 F 100 0.36 40.50 40.67 99.59 1.73

3 L 3.18 H 100 0.71 40.50 40.63 99.67 1.89

Practical Circuits Used for Charging Capacitor Banks

The circuits discussed so far consisted of

simple bridge rectifier, with current limited

by an impedance. In practice, there are twocircuits that are commonly used for charging

capacitors. They are as follows:

1. Voltage doubler circuit

2. Constant current charging circuit

1. Voltage Doubler

The voltage doubler or multiplier is used forraising the dc voltage into some multiples of

the peak AC voltage available from the

transformer. The advantage of doubler

(shown in Figure 11) is that it enables the

user to charge the capacitor to twice the peak

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voltage. Secondly, it also provides inherent

short circuit protection.In the positive half cycle, first Vac

alongwith Vc charge Vc throughD2 when Vac

and Vc polarities are additive. When Vc has

discharged, Vac charges Vc and Vc, provided

(Vc+ Vc ) is less than Vm. In negative cycle,

Vac charges only Vc through D1. The

equation for charging can be written as

follows.

When Vac is rising from negative

maximum to positive maximum (i.e from

270-0-90) ifVac> (Vc + Vc + Vf) then

Vac(n) =I(n)(R +Rf) + Vf+ Vc(n 1) +

Vc(n 1) +1 1

+

CI n T

CI n T ( ) ( ) (19)

Solving forI(n), we get

I nV n V n V n V

R RT

C

T

C

ac c c f

f

( )( ) ( ) ( )

( )

= +

+ + +

1 1

(20)

I(n) = 0, if |Vac(n) Vc(n 1)| Vc(n 1)

+ Vf (21)

and

V n V nI n T

Cc c( ) ( )

| ( )|= +1

(22)

V n V nI n T

Cc c = +

( ) ( )

| ( )|1

(23)

When Vac is falling from positive

maximum to negative maximum, (i.e from

90-180-270)

Vac(n) = Vc(n 1) +1

CI n T ( )

+I(n)(R +Rf) + Vf (24)

SoI(n) is given as

I(n) =V n V n V

R RT

C

ac c f

f

( ) ( )

+ +

1

(25)

Note that I(n) is positive from 270-0-

90 and negative from 90-180-270.

Figure 11. Voltage doubler circuit.

Table 8. Charging by voltage doubler,R=0, C=100 F, Voltage source: 1000 sin (100t).

S.No. C

(F)

C

(F)

Tac

(second)

Ep

(joule)

Peak current

(ampere)

Ec

(joules)

Charging

efficiency%

1 1 100 4.64 162.88 0.61 162 99.46

2 5 100 0.94 164.26 0.13 162 98.623 10 100 0.48 166.31 5.27 162 97.41

4 20 100 0.26 172.16 9.58 162 94.10

5 50 100 0.12 185.65 17.61 162 87.26

R

I

C'

+ -

C

+

-

Vc

acV

Vc'

D1

2D

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Table 8 shows the performance of a

charging circuit based on doubler usingvarious values of charging capacitor C. The

efficiency and charging time are calculated at

90% of maximum value, i.e. 1800 volt. The

efficiency is again very high as current incharging capacitor is bi-directional.

Table 9. Charging by voltage doubler at variable frequency, R=0, C=100 F, voltage source: 1000 sin

(2ft).

S.No. C'

(F)

C

(F)

Frequency

(Hz)

Tac

(second)

Ep

(joule)

Peak current

(ampere)

Ec

(joule)

Charging

efficiency%

1 1 100 10 23.22 163.69 0.12 162 98.97

2 1 100 50 4.64 162.92 0.61 162 99.44

3 1 100 100 2.32 163.05 1.20 162 99.36

4 1 100 1000 0.23 163.12 12.33 162 99.32

5 1 100 10000 0.02 185.57 119.68 162 87.30

Charging with doubler circuit with

variable frequency is given in Table 9. If thefrequency is changed, it is observed that the

charging rate is faster with increasing

frequency. The reason is same, i.e. reduction

in impedance of capacitors, causing higher

charging currents. Efficiency is again

unchanged, but reduces at higher frequency

due to dissipation in diodes.

2. Constant Current Charging Circuit

Capacitor banks can be charged by the

constant current charging circuit5 shown in

Figure 12. Here,L and C are in resonance atthe source frequency. As a result, the current

Icharging the capacitor bank is found to be

constant. Actually, it can be verified that ifthe load is a resistor instead of capacitor

bank, the current is perfectly constant.

However, in the case of capacitive load,

initially capacitor Cacts like a short circuit,

so the voltage across L is zero. As the

capacitor charges, this voltage goes on

increasing. As a result, the current flows aslong as the voltage across L exceeds Vc and

as such, is not absolutely continuous. Still,

the average current charging the capacitor isfound to be constant and the capacitor is

charged linearly on longer time scale as

compared to the cycle time of the source.

-

+

R

i

C'

L

i+I

I

-

+

C

+ -V

c

acV

Vc'

1

3 4

2

D

D

D

D

Figure 12. Constant current charging circuit.

The analysis of this circuit can be carried

out as follows.

When |Vl| > Vc + 2Vf,

Vac(n) =I(n)(R + 2Rf) + 2Vf+ Vc(n 1) +

Vc (n 1) +| ( )|I n T

C

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| ( )|( )

( )|I n T

Ci n R

i n T

C

+ +

(26)

Voltage across inductor Vl can be written

as difference of source voltage and voltage

on series capacitor

Vl(n) = Vac(n) Vc(n)- I(n-1)R (27)

And current in inductor i(n) is

i n i nV n T

L

l( ) ( )( )

= +1

(28)

SoI(n) is given as

I n

V n V V n

V n i n R TC

R RT

C

T

C

ac f c

c

f

( )

( ) ( )

( ) ( )

( )

=

+

+ + +

2 1

1 1

2

(29)

andI(n) = 0, if

|Vl| = Vc + 2Vf (30)

Again, I(n) is positive from 270-0-90

and negative from 90-180-270.The capacitor voltages are given as

Vc(n) = Vc(n 1) +| ( )|I n T

C

(31)

Vc (n)=Vc(n 1)+[i(n) +I(n)]'C

T(32)

Equations 27 to 32 can be used to

calculate charging of capacitor bank.

