AC VOLTAGE CONTROLLER CIRCUITS (RMS VOLTAGE CONTROLLERS)

Transcription

1 AC TAGE CNTRER CRCUT (RM TAGE CNTRER) AC voltage controllers (ac line voltage controllers) are eployed to vary the RM value of the alternating voltage applied to a load circuit by introducing Thyristors between the load and a constant voltage ac source. The RM 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. n 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 RM value of the ac output voltage and the ac power flow to the load is controlled by varying (adjusting) the trigger angle (RM) AC nput o lta g e f s s f s AC oltage Controller a ria b le AC RM /P o lta g e f There are two different types of thyristor control used in practice to control the ac power flow n-ff control Phase control These are the two ac output voltage control techniques. n n-ff 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). PHAE CNTR n 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 reaining part of input half cycle to disconnect the ac supply fro the load. By controlling the phase angle or the trigger angle (delay angle), the output RM voltage across the load can be controlled. The trigger delay angle is defined as the phase angle (the value of ωt) at which the thyristor turns on and the load current begins to flow.

2 Thyristor ac voltage controllers use ac line coutation or ac phase coutation. Thyristors in ac voltage controllers are line coutated (phase coutated) since the input supply is ac. When the input ac voltage reverses and becoes negative during the negative half cycle the current flowing through the conducting thyristor decreases and falls to zero. Thus the N 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 norally used. For applications upto 4Hz, if Triacs are available to eet the voltage and current ratings of a particular application, Triacs are ore coonly used. Due to ac line coutation or natural coutation, there is no need of extra coutation circuitry or coponents and the circuits for ac voltage controllers are very siple. Due to the nature of the output wavefors, the analysis, derivations of expressions for perforance paraeters are not siple, especially for the phase controlled ac voltage controllers with R load. But however ost of the practical loads are of the R type and hence R load should be considered in the analysis and design of ac voltage controller circuits. TYPE F AC TAGE CNTRER The ac voltage controllers are classified into two types based on the type of input ac supply applied to the circuit. ingle Phase AC Controllers. Three Phase AC Controllers. ingle phase ac controllers operate with single phase ac supply voltage of 3 RM at 5Hz in our country. Three phase ac controllers operate with 3 phase ac supply of 4 RM at 5Hz supply frequency. Each type of controller ay be sub divided into Uni-directional or half wave ac controller. Bi-directional or full wave ac controller. n brief different types of ac voltage controllers are ingle phase half wave ac voltage controller (uni-directional controller). ingle 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). APPCATN F AC TAGE CNTRER ighting / lluination control in ac power circuits. nduction heating. ndustrial heating & Doestic heating. Transforer tap changing (on load transforer tap changing). peed control of induction otors (single phase and poly phase ac induction otor control). AC agnet controls.

3 PRNCPE F N-FF CNTR TECHNQUE (NTEGRA CYCE CNTR) 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 T and T are turned on by applying appropriate gate trigger pulses to connect the input ac supply to the load for n nuber of input cycles during the tie interval t N. The thyristor switches T and T are turned off by blocking the gate trigger pulses for nuber of input cycles during the tie interval t FF. The ac controller N tie t N usually consists of an integral nuber of input cycles. R R oad Resistance Fig.: ingle phase full wave AC voltage controller circuit

4 s n wt o i o wt i g Gate pulse of T wt i g Gate pulse of T wt Fig.: Wavefors Exaple Referring to the wavefors of N-FF control technique in the above diagra, n Two input cycles. Thyristors are turned N during t N for two input cycles. ne input cycle. Thyristors are turned FF during t FF for one input cycle Fig.: Power Factor

5 Thyristors are turned N precisely at the zero voltage crossings of the input supply. The thyristor T is turned on at the beginning of each positive half cycle by applying the gate trigger pulses to T as shown, during the N tie t N. The load current flows in the positive direction, which is the downward direction as shown in the circuit diagra when T conducts. The thyristor T is turned on at the beginning of each negative half cycle, by applying gating signal to the gate of T, during t N. The load current flows in the reverse direction, which is the upward direction when T 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 echanical inertia and high theral tie constant (ndustrial heating and speed control of ac otors). Due to zero voltage and zero current switching of Thyristors, the haronics generated by switching actions are reduced. For a sine wave input supply voltage, v sinωt sinωt s RM value of input ac supply RM phase supply voltage. f the input ac supply is connected to load for n nuber of input cycles and disconnected for nuber of input cycles, then t n T, t T N FF Where T input cycle tie (tie period) and f f input supply frequency. t N controller on tie n T. t controller off tie T. FF T utput tie period ( t t ) ( nt T) N + +. FF

