# AC Voltage Controller Circuits (RMS Voltage Controllers)

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AC VOLTAGE CONTROLLER CIRCUITS (RMS VOLTAGE CONTROLLERS)

AC voltage controllers (ac line voltage controllers) are employed to vary the RMS value of the alternating voltage applied to a load circuit by introducing Thyristors between the load and a constant voltage ac source. The RMS value of alternating voltage applied to a load circuit is controlled by controlling the triggering angle of the Thyristors in the ac voltage controller circuits.

In brief, an ac voltage controller is a type of thyristor power converter which is used to convert a fixed voltage, fixed frequency ac input supply to obtain a variable voltage ac output. The RMS value of the ac output voltage and the ac power flow to the load is controlled by varying (adjusting) the trigger angle ‘α’

AC

V

Input

s

Voltage

fs

fs

AC Voltage Controller

V0(RMS)

Variable AC RMSO/P Voltage

fS

There are two different types of thyristor control used in practice to control the ac power flow

• On-Off control • Phase control

These are the two ac output voltage control techniques. In On-Off control technique Thyristors are used as switches to connect the load circuit to the ac supply (source) for a few cycles of the input ac supply and then to disconnect it for few input cycles. The Thyristors thus act as a high speed contactor (or high speed ac switch).

PHASE CONTROL In phase control the Thyristors are used as switches to connect the load circuit to

the input ac supply, for a part of every input cycle. That is the ac supply voltage is chopped using Thyristors during a part of each input cycle.

The thyristor switch is turned on for a part of every half cycle, so that input supply voltage appears across the load and then turned off during the remaining part of input half cycle to disconnect the ac supply from the load.

By controlling the phase angle or the trigger angle ‘α’ (delay angle), the output RMS voltage across the load can be controlled.

The trigger delay angle ‘α’ is defined as the phase angle (the value of ωt) at which the thyristor turns on and the load current begins to flow.

Thyristor ac voltage controllers use ac line commutation or ac phase commutation. Thyristors in ac voltage controllers are line commutated (phase commutated) since the input supply is ac. When the input ac voltage reverses and becomes negative during the negative half cycle the current flowing through the conducting thyristor decreases and

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falls to zero. Thus the ON thyristor naturally turns off, when the device current falls to zero.

Phase control Thyristors which are relatively inexpensive, converter grade Thyristors which are slower than fast switching inverter grade Thyristors are normally used.

For applications upto 400Hz, if Triacs are available to meet the voltage and current ratings of a particular application, Triacs are more commonly used.

Due to ac line commutation or natural commutation, there is no need of extra commutation circuitry or components and the circuits for ac voltage controllers are very simple.

Due to the nature of the output waveforms, the analysis, derivations of expressions for performance parameters are not simple, especially for the phase controlled ac voltage controllers with RL load. But however most of the practical loads are of the RL type and hence RL load should be considered in the analysis and design of ac voltage controller circuits.

TYPE OF AC VOLTAGE CONTROLLERS The ac voltage controllers are classified into two types based on the type of input

ac supply applied to the circuit. • Single Phase AC Controllers. • Three Phase AC Controllers.

Single phase ac controllers operate with single phase ac supply voltage of 230V RMS at 50Hz in our country. Three phase ac controllers operate with 3 phase ac supply of 400V RMS at 50Hz supply frequency.

Each type of controller may be sub divided into • Uni-directional or half wave ac controller. • Bi-directional or full wave ac controller.

In brief different types of ac voltage controllers are • Single phase half wave ac voltage controller (uni-directional controller). • Single phase full wave ac voltage controller (bi-directional controller). • Three phase half wave ac voltage controller (uni-directional controller). • Three phase full wave ac voltage controller (bi-directional controller).

APPLICATIONS OF AC VOLTAGE CONTROLLERS • Lighting / Illumination control in ac power circuits. • Induction heating. • Industrial heating & Domestic heating. • Transformer tap changing (on load transformer tap changing). • Speed control of induction motors (single phase and poly phase ac induction motor control). • AC magnet controls.

