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Ressonância em Transformadores com Terceira Harmônica - Prof. Silva, Resumos de Análise Complexa

Centro Universitário e Faculdade (Uniprojecão)Análise Complexa
Prof. Patrícia Silva

Este documento aborda o fenômeno da ressonância em transformadores elétricos, com foco especial na presença da terceira harmônica. Ele apresenta uma análise detalhada da equação e = |v|ω, que descreve a relação entre a tensão induzida (e), o módulo da velocidade angular (|v|) e a frequência angular (ω). O texto também discute a condição de ressonância, que ocorre quando a frequência de excitação é igual à frequência de ressonância do circuito, dada por f = 2π/√(lc). Além disso, o documento inclui uma figura que ilustra um transformador ressonante com a presença da terceira harmônica. Essa informação é relevante para o entendimento dos fenômenos eletromagnéticos em sistemas elétricos e pode ser útil para estudantes de engenharia elétrica, física e áreas afins.

Tipologia: Resumos

2022

Compartilhado em 16/06/2022

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7
High Voltage Generators
for Testing
7.0 Generation of High Voltages
The power systems engineers is interested in high voltages primarily for power transmission, and secondly for
testing of his equipment used in power transmission. In this chapter we are interested in generating high
voltages for testing of insulation. Thus generation has to be carried out in the testing laboratory. In many testing
laboratories, the primary source of power is at low voltage (400 V three phase or 230 V single phase, at 50 Hz).
Thus we need to be able to obtain the high voltage from this. Since insulation is usually being tested, the
impedances involved are extremely high (order of MDQGWKHFXUrents small (less than an ampere). Therefore
high voltage testing does not usually require high power. Thus special methods may be used which are not
applicable when generating high voltage in high power applications.
7.1 Generation of High Alternating Voltages
Single transformer test units are made for high alternating voltages up to about 200 kV.
However, for high voltages to reduce the cost (insulation cost increases rapidly with voltage) and make
transportation easier, a cascade arrangement of several transformers is used.
7.1.1 Cascade arrangement of transformers
Figure 7.1 shows a typical cascade arrangement of transformers used to obtain up to 300 kV from three units
each rated at 100 kV insulation. The low voltage winding is connected to the primary of the first transformer,
and this is connected to the transformer tank which is earthed. One end of the high voltage winding is also
earthed through the tank. The high voltage end and a tapping near this end is taken out at the top of the
transformer through a bushing, and forms the primary of the second transformer. One end of this winding is
connected to the tank of the second transformer to maintain the tank at high voltage. The secondary of this
transformer too has one end connected to the tank and at the other end the next cascaded transformer is fed.
This cascade arrangement can be continued further if a still higher voltage is required.
Fi
g
ure 7.1 - Cascade arran
g
ement of transformers
Insulating
Pedestal
Insulating
Pedestal
1 kV
99 kV
100 kV 1 kV
199 kV
200 kV 1 kV
200 kV
100 kV
hv output
300 kV
bushing
E
99 kV
100 kV
199 kV
200 kV
pf3
pf4
pf5
pf8
pf9

Pré-visualização parcial do texto

Baixe Ressonância em Transformadores com Terceira Harmônica - Prof. Silva e outras Resumos em PDF para Análise Complexa, somente na Docsity!

 High Voltage Generators

for Testing 

7.0 Generation of High Voltages

The power systems engineers is interested in high voltages primarily for power transmission, and secondly for

testing of his equipment used in power transmission. In this chapter we are interested in generating high

voltages for testing of insulation. Thus generation has to be carried out in the testing laboratory. In many testing

laboratories, the primary source of power is at low voltage (400 V three phase or 230 V single phase, at 50 Hz).

Thus we need to be able to obtain the high voltage from this. Since insulation is usually being tested, the

impedances involved are extremely high (order of M DQGWKHFXUrents small (less than an ampere). Therefore

high voltage testing does not usually require high power. Thus special methods may be used which are not

applicable when generating high voltage in high power applications.

7.1 Generation of High Alternating Voltages

Single transformer test units are made for high alternating voltages up to about 200 kV.

However, for high voltages to reduce the cost (insulation cost increases rapidly with voltage) and make

transportation easier, a cascade arrangement of several transformers is used.