6 We can show that, tn utput RM voltage ( RM ) i ( RM ) T t T N Where irm is the RM input supply voltage. T DERE AN EXPREN FR THE RM AUE F UTPUT TAGE, FR N-FF CNTR METHD. utput RM voltage in ωtd. ( ωt) RM ω t N ω T ωt t N in td t RM ωt ω. ( ω ) ω ubstituting for Cosθ in θ ωt N Cos t ω d t RM ωt ( ω ) ωtn ωtn. RM d ωt Cos ωtd( ωt) ωt ωt ωtn inωt ( ωt) RM ω tn sin ωtn sin ωt ( ωtn ) RM Now t N An integral nuber of input cycles; Hence t T, T,3 T,4 T,5 T,... & ω,4,6,8,,... N t N Where T is the input supply tie period (T input cycle tie period). Thus we note that sin ω t N RM ω t t ω T T N N

7 N RM i( RM ) T t t T N Where irm RM value of input supply voltage; tn t nt n T t + t nt + T n+ N N FF n k ( + n) RM k duty cycle (d). PERFRMANCE PARAMETER F AC TAGE CNTRER RM utput (oad) oltage n sin. RM ωtd( ωt) ( n+ ) n RM i( RM ) k k ( + n) k k i( RM ) RM Where irm RM value of input supply voltage. Duty Cycle tn tn nt k T t + t + n T N FF Where, k n ( + n) duty cycle (d). RM oad Current RM utput AC (oad) Power RM RM Z R ; for a resistive load Z R. P R RM

8 nput Power Factor P output load power P PF A input supply volt aperes R RM PF irm inrm ; in( RM ) RM input supply current. The input supply current is sae as the load current in Hence, RM supply current RM load current; in RM ( RM ). R RM RM irm k PF i RM in RM i RM i RM k PF k n + n The Average Current of Thyristor T( Avg) i T Wavefor of Thyristor Current n 3 ωt n sin ωtd. t T Avg + ( n) n sin ωtd. t T Avg + ( n) ( ω ) n cosωt + ( n) T Avg n + n ( ω ) [ cos + cos ] T Avg

9 n ( ) + ( + n) T Avg n + n [ ] T Avg n k. ( + n) T Avg tn k duty cycle ( t + t ) ( n+ ) N FF n n k. ( + n) T Avg, Where R axiu or peak thyristor current. RM Current of Thyristor T( RM) n sin ω. ( ω ) td t T RM ( n+ ) n sin. T RM ωtd( ωt) ( n+ ) ( cosωt ) n T RM d( ωt) ( n+ ) n T( RM) d t td t 4 ( n+ ) 4 ( n ) T RM ( ω ) cos ω. ( ω ) n sin ωt ( ωt) + ( n ) T RM n sin sin ( ) 4 +

10 { } n T RM 4 n+ n 4 ( n+ ) 4( n+ ) T RM n n ( + n) T RM k T RM k PRBEM. A single phase full wave ac voltage controller working on N-FF control technique has supply voltage of 3, RM 5Hz, load 5Ω. The controller is N for 3 cycles and off for 4 cycles. Calculate N & FF tie intervals. RM output voltage. nput P.F. Average and RM thyristor currents. 3, in RM, 35.69, T.sec f 5Hz, T s. n nuber of input cycles during which controller is N; n 3. nuber of input cycles during which controller is FF; 4. t n T 3 s 6s.6sec N tn n T.6sec controller N tie. t T 4 s 8s.8sec t FF FF T.8sec controller FF tie. Duty cycle n 3 k.485 ( + n) ( 4 + 3) RM output voltage i( RM ) ( + n) RM n

11 ( + ) RM RM 5.57 RM RM RM RM A Z R 5Ω RM P R W nput Power Factor PF. k n 3 PF 7 ( + n).485 PF Average Thyristor Current Rating n k T( Avg) + n where R A Peak (axiu) thyristor current. T Avg A T Avg RM Current Rating of Thyristor n k T RM + n A T RM

12 PRNCPE F AC PHAE CNTR The basic principle of ac phase control technique is explained with reference to a single phase half wave ac voltage controller (unidirectional controller) circuit shown in the below figure. The half wave ac controller uses one thyristor and one diode connected in parallel across each other in opposite direction that is anode of thyristor T is connected to the cathode of diode D and the cathode of T is connected to the anode of D. The output voltage across the load resistor R and hence the ac power flow to the load is controlled by varying the trigger angle. The trigger angle or the delay angle refers to the value of ωt or the instant T is triggered to turn it N, by applying a suitable gate trigger at which the thyristor pulse between the gate and cathode lead. The thyristor T is forward biased during the positive half cycle of input ac supply. t can be triggered and ade to conduct by applying a suitable gate trigger pulse only during the positive half cycle of input supply. When T is triggered it conducts and the load current flows through the thyristor T, the load and through the transforer secondary winding. By assuing T as an ideal thyristor switch it can be considered as a closed switch when it is N during the period ωt to radians. The output voltage across the load follows the input supply voltage when the thyristor T is turned-on and when it conducts fro ωt to radians. When the input supply voltage decreases to zero at ωt, for a resistive load the load current also falls to zero at ωt and hence the thyristor T turns off at ωt. Between the tie period ωt to, when the supply voltage reverses and becoes negative the diode D becoes forward biased and hence turns N and conducts. The load current flows in the opposite direction during ωt to radians when D is N and the output voltage follows the negative half cycle of input supply. Fig.: Halfwave AC phase controller (Unidirectional Controller)