PRINCIPLE OF ON-OFF CONTROL TECHNIQUE (INTEGRAL CYCLE CONTROL)

The basic principle of on-off control technique is explained with reference to a single phase full wave ac voltage controller circuit shown below. The thyristor switches T1 and T2 are turned on by applying appropriate gate trigger pulses to connect the input ac supply to the load for ‘n’ number of input cycles during the time interval tON . The

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thyristor switches T1 and T2 are turned off by blocking the gate trigger pulses for ‘m’ number of input cycles during the time interval tOFF . The ac controller ON time tON usually consists of an integral number of input cycles.

R = RL = Load Resistance Fig.: Single phase full wave AC voltage controller circuit

Vs

n

m

wt

Vo io

wt

ig1

Gate pulse of T1

wt

ig2

Gate pulse of T2

wt

Fig.: Waveforms Example Referring to the waveforms of ON-OFF control technique in the above diagram,

n = Two input cycles. Thyristors are turned ON during tON for two input cycles.

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Fig.: Power Factor

Thyristors are turned ON precisely at the zero voltage crossings of the input

supply. The thyristor T1 is turned on at the beginning of each positive half cycle by

applying the gate trigger pulses to T1 as shown, during the ON time tON . The load current

flows in the positive direction, which is the downward direction as shown in the circuit

diagram when T1 conducts. The thyristor T2 is turned on at the beginning of each

negative half cycle, by applying gating signal to the gate of T2 , during tON . The load

current flows in the reverse direction, which is the upward direction when T2 conducts.

Thus we obtain a bi-directional load current flow (alternating load current flow) in a ac voltage controller circuit, by triggering the thyristors alternately.

This type of control is used in applications which have high mechanical inertia and high thermal time constant (Industrial heating and speed control of ac motors). Due to zero voltage and zero current switching of Thyristors, the harmonics generated by switching actions are reduced.

For a sine wave input supply voltage,

= vs V= m sinωt 2VS sinωt

V = RMS value of input ac supply = Vm = RMS phase supply voltage.

S

2

If the input ac supply is connected to load for ‘n’ number of input cycles and disconnected for ‘m’ number of input cycles, then

tON = n ×T , tOFF = m ×T

Where T = 1 = input cycle time (time period) and f

f = input supply frequency. tON = controller on time = n ×T . tOFF = controller off time = m ×T .

TO = Output time period = (tON + tOFF ) =(nT + mT ) .

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We can show that,

Output RMS volta= ge V V= tON V tON

O(RMS ) i(RMS ) T

ST

O

O

Where Vi(RMS) is the RMS input supply voltage = VS .

TO DERIVE AN EXPRESSION FOR THE RMS VALUE OF OUTPUT VOLTAGE, FOR ON-OFF CONTROL METHOD.

Output RMS voltage VO(RMS) =

∫ ( ) 1 ωtON 2 2

Vm Sin ωt.d ωt ωTO ωt=0

VO(RMS ) =

∫ ( ) Vm2

ωtON

Sin2ωt.d

ωt

ωTO 0

Substituting for

Sin2θ = 1− Cos2θ 2

VO(RMS ) =

∫ Vm2 ωtON 1− Cos2ωt d (ωt )

ωTO 0 2

= VO(RMS )

∫ ∫ Vm2

ωtON

ωtON

d (ωt ) − Cos2ωt.d (ωt )

2ωTO 0

0

= VO(RMS )

Vm2 (ωt ) ωtON − Sin2ωt

2ωTO

0

2

ωtON

0

= VO(RMS )

Vm2

(ωtON

−

0)

−

sin

2ωtON

− sin

0

2ωTO

2

Now tON = An integral number of input cycles; Hence tON = T , 2T ,3T , 4T ,5T ,..... & ωtON = 2π , 4π , 6π ,8π ,10π ,......