7.1.1 Cascade arrangement of transformers

Figure 7.1 shows a typical cascade arrangement of transformers used to obtain up to 300 kV from three units

each rated at 100 kV insulation. The low voltage winding is connected to the primary of the first transformer,

and this is connected to the transformer tank which is earthed. One end of the high voltage winding is also

earthed through the tank. The high voltage end and a tapping near this end is taken out at the top of the

transformer through a bushing, and forms the primary of the second transformer. One end of this winding is

connected to the tank of the second transformer to maintain the tank at high voltage. The secondary of this

transformer too has one end connected to the tank and at the other end the next cascaded transformer is fed.

This cascade arrangement can be continued further if a still higher voltage is required.

Figure 7.1 - Cascade arrangement of transformers

Insulating

Pedestal

Insulating

Pedestal

1 kV

99 kV

100 kV

1 kV

199 kV

200 kV

1 kV

200 kV

100 kV

hv output

300 kV

bushing

E

99 kV

100 kV

199 kV

200 kV

 High Voltage Engineering - J R Lucas 2001

In the cascade arrangement shown, each transformer needs only to be insulated for 100 kV, and hence the

transformer can be relatively small. If a 300 kV transformer had to be used instead, the size would be

massive. High voltage transformers for testing purposes are designed purposely to have a poor regulation.

This is to ensure that when the secondary of the transformer is short circuited (as will commonly happen in

flash-over tests of insulation), the current would not increase to too high a value and to reduce the cost. In

practice, an additional series resistance (commonly a water resistance) is also used in such cases to limit the

current and prevent possible damage to the transformer.

What is shown in the cascade transformer arrangement is the basic principle involved. The actual arrangement

could be different for practical reasons.

7.1.2 Resonant Transformers

The resonance principle of a series tuned L-C circuit can be made use of to obtain a higher voltage with a given

transformer.

Let R represent the equivalent parallel resistance across the coil and the

device under test. The current i would be given by

Since R is usually very large, the Q factor of the circuit (Q = R/L& would be very large, and the output voltage

would be given by

It can thus be seen that a much larger value that the input can be obtained across the device under test in the

resonant principle.

Figure 7.3 shows the application of the resonance principle at power frequency.

{FFFFFFFFFFFI
G
G
G
E
G
GFFOFF&
GFFNFF
GG
GG
GLFFFFFFNFFFFFF{
GGG
GHOIY
GGGG
GGGG
G/GG5G
GGG
JNK
GG
{FFFFFFFFFFFOFFFFFFOFFFFFF{

Figure 7.2 - Resonance circuit

atresonance

j L

E.R

=-

R+j L- LCR

  • LCR.E

i.e. v =

R+j L

j LR

sothat v=i.

2

2

ω ω

ω

ω

ω

ω

ω

ω

ω R+j L

j LR

+

j C

1

E

i=

= E. Q

L

R

|v|=E.

ω

L C

1

atresonance ω = 2 πf =

FFFFFFFFFFFFFFFFFFIFFFFFFFFFFFFFFFFFFFFFFFI
G
G G
FFOFF&
FFNFFG
G
GG
FFFFFFG
LFFFFFFFFFFFFFF HOI

50 Hz G7HVW

GG

supply device

JNK

L air-cored

FFFFFF G

coil

G
GG
FFFFFFFFFFFFFFFFFFOFFFFFFFFFFFFFFFFFFFNFFFFFFFFFF
FFFFFFFK

air-cored FOF

F

Figure 7.3 - resonant transformer

 High Voltage Engineering - J R Lucas 2001

The supply is usually 50 Hz to the primary of the high voltage testing transformer. [In the circuit shown, C

2

includes the capacitance of the sphere gap used for measurement.] The primary circuit of the tesla transformer

also contains a trigger spark gap. Since the supply to the primary of the tesla transformer is alternating, the

capacitor C

1

is charged up to some maximum voltage, which depends upon the secondary side of the supply

transformer, and upon the setting of the trigger gap.