13 Equations nput AC upply oltage across the Transforer econdary Winding. vs sinωt in( RM ) RM value of secondary supply voltage. utput oad oltage v o v ; for ω t to v v sinωt ; for ωt to. o utput oad Current i i o o vo sinωt i ; for ωt to. R R i ; for ω t to. T DERE AN EXPREN FR RM UTPUT TAGE RM sin td. RM ω ( ωt) cosωt. d RM ( ωt)

14 ( cos t). d RM ω ( ωt) 4 d( ωt) cos ωtd. ωt RM sin ωt ( ωt) RM sin ωt ( ) RM sin 4 sin ;sin4 RM sin + ( ) RM sin + ( ) RM sin + ( ) RM sin + ( i RM ) RM sin + ( ) RM Where, irm RM value of input supply voltage (across the transforer secondary winding). Note: utput RM voltage across the load is controlled by changing ' ' as indicated by the expression for RM

15 PT F RM ERU TRGGER ANGE FR A NGE PHAE HAF-WAE AC TAGE CNTRER (UNDRECTNA CNTRER) sin + ( ) RM sin + ( ) RM By using the expression for RM we can obtain the control characteristics, which is the plot of RM output voltage RM versus the trigger angle. A typical control characteristic of single phase half-wave phase controlled ac voltage controller is as shown below Trigger angle in degrees Trigger angle in radians 3 ; ( ) ; ( ) ; ( 3 ) 6 ; ( 4 ) ; ( 5 ) ;( 6 ) RM (RM) % 7.7% 6% % 6 8 Trigger angle in degrees

16 Fig.: Control characteristics of single phase half-wave phase controlled ac voltage controller Note: We can observe fro the control characteristics and the table given above that the range of RM output voltage control is fro % of to 7.7% of when we vary the trigger angle fro zero to 8 degrees. Thus the half wave ac controller has the draw back of liited range RM output voltage control. T CACUATE THE AERAGE AUE (DC AUE) F UTPUT TAGE sin ωtd. dc ( ωt) td t dc sin ω. ( ω ) dc cosωt [ cos + cos dc ] ; cos dc [ cos ] ; dc When ' ' is varied fro to. dc varies fro to Hence ( cos ) DADANTAGE F NGE PHAE HAF WAE AC TAGE CNTRER. The output load voltage has a DC coponent because the two halves of the output voltage wavefor are not syetrical with respect to level. The input supply current wavefor also has a DC coponent (average value) which can result in the proble of core saturation of the input supply transforer. The half wave ac voltage controller using a single thyristor and a single diode provides control on the thyristor only in one half cycle of the input supply. Hence ac power flow to the load can be controlled only in one half cycle. Half wave ac voltage controller gives liited range of RM output voltage control. Because the RM value of ac output voltage can be varied fro a axiu of % of at a trigger angle to a low of 7.7% of at Radians. These drawbacks of single phase half wave ac voltage controller can be over coe by using a single phase full wave ac voltage controller.

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18 APPCATN F RM TAGE CNTRER peed control of induction otor (polyphase ac induction otor). Heater control circuits (industrial heating). Welding power control. nduction heating. n load transforer tap changing. ighting control in ac circuits. Ac agnet controls. Proble. A single phase half-wave ac voltage controller has a load resistance R 5Ω, input ac supply voltage is 3 RM at 5Hz. The input supply transforer has a turns ratio of :. f the thyristor T is triggered at 6. Calculate RM output voltage. utput power. RM load current and average load current. nput power factor. Average and RM thyristor current. Given, p 3, RM priary supply voltage. f nput supply frequency 5Hz. R 5Ω 6 radians. 3 RM secondary voltage. p Np N Therefore 3 Where, p N p Nuber of turns in the priary winding.

19 N Nuber of turns in the secondary winding. RM alue of utput (oad) oltage RM td t RM sin ω. ( ω ) We have obtained the expression for RM as sin + ( ) RM RM sin RM [ ] RM RM oad Current RM RM utput oad Power P RM Aps R 5 P R Watts RM P KW nput Power Factor P PF RM secondary supply voltage 3. RM secondary supply current RM load current Aps RM

20 W PF.9498 ( ) W Average utput (oad) oltage sin td. dc ω ( ωt) We have obtained the expression for the average / DC output voltage as, [ cos ] dc cos( 6 ) [.5 ] dc dc olts [ ] Average DC oad Current dc dc Aps R 5 Average & RM Thyristor Currents i T ( + ) 3 ωt Fig.: Thyristor Current Wavefor Referring to the thyristor current wavefor of a single phase half-wave ac voltage controller circuit, we can calculate the average thyristor current T( Avg) as sin td. T Avg ω ( ωt)