Where T is the input supply time period (T = input cycle time period). Thus we note that sin 2ωtON = 0

= V = Vm2 ω tON Vm tON

O(RMS )

2ω TO

2 TO

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= V V= tON V tON

O(RMS ) i(RMS ) T

ST

O

O

Where V = V=m V = RMS value of input supply voltage;

i(RMS )

2S

= tON

= tON

nT

n

= = k = duty cycle (d).

TO tON + tOFF nT + mT (n + m)

n

= VO(RMS) V= S (m + n) VS k

PERFORMANCE PARAMETERS OF AC VOLTAGE CONTROLLERS

• RMS Output (Load) Voltage

∫ 1

V

=

n

2π

2

V 2 sin2 ωt.d (ωt )

O(RMS) 2π (n + m) 0 m

( ) = V

Vm

n = V=k V

k

O(RMS ) 2 m + n

i(RMS )

S

= VO(RMS ) V= i(RMS ) k VS k

Where VS = Vi(RMS) = RMS value of input supply voltage.

• Duty Cycle

=k t= ON

tON = nT

TO (tON + tOFF ) (m + n)T

Where, k = n = duty cycle (d).

(m+ n)

• RMS Load Current

I= O(RMS )

VO(RMS ) =

Z

VO(RMS ) ; RL

for a resistive load Z = RL .

• Output AC (Load) Power = PO IO2(RMS ) × RL

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• Input Power Factor P=F P=O output load power = PO VA input supply volt amperes VS IS

IO2(RMS ) × RL

PF =

;

Vi(RMS ) × Iin(RMS )

= IS I= in(RMS) RMS input supply current.

The input supply current is same as the load current I=in I=O IL

Hence, RMS supply current = RMS load current; Iin(RMS) = IO(RMS) .

IO2(RMS ) × RL

VO(RMS ) Vi(RMS ) k

= PF

== = k

Vi(RMS ) × Iin(RMS ) Vi(RMS )

Vi(RMS )

P=F =k

n m+ n

• The Average Current of Thyristor IT(Avg)

Waveform of Thyristor Current

iT n m Im

0

π

2π

3π

ωt

∫ I

=

n

π

I sinωt.d (ωt )

T ( Avg) 2π ( m + n) 0 m

∫ I

=

nIm

π

sinωt.d (ωt )

T( Avg) 2π ( m + n) 0

nI

π

= IT(Avg) 2π (mm+ n) − cosωt 0

I = ( ) nIm [− cosπ + cos 0] T Avg 2π (m + n)

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= IT ( Avg)

2π

nIm

(m +

n

)

−

(

−1)

+

1

IT(Avg) = 2π (mn + n) [2Im ]

= I = Imn k.Im

T(Avg) π (m + n) π

=k d= uty cycle = tON n

(tON + tOFF ) (n + m)

= I = Imn k.Im ,

T(Avg) π (m + n) π

Where I = Vm = maximum or peak thyristor current. m RL

• RMS Current of Thyristor IT(RMS)

∫ 1

I

=

n

π

2

I 2 sin2 ωt.d (ωt )

T(RMS) 2π (n + m) 0 m

∫ 1

I

=

nI m2

π

2

sin2 ωt.d (ωt )

T(RMS) 2π (n + m) 0

∫ 1

I

=

nI m2

π

(1

−

cos

2ω

t

)

d

(

ω

t

)

2

T(RMS) 2π (n + m) 0

2

∫ ∫ = IT(RMS)

1

nIm2

π

π

2

d (ωt ) − cos 2ωt.d (ωt )

4π (n + m) 0

0

= I nIm2

(ωt )

π sin 2ωt

−

T(RMS) 4π (n + m)

0 2

1

π 2 0

1

= I

nI m2

(π

− 0) − sin 2π

− sin 0

2

T(RMS) 4π (n + m)

2

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= IT (RMS )

1

nI m2

2

{π − 0 − 0}

4π (n + m)

= IT (RMS )

1

nI 2π 2

= 4π (nm+ m)

1

nI 2 2

4

(

n

m

+

m

)

= I I= m n Im k

T(RMS) 2 (m + n) 2

IT (RMS ) = I2m k

PROBLEM 1. A single phase full wave ac voltage controller working on ON-OFF control technique has supply voltage of 230V, RMS 50Hz, load = 50Ω. The controller is ON for 30 cycles and off for 40 cycles. Calculate • ON & OFF time intervals. • RMS output voltage. • Input P.F. • Average and RMS thyristor currents.