At this voltage, the trigger gap breaks down, the capacitor C

1

discharges, and a train of damped oscillations of

high frequency is produced in the circuit containing C

1

, the spark gap and the primary winding of the tesla

transformer. During the time taken for this train of oscillations to die away, the spark gap is conducting, due to

the formation of an arc across it. This charge and discharge of capacitor C

1

takes place twice in one voltage

cycle. Thus there will be a hundred of these trains of damped oscillations per second. The frequency of

oscillations themselves is very high (about 100 kHz usually), its actual value depending upon the inductance and

capacitance of the oscillatory circuit.

The circuit parameters are generally such that the resonant frequencies of the two sides are the same.

The expression for the voltage variation being obtained as the solution to a fourth order differential equation.

The solution to the differential equation will generally be in conjugate pairs.

Thus the solution can be written in the form

If the two undamped frequencies are equal (corresponding to L

1

C

1

= L

1

C

1

), then the damped resonant

frequencies are nearly equal (&

2

The exponential decays of the components of the voltage depends on the resistance values.

If amplitudes A

1

and A

2

are equal, and the decays also equal, then the summation in v would have the form

If &

2

, then (&

1

2

&VRWKDWWKHVXPRIWKHWZRVLQHWHUPVUHSUHVHQWVDSURGXFWRIWHUPVRQHRI

which is of very nearly the resonant frequency, and the other with a frequency equal to the difference frequency

between the primary and the secondary resonance frequencies. If the magnitudes and decays were not

considered equal, the above result will be modified by the constants A

1

and A

2

, and the exponential decays e

-a1 t

and e

-a2 t

The energy tends to get transferred from primary to the secondary and vice versa, so that the voltage of primary

is minimum when the secondary voltage is maximum and vice versa. Oscillation would occur which would be

damped out due to the resistance in the circuit.

ω

2

1 1 2 2

1

L

C

=

L

C

≈

where αis complex

α α α α

, e

+D e

+C e

+B e

v = A

- t - t - t - t

1 2 3 4

= a+j , =a-j ; .... etc

1 2

ω α

ω α

, , , , are constants

A

,

A

,

a

,

a

where

sin sin

1 2

1 2 1 2 1 2

φ φ ω ω

φ ω

φ ω

( t+ ) A e

( t+ )+ Ae

v =

2

2

t a

-

2 1

1

t a

-

1

1 2

2

( - .t+ - )

.

2

( + .t+ + )

( t+ )+ ( t+ )= 2

1 2

1 2 1 2

1 2

2

2 1

1

φ φ ω ω

φ φ ω ω

φ ω

φ ω

sin sin sin cos

Generation of High Voltages for Testing

What we require is a single series of short duration pulses. This can be done by preventing the energy from

travelling backwards and forwards in the tesla transformer by quenching the trigger gap by air blast cooling.

When the primary voltage is zero, the blast of air removes the spark in the primary gap so that the energy is

confined to the secondary. Figure 7.6 (a) shows the primary and secondary voltage waveforms without

quenching and figure 7.6 (b) shows the corresponding waveforms with quenching.

Figure 7.6 - Voltage waveforms across tesla transformer

Primary voltage

Secondary voltage

Primary voltage

Secondary voltage

(a)

(b)

Generation of High Voltages for Testing

Cockroft-Walton Circuit

When more than doubling of the voltage is required, the Cockroft-Walton voltage multiplier circuit is commonly

used. The circuit is shown in figure 7.9.

Let V

max

be the peak value of the secondary voltage of the high voltage transformer. To analyze the behaviour,

let us consider that charging of capacitors actually takes place stage by stage rather than somewhat

simultaneously. This assumption will not invalidate the result but will make analysis easier to follow. Consider

the first part of the circuit containing the diode D

1

, the capacitor C

1

, and the secondary winding. During the first

negative half cycle of the applied voltage, the capacitor C

1

charge up to voltage V

max

. Since during the positive

half cycle which follows, the diode D

1

is reverse biassed, the capacitor C

1

will not discharge (or will not charge

up in the other direction) and the peak of this half cycle, the point a will be at 2 V

max

. During the following

cycles, the potential at a will vary between 0 and 2 V

max

, depending on whether the secondary voltage and the

capacitor voltage are opposing or assisting.