21 sin td. T Avg ω ( ωt) ( cosωt) T Avg cos( ) cos T Avg + [ + cos ] T Avg Where, R Peak thyristor current Peak load current Aps R [ + cos ] T Avg + T Avg T Avg 3 cos [ ].553 Aps T Avg RM thyristor current T( RM) can be calculated by using the expression sin td. T RM ω ( ωt) ( ωt) cos. d T RM ( ωt) d( t) cos td. T RM ω ω ( ωt) 4

22 sin ωt T( RM) ( ωt) 4 sin sin 4 ( ) T RM sin 4 + ( ) T RM sin + ( ) T RM T RM sin ( ) T RM A T RM.9746 Aps T RM NGE PHAE FU WAE AC TAGE CNTRER (AC REGUATR) R RM TAGE CNTRER WTH RETE AD ingle phase full wave ac voltage controller circuit using two CRs or a single triac is generally used in ost of the ac control applications. The ac power flow to the load can be controlled in both the half cycles by varying the trigger angle ' '. The RM value of load voltage can be varied by varying the trigger angle ' '. The input supply current is alternating in the case of a full wave ac voltage controller and due to the syetrical nature of the input supply current wavefor there is no dc coponent of input supply current i.e., the average value of the input supply current is zero. A single phase full wave ac voltage controller with a resistive load is shown in the figure below. t is possible to control the ac power flow to the load in both the half

23 cycles by adjusting the trigger angle ' '. Hence the full wave ac voltage controller is also referred to as to a bi-directional controller. Fig.: ingle phase full wave ac voltage controller (Bi-directional Controller) using CRs The thyristor T is forward biased during the positive half cycle of the input supply voltage. The thyristor T is triggered at a delay angle of ' ' ( radians). Considering the N thyristor T as an ideal closed switch the input supply voltage appears across the load resistor R and the output voltage v v during ωt to radians. The load current flows through the N thyristor T and through the load resistor R in the downward direction during the conduction tie of T fro ωt to radians. At ωt, when the input voltage falls to zero the thyristor current (which is flowing through the load resistor R ) falls to zero and hence T naturally turns off. No current flows in the circuit during ωt to ( + ). The thyristor T is forward biased during the negative cycle of input supply and when thyristor T is triggered at a delay angle ( + ), the output voltage follows the negative halfcycle of input fro ωt ( + ) to. When T is N, the load current flows in the reverse direction (upward direction) through T during ωt ( + ) to radians. The tie interval (spacing) between the gate trigger pulses of T and T is kept at radians or 8. At ωt the input supply voltage falls to zero and hence the load current also falls to zero and thyristor T turn off naturally. nstead of using two CR s in parallel, a Triac can be used for full wave ac voltage control.

24 Fig.: ingle phase full wave ac voltage controller (Bi-directional Controller) using TRAC Fig: Wavefors of single phase full wave ac voltage controller EQUATN nput supply voltage v sinωt sinωt; utput voltage across the load resistor R ; v v sinωt ; for ωt to and ωt ( + ) to utput load current v sinωt i sinωt ; R R

25 for ωt to and t ( + ) to ω T DERE AN EXPREN FR THE RM AUE F UTPUT (AD) TAGE The RM value of output voltage (load voltage) can be found using the expression RM v d t ; ( ω RM ) For a full wave ac voltage controller, we can see that the two half cycles of output voltage wavefors are syetrical and the output pulse tie period (or output pulse repetition tie) is radians. Hence we can also calculate the RM output voltage by using the expression given below. RM sin ωtd. ωt RM. ( ω ) v d t ; v v sinωt; For ωt to and t ( + ) to ω Hence, ( sin t) d( t) ( sin t RM ω ω ω ) d( ωt) + + sin ωtd. ( ωt) sin ωtd. ( ωt) + + cosωt cosωt d( ωt) d( ωt) + + d( ωt) cos ωtd. ( ωt) + d( ωt) cos ωtd. ( ωt) + + t t + 4 sin sin t ( t) ω ω ω ω ( ) ( ) ( sin sin ) ( sin 4 sin ( ) )

26 sin sin 4 + ( ) ( ) ( ( ) ) sin sin ( + ) ( ) ( + ) sin sin ( ) sin sin.cos cos.sin ( ) ( ) sin & cos Therefore, 4 sin sin + + RM ( ) sin 4 + ( ) sin RM 4 + Taking the square root, we get ( ) sin RM + ( ) sin RM + ( ) sin RM + sin + ( ) RM sin + ( ) RM