Vin(RMS) = 230V ,

Vm =2 × 230V =325.269 V, Vm = 325.269V ,

11

T= =

= 0.02sec ,

f 50Hz

T = 20ms .

n = number of input cycles during which controller is ON; n = 30 . m = number of input cycles during which controller is OFF; m = 40 .

tON =n ×T =30× 20ms =600ms =0.6sec

tON =n ×T =0.6sec = controller ON time.

tOFF = m ×T = 40× 20ms = 800ms = 0.8sec tOFF = m ×T = 0.8sec = controller OFF time.

n

30

Duty c= ycle k = = 0.4285

(m + n) (40 + 30)

RMS output voltage VO= (RMS )

Vi(RMS ) ×

n

(m+ n)

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30

3

VO(RMS) = 230V × (30 + 40) = 230 7

VO(RM=S) 230V 0.4285=7 230 × 0.65465

VO(RMS) = 150.570V

VO(RMS ) VO(RMS ) 150.570V

I= O(RMS )

= Z

= R

= 3.0114A 50Ω

L

P=O IO2(RMS) × R=L 3.01142 × 5=0 453.426498W

Input Power Factor P.F = k

n

30

= PF

= = 0.4285

(m + n) 70

PF = 0.654653

Average Thyristor Current Rating IT(Avg) = Iπm × m n+ n = k ×πIm

where

I= V=m m RL

2 × 230 325.269

=

50

50

Im = 6.505382A = Peak (maximum) thyristor current.

= IT(Avg) 6.50π5382 × 73 IT( Avg) = 0.88745A

RMS Current Rating of Thyristor

I= Im

n =

I= m k

6.505382 ×

3

T(RMS) 2 (m + n) 2

2

7

IT(RMS) = 2.129386 A

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AC VOLTAGE CONTROLLER CIRCUITS (RMS VOLTAGE CONTROLLERS)

AC voltage controllers (ac line voltage controllers) are employed to vary the RMS value of the alternating voltage applied to a load circuit by introducing Thyristors between the load and a constant voltage ac source. The RMS value of alternating voltage applied to a load circuit is controlled by controlling the triggering angle of the Thyristors in the ac voltage controller circuits.

In brief, an ac voltage controller is a type of thyristor power converter which is used to convert a fixed voltage, fixed frequency ac input supply to obtain a variable voltage ac output. The RMS value of the ac output voltage and the ac power flow to the load is controlled by varying (adjusting) the trigger angle ‘α’

AC

V

Input

s

Voltage

fs

fs

AC Voltage Controller

V0(RMS)

Variable AC RMSO/P Voltage

fS

There are two different types of thyristor control used in practice to control the ac power flow

• On-Off control • Phase control

These are the two ac output voltage control techniques. In On-Off control technique Thyristors are used as switches to connect the load circuit to the ac supply (source) for a few cycles of the input ac supply and then to disconnect it for few input cycles. The Thyristors thus act as a high speed contactor (or high speed ac switch).

PHASE CONTROL In phase control the Thyristors are used as switches to connect the load circuit to

the input ac supply, for a part of every input cycle. That is the ac supply voltage is chopped using Thyristors during a part of each input cycle.

The thyristor switch is turned on for a part of every half cycle, so that input supply voltage appears across the load and then turned off during the remaining part of input half cycle to disconnect the ac supply from the load.

By controlling the phase angle or the trigger angle ‘α’ (delay angle), the output RMS voltage across the load can be controlled.

The trigger delay angle ‘α’ is defined as the phase angle (the value of ωt) at which the thyristor turns on and the load current begins to flow.