Initially, capacitor C

2

would be uncharged, and the voltage at b would be zero. Thus as the voltage at a varies

between 0 and 2 V

max

, the diode D

2

is forward biassed, and the capacitor C

2

would charge to 2 V

max

. Once the

voltage at b has reached 2 V

max

, the voltage at a would be less than or equal to the voltage at b. Thus once C

2

has charged up, this diode too would be reverse biassed and the capacitor C

2

would not discharge. The voltage

at b would now remain constant at 2 V

max

. C

3

is also initially assumed uncharged. Since the voltage at a varies

between 0 and 2 V

max

, the diode D

3

would initially be forward biassed for almost the whole cycle. Thus the

capacitor C

3

charges until it reaches 2 V

max

when b is 2 V

max

and a is 0. As the voltage at a again increases to 2

V

max

, the voltage at c increases, and thus the diode D

3

is reverse biassed and C

3

would not discharge. Now as a

reaches 2 V

max

the voltage at c rises to 4 V

max

, as C

3

has not discharged.

Thus after charging up has taken place, the voltage at c varies between 2 V

max

and 4 V

max

. Assuming C

4

also to

be initially uncharged, since the voltage at b is a constant at 2 V

max

and the voltage at c varies between 2 V

max

and 4 V

max

initially, during most of the cycle, the diode D

4

is forward biassed and C

4

charges up to the maximum

difference between d and b (i.e. to 2 V

max

). This occurs when the voltage at c is 4 V

max

and the voltage at d

would now be 4 V

max

. As the voltage at c falls from 4 V

max

to 2 V

max

, since the capacitor C

4

has charged up it

would not discharge, since there is no discharge path. Thus once the capacitors are charged up the voltage at d

remains constant at 4 V

max

d

HFFFFFFFFFFFFFFFFFFFFFNFFFFFFFFF+
GG
FOFG
D G
4 G
GFOF&
HFFFFFFFFFFFFFFFFFFFFMFFNF
GFOF G
G'G
GG
FOF&GG

FNF b LFFFFFFFFFFFFFFFFFFFFFME

GGG
GFOFG
G'G
GG
GGG

a LFFFFFFFFFFFFFFFFFFFFMDG

GGG
FOF&GG
FNFGFOF&
GGFNF
FFFFFFFFFFFFOF G
D G
V 1 G

a.c. supply max GG

GG
GG
FFFFFFFFFFFGG
JFFFFFFFFFFFFFFFFFFFFOFFFFFFFFFFFFFFFFFFFFFPFFFFFFF
0 0 FOF
F(

Figure 7.9 - Cockroft-Walton Circuit

 High Voltage Engineering - J R Lucas 2001

This sequence of voltages gained is shown in Table 7.1.

Cycle

Location

T/

T

3T/

2T

5T/

3T

7T/

4T

a 0 2 Vm 0 2 Vm 0 2 Vm 0 2 Vm 0

b 0 2 Vm 2 Vm 2 Vm 2 Vm 2 Vm 2 Vm 2 Vm 2 Vm

c 0 0 2 Vm 4 Vm 2 Vm 4 Vm 2 Vm 4 Vm 2 Vm

d 0 0 0 4 Vm 4 Vm 4 Vm 4 Vm 4 Vm 4 Vm

Table 1

When the generator is used for a test, or when it is loaded, a current is drawn from the generator, and the

capacitors lose some of their charge to the load, and the voltage falls slightly depending on the load. As the

voltage across any of the capacitors drops, then at some point in the applied alternating voltage cycle, the

corresponding diode would become forward biassed and charging up of the capacitor would once again result.

Thus when a load is connected, there would be a small ripple in the output voltage.

7.2.3 Electrostatic generators

Electrostatic generators using the principle of charge transfer can give very high direct voltages. The basic

principle involved is that the charge is placed on a carrier, either insulating or an isolated conductor, and raised

to the required potential by being mechanically moved through the electrostatic field.

Van de Graeff generator

The Van de Graeff generator is one of the methods used to obtain very high voltages. However they cannot

supply much currents and the power output is restricted to a few kilowatt, and their use is restricted to low

current applications.

Figure 7.10 - Van de Graeff Generator

                    • + + + + + +

positive

                    • + + + + +

−

−

positive

insulating

belt

moving

driver

pulley

corona spray

device

− − − − − − − − − − − − −