27 sin + irm ( ) RM sin + ( ) RM Maxiu RM voltage will be applied to the load when, in that case the full sine wave appears across the load. RM load voltage will be the sae as the RM supply voltage. When is increased the RM load voltage decreases. sin ( RM ) + ( ) RM + RM irm The output control characteristic for a single phase full wave ac voltage controller with resistive load can be obtained by plotting the equation for RM CNTR CHARACTERTC F NGE PHAE FU-WAE AC TAGE CNTRER WTH RETE AD The control characteristic is the plot of RM output voltage RM versus the trigger angle ; which can be obtained by using the expression for the RM output voltage of a full-wave ac controller with resistive load. sin + ( ) RM ; Where RM value of input supply voltage Trigger angle in degrees Trigger angle in radians ; ( ) 6 ;( 6) ;( 3 ) RM % % 98.54% 89.69% 7.7%

28 ; ( 4 ) ; 6 ( 5 6) 8 ;( 6 ) % 6.98% (RM) Trigger angle in degrees We can notice fro the figure, that we obtain a uch better output control characteristic by using a single phase full wave ac voltage controller. The RM output voltage can be varied fro a axiu of % at to a iniu of at 8. Thus we get a full range output voltage control by using a single phase full wave ac voltage controller. Need For solation n the single phase full wave ac voltage controller circuit using two CRs or Thyristors T and T in parallel, the gating circuits (gate trigger pulse generating circuits) of Thyristors T and T ust be isolated. Figure shows a pulse transforer with two separate windings to provide isolation between the gating signals of T and T. Gate Trigger Pulse Generator G K G K Fig.: Pulse Transforer

29 NGE PHAE FU-WAE AC TAGE CNTRER WTH CMMN CATHDE t is possible to design a single phase full wave ac controller with a coon cathode configuration by having a coon cathode point for T and T & by adding two diodes in a full wave ac controller circuit as shown in the figure below Fig.: ingle phase full wave ac controller with coon cathode (Bidirectional controller in coon cathode configuration) Thyristor T and diode D are forward biased during the positive half cycle of input supply. When thyristor T is triggered at a delay angle, Thyristor T and diode D conduct together fro ωt to during the positive half cycle. The thyristor T and diode D are forward biased during the negative half cycle of input supply, when trigged at a delay angle, thyristor T and diode D conduct together during the negative half cycle fro ωt ( + ) to. n this circuit as there is one single coon cathode point, routing of the gate trigger pulses to the thyristor gates of T and T is sipler and only one isolation circuit is required. But due to the need of two power diodes the costs of the devices increase. As there are two power devices conducting at the sae tie the voltage drop across the N devices increases and the N state conducting losses of devices increase and hence the efficiency decreases. NGE PHAE FU WAE AC TAGE CNTRER UNG A NGE THYRTR

30 D D 3 + T AC upply D 4 D R - A single phase full wave ac controller can also be ipleented with one thyristor and four diodes connected in a full wave bridge configuration as shown in the above figure. The four diodes act as a bridge full wave rectifier. The voltage across the thyristor T and current through thyristor T are always unidirectional. When T is triggered at ωt, during the positive half cycle ( ), the load current flows through D, T, diode D and through the load. With a resistive load, the thyristor current (flowing through the N thyristor T ), the load current falls to zero at ωt, when the input supply voltage decreases to zero at ωt, the thyristor naturally turns FF. n the negative half cycle, diodes D 3 & D 4 are forward biased during ωt to radians. When T is triggered at ωt ( + ), the load current flows in the opposite direction (upward direction) through the load, through D 3, T and D 4. Thus D 3, D4 and T conduct together during the negative half cycle to supply the load power. When the input supply voltage becoes zero at ωt, the thyristor current (load current) falls to zero at ωt and the thyristor T naturally turns FF. The wavefors and the expression for the RM output voltage are the sae as discussed earlier for the single phase full wave ac controller. But however if there is a large inductance in the load circuit, thyristor T ay not be turned FF at the zero crossing points, in every half cycle of input voltage and this ay result in a loss of output control. This would require detection of the zero crossing of the load current wavefor in order to ensure guaranteed turn off of the conducting thyristor before triggering the thyristor in the next half cycle, so that we gain control on the output voltage. n this full wave ac controller circuit using a single thyristor, as there are three power devices conducting together at the sae tie there is ore conduction voltage drop and an increase in the N state conduction losses and hence efficiency is also reduced. The diode bridge rectifier and thyristor (or a power transistor) act together as a bidirectional switch which is coercially available as a single device odule and it

31 has relatively low N state conduction loss. t can be used for bidirectional load current control and for controlling the RM output voltage. NGE PHAE FU WAE AC TAGE CNTRER (BDRECTNA CNTRER) WTH R AD n this section we will discuss the operation and perforance of a single phase full wave ac voltage controller with R load. n practice ost of the loads are of R type. For exaple if we consider a single phase full wave ac voltage controller controlling the speed of a single phase ac induction otor, the load which is the induction otor winding is an R type of load, where R represents the otor winding resistance and represents the otor winding inductance. A single phase full wave ac voltage controller circuit (bidirectional controller) with an R load using two thyristors T and T ( T and T are two CRs) connected in parallel is shown in the figure below. n place of two thyristors a single Triac can be used to ipleent a full wave ac controller, if a suitable Traic is available for the desired RM load current and the RM output voltage ratings. Fig: ingle phase full wave ac voltage controller with R load The thyristor T is forward biased during the positive half cycle of input supply. et us assue that T is triggered at ωt, by applying a suitable gate trigger pulse to T during the positive half cycle of input supply. The output voltage across the load follows the input supply voltage when T is N. The load current i flows through the thyristor T and through the load in the downward direction. This load current pulse flowing through T can be considered as the positive current pulse.