Thyristor ac voltage controllers use ac line commutation or ac phase commutation. Thyristors in ac voltage controllers are line commutated (phase commutated) since the input supply is ac. When the input ac voltage reverses and becomes negative during the negative half cycle the current flowing through the conducting thyristor decreases and

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falls to zero. Thus the ON thyristor naturally turns off, when the device current falls to zero.

Phase control Thyristors which are relatively inexpensive, converter grade Thyristors which are slower than fast switching inverter grade Thyristors are normally used.

For applications upto 400Hz, if Triacs are available to meet the voltage and current ratings of a particular application, Triacs are more commonly used.

Due to ac line commutation or natural commutation, there is no need of extra commutation circuitry or components and the circuits for ac voltage controllers are very simple.

Due to the nature of the output waveforms, the analysis, derivations of expressions for performance parameters are not simple, especially for the phase controlled ac voltage controllers with RL load. But however most of the practical loads are of the RL type and hence RL load should be considered in the analysis and design of ac voltage controller circuits.

TYPE OF AC VOLTAGE CONTROLLERS The ac voltage controllers are classified into two types based on the type of input

ac supply applied to the circuit. • Single Phase AC Controllers. • Three Phase AC Controllers.

Single phase ac controllers operate with single phase ac supply voltage of 230V RMS at 50Hz in our country. Three phase ac controllers operate with 3 phase ac supply of 400V RMS at 50Hz supply frequency.

Each type of controller may be sub divided into • Uni-directional or half wave ac controller. • Bi-directional or full wave ac controller.

In brief different types of ac voltage controllers are • Single phase half wave ac voltage controller (uni-directional controller). • Single phase full wave ac voltage controller (bi-directional controller). • Three phase half wave ac voltage controller (uni-directional controller). • Three phase full wave ac voltage controller (bi-directional controller).

APPLICATIONS OF AC VOLTAGE CONTROLLERS • Lighting / Illumination control in ac power circuits. • Induction heating. • Industrial heating & Domestic heating. • Transformer tap changing (on load transformer tap changing). • Speed control of induction motors (single phase and poly phase ac induction motor control). • AC magnet controls.

PRINCIPLE OF ON-OFF CONTROL TECHNIQUE (INTEGRAL CYCLE CONTROL)

The basic principle of on-off control technique is explained with reference to a single phase full wave ac voltage controller circuit shown below. The thyristor switches T1 and T2 are turned on by applying appropriate gate trigger pulses to connect the input ac supply to the load for ‘n’ number of input cycles during the time interval tON . The

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thyristor switches T1 and T2 are turned off by blocking the gate trigger pulses for ‘m’ number of input cycles during the time interval tOFF . The ac controller ON time tON usually consists of an integral number of input cycles.

R = RL = Load Resistance Fig.: Single phase full wave AC voltage controller circuit

Vs

n

m

wt

Vo io

wt

ig1

Gate pulse of T1

wt

ig2

Gate pulse of T2

wt

Fig.: Waveforms Example Referring to the waveforms of ON-OFF control technique in the above diagram,

n = Two input cycles. Thyristors are turned ON during tON for two input cycles.

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Fig.: Power Factor

Thyristors are turned ON precisely at the zero voltage crossings of the input

supply. The thyristor T1 is turned on at the beginning of each positive half cycle by

applying the gate trigger pulses to T1 as shown, during the ON time tON . The load current

flows in the positive direction, which is the downward direction as shown in the circuit

diagram when T1 conducts. The thyristor T2 is turned on at the beginning of each

negative half cycle, by applying gating signal to the gate of T2 , during tON . The load

current flows in the reverse direction, which is the upward direction when T2 conducts.

Thus we obtain a bi-directional load current flow (alternating load current flow) in a ac voltage controller circuit, by triggering the thyristors alternately.

This type of control is used in applications which have high mechanical inertia and high thermal time constant (Industrial heating and speed control of ac motors). Due to zero voltage and zero current switching of Thyristors, the harmonics generated by switching actions are reduced.

For a sine wave input supply voltage,

= vs V= m sinωt 2VS sinωt

V = RMS value of input ac supply = Vm = RMS phase supply voltage.