Alternatively, an approximate solution of

this circuit can be found as follows.

Source voltage = Vmsin(t)

So Vrms =Vm

2

The current charging the capacitor bank

is I and can be calculated from this voltageas-

RMS value ofI=Irms =V

Xc

rms

Peak value ofI=Im =Irms 2

Average value ofI=Idc = kIm

Charging time to voltage V=VC

Idc

For sinusoidal current, the value of k

relatingIdc withIm is 2/ or 0.637. However,as the charging current is not flowing

continuously, this value is somewhat less.

Some examples are listed in Table 10 in

which capacitor bank is of 100 F, beingcharged by 50 Hz source with peak value of

1000 volt. It can be seen that the ratio ofIdc

to Im is varying from 0.566 to 0.486. A

typical charging pattern of capacitor is

shown in Figure 13 along with the voltagebeing developed across the inductor.

Table 10. Constant current charging examples,R=0, C=100 F, voltage source: 1000 sin (100t).

Irms

(A)

Im

(A)

Idc

(A)

t

(s)

S.No. C

(F)

L

(H)

(calculated)

t(actual)

(s)

Ep

(joule)

Efficiency

%

Idc

(actual)

(A)

Idc/Im

1 0.1 101.32 0.02 0.03 0.02 4.5 5.07 40.58 99.79 0.02 0.57

2 0.2 50.66 0.04 0.06 0.04 2.25 2.54 40.74 99.42 0.04 0.56

3 0.5 20.26 0.11 0.16 0.1 0.9 1.02 40.91 98.99 0.09 0.564 1 10.13 0.22 0.31 0.2 0.45 0.51 40.80 99.26 0.18 0.56

5 2 5.07 0.44 0.63 0.4 0.23 0.26 41.89 96.68 0.35 0.55

6 5 2.03 1.11 1.57 1 0.09 0.11 45.64 88.75 0.83 0.53

7 10 1.01 2.22 3.14 2 0.05 0.06 51.66 78.40 1.53 0.49

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Figure 13. Constant current charging for capacitor of 100 F, with L=10 h,C= 1 F, voltage source = 1000 sin (100t).

An interesting aspect of this circuit is that

it can charge the capacitor bank to a voltage

much higher than the peak voltage of the

source, as decided by the Q of the circuit and

reverse breakdown voltage of diode. Ofcourse, for charging to higher voltages, theL,

C and diodes should be of appropriateratings and should withstand the high

voltage.

It should also be noted that thecalculation of resonant frequency is not very

critical. The charging is fairly linear over abroad range of frequency,L and C values, ofaround 20%.

Beyond this range, if frequency or L is

reduced, the charging voltage will reduce, as

inductance starts behaving like a short

circuit. Similarly if C is reduced, the

charging becomes slower as C goes towardsopen circuit. On the contrary, if frequency orL are increased, the inductance looks like an

open circuit and the performance of this

circuit approaches the one explained in Case

2, i.e. charging with capacitor. If C isincreased, the charging will become faster

and performance will be as per Case 2.

Conclusion

In this paper, a mathematical analysis of

various methods of charging capacitor banks

for pulsed power applications has been

carried out. The algorithms developed for

various methods were tested using PSPICE.

Using these algorithms, one can calculate

charging time, efficiency, peak current etc.

for various configurations by either using

spreadsheet like MS Excel or writing simple

programs in any higher level language like

Fortran, Basic or C. This method is

particularly helpful for calculation of

parameters for large capacitor banks with

charging times of several seconds, wherecommercial softwares can become unstable.

Acknowledgement

The authors wish to express their sincere

thanks to Dr P. D. Gupta for his constant

encouragement and useful suggestions and to

Shri U. Nundy for his critical reading of the

manuscript.

References

1. M.H.Rashid, Power electronics: circuits,devices and applications, Prentice-Hall,

Englewoods Cliff, 1993.2. W.Koechner, Solid-state laser engineering,

Springer-Verlag, Heidelberg, 1988.

3. G.N.Glasoe and J.V.Lebacqz, Pulsegenerators, McGraw Hill, New York, 1948.

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4. B.R.Hayworth, Constant power charging

supplies for high voltage energy transfer,Technical note #109, Capacitor Specialists

Inc., April 1975.

5. H.K.Jennings, Charging large capacitor

banks in thermonuclear research, ElectricalEngineering, Vol.80, pp 419-421, 1961.

CAPACITOR BANKS FOR HIGH POWER LASERS

High power Nd:glass laser chains are used for inertial confinement fusion

research. Laser amplifier stages in such systems are pumped optically by

discharging a capacitor bank into xenon flashlamps. The charging of the

bank takes few seconds and discharge in flashlamps is done in ~ 400 s. Atypical laser system consists of an oscillator, followed by several stages of

amplifiers, each pumped by its own capacitor bank. The laser pulse

generated by the oscillator is amplified by several magnitudes and incident

on a target. National Ignition Facility (NIF) in Lawrence Livermore

National Laboratory, U.S.A. has largest laser facility, with 192 beams giving

a total laser energy of 1.8 MJ. The corresponding capacitor bank for this

laser is 330 MJ.

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