32 Due to the inductance in the load, the load current i flowing through T would not fall to zero at ωt, when the input supply voltage starts to becoe negative. The thyristor T will continue to conduct the load current until all the inductive energy stored in the load inductor is copletely utilized and the load current through T falls to zero at ωt β, where β is referred to as the Extinction angle, (the value of ω t ) at which the load current falls to zero. The extinction angle β is easured fro the point of the beginning of the positive half cycle of input supply to the point where the load current falls to zero. The thyristor T conducts fro ωt to β. The conduction angle of T is δ ( β ), which depends on the delay angle and the load ipedance angle φ. The wavefors of the input supply voltage, the gate trigger pulses of T and T, the thyristor current, the load current and the load voltage wavefors appear as shown in the figure below. Fig.: nput supply voltage & Thyristor current wavefors β is the extinction angle which depends upon the load inductance value.

33 Fig.: Gating ignals

34 Wavefors of single phase full wave ac voltage controller with R load for > φ. Discontinuous load current operation occurs for φ β < + ; i.e., ( β ) <, conduction angle <. > and Fig.: Wavefors of nput supply voltage, oad Current, oad oltage and Thyristor oltage across T Note The RM value of the output voltage and the load current ay be varied by varying the trigger angle. This circuit, AC RM voltage controller can be used to regulate the RM voltage across the terinals of an ac otor (induction otor). t can be used to control the teperature of a furnace by varying the RM output voltage.

35 For very large load inductance the CR ay fail to coutate, after it is triggered and the load voltage will be a full sine wave (siilar to the applied input supply voltage and the output control will be lost) as long as the gating signals are applied to the thyristors T and T. The load current wavefor will appear as a full continuous sine wave and the load current wavefor lags behind the output sine wave by the load power factor angle φ. T DERE AN EXPREN FR THE UTPUT (NDUCTE AD) CURRENT, DURNG ωt to β WHEN THYRTR T CNDUCT Considering sinusoidal input supply voltage we can write the expression for the supply voltage as v sinωt instantaneous value of the input supply voltage. et us assue that the thyristor T is triggered by applying the gating signal to T at ωt. The load current which flows through the thyristor T during ωt to β can be found fro the equation di + Ri sinωt dt ; The solution of the above differential equation gives the general expression for the output load current which is of the for sin ( ω φ) + ; Z t i t τ Ae Where axiu or peak value of input supply voltage. ( ω ) Z R + oad ipedance. ω φ tan R oad ipedance angle (power factor angle of load). τ oad circuit tie constant. R Therefore the general expression for the output load current is given by the equation R t i sin ( ωt φ) + Ae ; Z

36 The value of the constant A can be deterined fro the initial condition. i.e. initial value of load current i, at ωt. Hence fro the equation for i equating i to zero and substituting ωt, we get i sin( φ) + Ae Z R t R t sin Z Therefore Ae ( φ ) A sin R t Z e + R t Z ( φ ) e sin ( φ ) A A R ( ωt) Z ω e sin ( φ ) By substituting ωt, we get the value of constant A as A R ( ) Z ω e sin ( φ ) ubstituting the value of constant A fro the above equation into the expression for i, we obtain R R( ) t ω i sin ( ωt φ) + e e sin ( φ) Z Z ; ( ω ) ( ) R t R i t e e + Z Z ω ω sin ( ω φ) sin ( φ) + Z Z R i t e ω t ω sin ( ω φ) sin ( φ) Therefore we obtain the final expression for the inductive load current of a single phase full wave ac voltage controller with R load as i sin ( ωt φ) sin ( φ) e Z R ( ω t ) ω ; Where ωt β.

37 The above expression also represents the thyristor current i T, during the conduction tie interval of thyristor T fro ωt to β. To Calculate Extinction Angle β The extinction angle β, which is the value of ω t at which the load current i falls to zero and T is turned off can be estiated by using the condition that i, at ωt β By using the above expression for the output load current, we can write As i sin sin Z ( β φ) ( φ) e R ( β ) ω Z we can write ( β φ) ( φ) Therefore we obtain the expression R ( β ) ω sin sin e sin ( β φ) sin ( φ) e R ( β ) ω The extinction angle β can be deterined fro this transcendental equation by using the iterative ethod of solution (trial and error ethod). After β is calculated, we can deterine the thyristor conduction angle δ ( β ). β is the extinction angle which depends upon the load inductance value. Conduction angle δ increases as is decreased for a known value of β. For δ < radians, i.e., for ( β ) < radians, for ( + ) > β the load current wavefor appears as a discontinuous current wavefor as shown in the figure. The output load current reains at zero during ωt β +. This is to referred to as discontinuous load current operation which occurs for β < ( + ). When the trigger angle is decreased and ade equal to the load ipedance angle φ i.e., when φ sin β φ, we obtain fro the expression for ( β φ) ; Therefore sin β φ radians. Extinction angle β ( + φ) ( + ) ; for the case when φ ; for the case when φ Conduction angle δ ( β ) radians 8 Each thyristor conducts for 8 ( radians ). T conducts fro ωt φ to ( + φ ) and provides a positive load current. T conducts fro ( + φ ) to ( + φ ) and provides a negative load current. Hence we obtain a continuous load current and