S

2

If the input ac supply is connected to load for ‘n’ number of input cycles and disconnected for ‘m’ number of input cycles, then

tON = n ×T , tOFF = m ×T

Where T = 1 = input cycle time (time period) and f

f = input supply frequency. tON = controller on time = n ×T . tOFF = controller off time = m ×T .

TO = Output time period = (tON + tOFF ) =(nT + mT ) .

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We can show that,

Output RMS volta= ge V V= tON V tON

O(RMS ) i(RMS ) T

ST

O

O

Where Vi(RMS) is the RMS input supply voltage = VS .

TO DERIVE AN EXPRESSION FOR THE RMS VALUE OF OUTPUT VOLTAGE, FOR ON-OFF CONTROL METHOD.

Output RMS voltage VO(RMS) =

∫ ( ) 1 ωtON 2 2

Vm Sin ωt.d ωt ωTO ωt=0

VO(RMS ) =

∫ ( ) Vm2

ωtON

Sin2ωt.d

ωt

ωTO 0

Substituting for

Sin2θ = 1− Cos2θ 2

VO(RMS ) =

∫ Vm2 ωtON 1− Cos2ωt d (ωt )

ωTO 0 2

= VO(RMS )

∫ ∫ Vm2

ωtON

ωtON

d (ωt ) − Cos2ωt.d (ωt )

2ωTO 0

0

= VO(RMS )

Vm2 (ωt ) ωtON − Sin2ωt

2ωTO

0

2

ωtON

0

= VO(RMS )

Vm2

(ωtON

−

0)

−

sin

2ωtON

− sin

0

2ωTO

2

Now tON = An integral number of input cycles; Hence tON = T , 2T ,3T , 4T ,5T ,..... & ωtON = 2π , 4π , 6π ,8π ,10π ,......

Where T is the input supply time period (T = input cycle time period). Thus we note that sin 2ωtON = 0

= V = Vm2 ω tON Vm tON

O(RMS )

2ω TO

2 TO

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= V V= tON V tON

O(RMS ) i(RMS ) T

ST

O

O

Where V = V=m V = RMS value of input supply voltage;

i(RMS )

2S

= tON

= tON

nT

n

= = k = duty cycle (d).

TO tON + tOFF nT + mT (n + m)

n

= VO(RMS) V= S (m + n) VS k

PERFORMANCE PARAMETERS OF AC VOLTAGE CONTROLLERS

• RMS Output (Load) Voltage

∫ 1

V

=

n

2π

2

V 2 sin2 ωt.d (ωt )

O(RMS) 2π (n + m) 0 m

( ) = V

Vm

n = V=k V

k

O(RMS ) 2 m + n

i(RMS )

S

= VO(RMS ) V= i(RMS ) k VS k

Where VS = Vi(RMS) = RMS value of input supply voltage.

• Duty Cycle

=k t= ON

tON = nT

TO (tON + tOFF ) (m + n)T

Where, k = n = duty cycle (d).

(m+ n)

• RMS Load Current

I= O(RMS )

VO(RMS ) =

Z

VO(RMS ) ; RL

for a resistive load Z = RL .

• Output AC (Load) Power = PO IO2(RMS ) × RL

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• Input Power Factor P=F P=O output load power = PO VA input supply volt amperes VS IS

IO2(RMS ) × RL

PF =

;

Vi(RMS ) × Iin(RMS )

= IS I= in(RMS) RMS input supply current.

The input supply current is same as the load current I=in I=O IL

Hence, RMS supply current = RMS load current; Iin(RMS) = IO(RMS) .