38 the output voltage wavefor appears as a continuous sine wave identical to the input supply voltage wavefor for trigger angle φ and the control on the output is lost. v v v φ 3 ωt φ φ φ i φ ωt Fig.: utput voltage and output current wavefors for a single phase full wave ac voltage controller with R load for φ Thus we observe that for trigger angle φ, the load current tends to flow continuously and we have continuous load current operation, without any break in the load current wavefor and we obtain output voltage wavefor which is a continuous sinusoidal wavefor identical to the input supply voltage wavefor. We loose the control on the output voltage for φ as the output voltage becoes equal to the input supply voltage and thus we obtain RM ; for φ Hence, RM output voltage RM input supply voltage for φ T DERE AN EXPREN FR RM UTPUT TAGE RM F A NGE PHAE FU-WAE AC TAGE CNTRER WTH R AD.

39 When >, the load current and load voltage wavefors becoe discontinuous as shown in the figure above. β sin td. RM ω ( ωt) utput vo sinωt, for ωt to β, when T is N. ( ωt) β cos RM d( ωt) β β d t td t RM ( ω ) cos ω. ( ω ) β sin ωt ( ωt) RM ( β ) RM β sin β sin + sin sin β + ( β ) RM sin sin β + ( β ) RM The RM output voltage across the load can be varied by changing the trigger angle. For a purely resistive load, therefore load power factor angle φ. ω φ tan ; R Extinction angle β radians 8

40 PERFRMANCE PARAMETER F A NGE PHAE FU WAE AC TAGE CNTRER WTH RETE AD sin + RM utput oltage ( ) RM input supply voltage. RM ; RM RM RM value of load current. R RM RM value of input supply current. utput load power nput Power Factor P R RM R R P RM RM PF RM Average Thyristor Current, RM sin PF ( ) + i T ( + ) 3 ωt Fig.: Thyristor Current Wavefor id t td t T Avg T ( ω ) sin ω. ( ω ) sin ωtd. ( ωt) cosωt T Avg T Avg [ cos + cos ] [ + cos ]

41 Maxiu Average Thyristor Current, for, T( Avg) RM Thyristor Current sin td. T RM ω ( ωt) sin + ( ) T RM Maxiu RM Thyristor Current, for, T( RM) n the case of a single phase full wave ac voltage controller circuit using a Triac with resistive load, the average thyristor current. Because the Triac T Avg conducts in both the half cycles and the thyristor current is alternating and we obtain a syetrical thyristor current wavefor which gives an average value of zero on integration. PERFRMANCE PARAMETER F A NGE PHAE FU WAE AC TAGE CNTRER WTH R- AD The Expression for the utput (oad) Current The expression for the output (load) current which flows through the thyristor, during ωt to β is given by R ( ω t ) ω i it sin ( ωt φ) sin ( φ) e Z ; for ωt β Where, Maxiu or peak value of input ac supply voltage. ( ω ) Z R + oad ipedance. ω φ tan R oad ipedance angle (load power factor angle). Thyristor trigger angle Delay angle. β Extinction angle of thyristor, (value of ω t ) at which the thyristor (load) current falls to zero. β is calculated by solving the equation

42 sin ( β φ) sin ( φ) e R ( β ) ω Thyristor Conduction Angle δ ( β ) Maxiu thyristor conduction angle δ ( β ) radians 8 for φ. RM utput oltage sin sin β + ( β ) RM The Average Thyristor Current β T Avg id T ( ωt) β R ( ω t ) ω sin ( t ) sin T Avg ω φ ( φ) e d( ωt) Z β β R ( ω t ) ω sin ( t ). d( t) sin T Avg ω φ ω ( φ) e d( ωt) Z Maxiu value of T( Avg) occur at. The thyristors should be rated for axiu T( Avg), where. Z RM Thyristor Current T( RM) β T RM id T ( ωt) Maxiu value of T( RM) occurs at. Thyristors should be rated for axiu T( RM) When a Triac is used in a single phase full wave ac voltage controller with R type of load, then and axiu T Avg T( RM)

43 PRBEM. A single phase full wave ac voltage controller supplies an R load. The input supply voltage is 3, RM at 5Hz. The load has H, R Ω, the delay angle of thyristors T and T are equal, where. Deterine 3 a. Conduction angle of the thyristor T. b. RM output voltage. c. The input power factor. Coent on the type of operation. Given s 3, f 5Hz, H, R Ω, 6, radians, ( ω ) ( ω ) oad pedance Z R ( f) ω Ω Z Ω A Z.488 ω oad pedance Angle φ tan R φ tan tan (.3459) Trigger Angle > φ. Hence the type of operation will be discontinuous load current operation, we get β < + β < ( 8 + 6) ; β < 4 Therefore the range of ( 8 < β < 4 ) β is fro 8 degrees to 4 degrees.