IO2(RMS ) × RL

VO(RMS ) Vi(RMS ) k

= PF

== = k

Vi(RMS ) × Iin(RMS ) Vi(RMS )

Vi(RMS )

P=F =k

n m+ n

• The Average Current of Thyristor IT(Avg)

Waveform of Thyristor Current

iT n m Im

0

π

2π

3π

ωt

∫ I

=

n

π

I sinωt.d (ωt )

T ( Avg) 2π ( m + n) 0 m

∫ I

=

nIm

π

sinωt.d (ωt )

T( Avg) 2π ( m + n) 0

nI

π

= IT(Avg) 2π (mm+ n) − cosωt 0

I = ( ) nIm [− cosπ + cos 0] T Avg 2π (m + n)

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= IT ( Avg)

2π

nIm

(m +

n

)

−

(

−1)

+

1

IT(Avg) = 2π (mn + n) [2Im ]

= I = Imn k.Im

T(Avg) π (m + n) π

=k d= uty cycle = tON n

(tON + tOFF ) (n + m)

= I = Imn k.Im ,

T(Avg) π (m + n) π

Where I = Vm = maximum or peak thyristor current. m RL

• RMS Current of Thyristor IT(RMS)

∫ 1

I

=

n

π

2

I 2 sin2 ωt.d (ωt )

T(RMS) 2π (n + m) 0 m

∫ 1

I

=

nI m2

π

2

sin2 ωt.d (ωt )

T(RMS) 2π (n + m) 0

∫ 1

I

=

nI m2

π

(1

−

cos

2ω

t

)

d

(

ω

t

)

2

T(RMS) 2π (n + m) 0

2

∫ ∫ = IT(RMS)

1

nIm2

π

π

2

d (ωt ) − cos 2ωt.d (ωt )

4π (n + m) 0

0

= I nIm2

(ωt )

π sin 2ωt

−

T(RMS) 4π (n + m)

0 2

1

π 2 0

1

= I

nI m2

(π

− 0) − sin 2π

− sin 0

2

T(RMS) 4π (n + m)

2

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= IT (RMS )

1

nI m2

2

{π − 0 − 0}

4π (n + m)

= IT (RMS )

1

nI 2π 2

= 4π (nm+ m)

1

nI 2 2

4

(

n

m

+

m

)

= I I= m n Im k

T(RMS) 2 (m + n) 2

IT (RMS ) = I2m k

PROBLEM 1. A single phase full wave ac voltage controller working on ON-OFF control technique has supply voltage of 230V, RMS 50Hz, load = 50Ω. The controller is ON for 30 cycles and off for 40 cycles. Calculate • ON & OFF time intervals. • RMS output voltage. • Input P.F. • Average and RMS thyristor currents.

Vin(RMS) = 230V ,

Vm =2 × 230V =325.269 V, Vm = 325.269V ,

11

T= =

= 0.02sec ,

f 50Hz

T = 20ms .

n = number of input cycles during which controller is ON; n = 30 . m = number of input cycles during which controller is OFF; m = 40 .

tON =n ×T =30× 20ms =600ms =0.6sec

tON =n ×T =0.6sec = controller ON time.

tOFF = m ×T = 40× 20ms = 800ms = 0.8sec tOFF = m ×T = 0.8sec = controller OFF time.

n

30

Duty c= ycle k = = 0.4285

(m + n) (40 + 30)

RMS output voltage VO= (RMS )

Vi(RMS ) ×

n

(m+ n)

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30

3

VO(RMS) = 230V × (30 + 40) = 230 7

VO(RM=S) 230V 0.4285=7 230 × 0.65465

VO(RMS) = 150.570V

VO(RMS ) VO(RMS ) 150.570V

I= O(RMS )

= Z

= R

= 3.0114A 50Ω

L

P=O IO2(RMS) × R=L 3.01142 × 5=0 453.426498W

Input Power Factor P.F = k

n

30

= PF

= = 0.4285

(m + n) 70

PF = 0.654653

Average Thyristor Current Rating IT(Avg) = Iπm × m n+ n = k ×πIm

where

I= V=m m RL

2 × 230 325.269

=

50

50

Im = 6.505382A = Peak (maximum) thyristor current.

= IT(Avg) 6.50π5382 × 73 IT( Avg) = 0.88745A

RMS Current Rating of Thyristor

I= Im

n =

I= m k

6.505382 ×

3

T(RMS) 2 (m + n) 2

2

7

IT(RMS) = 2.129386 A

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