44 Extinction Angle β is calculated by using the equation sin ( β φ) sin ( φ) e R ( β ) ω n the exponential ter the value of and β should be substituted in radians. Hence R ( βrad Rad ) ω sin ( β φ) sin ( φ) e ; Rad 3 Assuing β 9 ; ( φ) ( ) ( β ) sin 7.44 sin e ( β ) 3.83( β ) sin e 8 radians, β Rad β Rad β 9 β H.: R.H.: Assuing β 83 ; ( β ) sin sin e 4.94 β Rad β ( β ) H.: ( β φ) sin sin sin R.H.: e Assuing β 8 β Rad β β 3 3

45 .H.: ( β φ) R.H.: sin sin e Assuing β 96 β Rad β H.: ( β φ) R.H.: sin sin e Assuing β 97 β Rad β H.: R.H.: sin β φ sin e Assuing β 97.4 β Rad β H.: sin ( β φ) sin ( ) R.H.: e 3.79 Conduction Angle δ ( β ) RM utput oltage sin sinβ RM ( β ) + RM RM sin 6 sin RM

46 nput Power Factor P PF RM RM A Z.488 P R W RM 3, RM P PF A single phase full wave controller has an input voltage of (RM) and a load resistance of 6 oh. The firing angle of thyristor is. Find a. RM output voltage b. Power output c. nput power factor d. Average and RM thyristor current. olution 9,, R 6Ω RM alue of utput oltage sin + sin8 + RM utput Current olts A R 6 oad Power P R P watts

47 nput Current is sae as oad Current Therefore 4.4 Aps nput upply olt-ap A Therefore oad Power nput olt-ap nput Power Factor.77( lag) Each Thyristor Conducts only for half a cycle Average thyristor current T( Avg) RM thyristor current T( RM) td t T Avg R sin ω. ( ω ) ( + cos ) ; R [ + cos9 ] 4.5 A 6 sin ωt d t T RM R ( ω ) ( cosω ) t R d ( ωt) sin R + sin R + sin8 + Aps 6

48 3. A single phase half wave ac regulator using one CR in anti-parallel with a diode feeds kw, 3 heater. Find load power for a firing angle of 45. power olution 45, 3 ; P KW W 4 At standard rs supply voltage of 3, the heater dissipates KW of output Therefore P R R Resistance of heater 3 R 5.9Ω P RM value of output voltage sin + ; for firing angle 45 sin olts 4 RM value of output current Aps R 5.9 oad Power P R Watts 4. Find the RM and average current flowing through the heater shown in figure. The delay angle of both the CRs is 45. CR + i o -φ ac CR kw, heater

49 olution 4 45, Resistance of heater R 48.4Ω R Resistance value of output voltage sin + sin olts 4 RM current flowing through heater Aps R 48.4 Average current flowing through the heater Avg 5. A single phase voltage controller is eployed for controlling the power flow fro, 5 Hz source into a load circuit consisting of R 4 Ω and ω 6 Ω. Calculate the following a. Control range of firing angle b. Maxiu value of RM load current c. Maxiu power and power factor d. Maxiu value of average and RM thyristor current. olution For control of output power, iniu angle of firing angle is equal to the load ipedance angle θ θ, load angle ω 6 θ tan tan 56.3 R 4 Maxiu possible value of is 8 Therefore control range of firing angle is 56.3 < < 8

50 Maxiu value of RM load current occurs when θ At this value of the Maxiu value of RM load current Aps Z P R W Maxiu Power nput olt-ap W Power Factor P nput A Average thyristor current will be axiu when θ and conduction angleγ 8. Therefore axiu value of average thyristor current + sin T Avg ( ωt θ) d( ωt) Z Note: sin ( ω θ) sin ( θ) At, i it t e Z it i sin t Z ( ω θ ) Z cos( ωt θ) T Avg + R ( ω t ) ω But θ, Z cos( + θ) + cos( θ) T Avg Z + Z Z cos( ) cos( ) [ ] T Avg Aps T Avg Z iilarly, axiu RM value occurs when and γ. Therefore axiu value of RM thyristor current + TM sin t d t Z ( ω θ) ( ω )

51 ( ωt θ) + cos TM d t Z ( ω ) TM TM ( ωt θ) sin t ω 4 Z 4 Z + [ ] + TM.5777 Aps Z 4 + 6