Analysis of Electric Machinery and Drive Systems Editor(s): Paul Krause, Oleg Wasynczuk, Scott Sudhoff, Steven Pekarek
434 11.1. INTRODUCTION A brief analysis of single- and three-phase semi-controlled bridge converters is pre- sented in this chapter. This type of converter is also commonly referred to as a line- commutated converter. The objective is to provide a basic background in converter operation without becoming overly involved. For this reason, only the constant-current operation is considered. A more detailed analysis of these and other converters can be found in References 1–4 . Finally, to set the stage for the analysis of dc and ac drive systems in later chapters, an average-value model of the three-phase semi-controlled bridge converter is derived. This model can be used to predict the average-value per- formance during steady-state and transient operating conditions. 11.2. SINGLE-PHASE LOAD COMMUTATED CONVERTER A single-phase line-commutated full-bridge converter is shown in Figure 11.2-1 . The ac source voltage and current are denoted e ga and i ga , respectively. The series inductance (commutating inductance) is denoted l c . The thyristors are numbered T 1 through T 4, and the associated gating or fi ring signals are denoted e f 1 through e f 4 . The converter output voltage and current are v d and i d . The following simplifying assumptions are Analysis of Electric Machinery and Drive Systems, Third Edition. Paul Krause, Oleg Wasynczuk, Scott Sudhoff, and Steven Pekarek. © 2013 Institute of Electrical and Electronics Engineers, Inc. Published 2013 by John Wiley & Sons, Inc. SEMI-CONTROLLED BRIDGE CONVERTERS 11 SINGLE-PHASE LOAD COMMUTATED CONVERTER 435 Figure 11.2-1. Single-phase full-bridge converter. made in this analysis: (1) the ac source contains only one frequency, (2) the output current i d is constant, (3) the thyristor is an infi nite impedance device when in the reverse bias mode (cathode positive) or when the gating signal to allow current fl ow has not occurred, and (4) when conducting, the voltage drop across the thyristor is negligibly small. Operation without Commutating Inductance or Firing Delay It is convenient to analyze converter operation in steps starting with the simplest case where the commutating inductance is not present and there is no fi ring delay. In this case, it can be assumed that the gating signals are always present, whereupon the thyris- tors will conduct whenever they become forward biased (anode positive) just as if they were diodes. Converter operation for constant i d with l c = 0 and without fi ring delay is depicted in Figure 11.2-2 . The thyristor in the upper part of the converter ( T1 or T3 ) that conducts is the one with the greatest anode voltage. Similarly, the thyristor that conducts in the lower part of the converter ( T 2 or T 4) is the one whose cathode voltage is the most negative. In this case, the converter operates as a full-wave rectifi er. Let us begin our analysis assuming that the source voltage may be described by eE ga g = 2cos θ (11.2-1) where θωφ ggg t=+ (11.2-2) In (11.2-2) , ω g and ϕ g are the radian frequency and phase of the source, respectively. We wish to compute the steady-state average-value of v d , which is defi ned as Vvd ddg = − ∫ 1 2 π θ π π (11.2-3) It is noted that the output voltage is made up of two identical π intervals per cycle of the source voltage. For the interval − π /2 ≤ θ g ≤ π /2 436 SEMI-CONTROLLED BRIDGE CONVERTERS ve dga = (11.2-4) Using symmetry and (11.2-1)–(11.2-4) , the average output voltage may be determined by fi nding the average of (11.2-3) over the interval − π /2 ≤ θ g ≤ π /2. Thus, the average- value of v d may be expressed VV E d E dd gg == = − ∫ 0 2 2 1 2 22 π θθ π π π cos / / (11.2-5) where E is the rms value of the source voltage. We will use V d 0 to denote the average output voltage without commutation inductance and without fi ring delay. Figure 11.2-2. Single-phase, full-bridge converter operation for constant output current without l c and fi ring delay. SINGLE-PHASE LOAD COMMUTATED CONVERTER 437 Operation with Commutating Inductance and without Firing Delay When l c is zero, the process of “current switching” from one thyristor to the other in either the upper or lower part of the converter ( T 1 to T 3 to T 1 to . . . , etc., and T 2 to T 4 to T 2 to . . . , etc.) takes place instantaneously. Instantaneous commutation cannot occur in practice since there is always some inductance between the source and the converter. The operation of the converter with commutating inductance and without fi ring delay is shown in Figure 11.2-3 . During commutation, the source is short- circuited simultaneously through T 1 and T 3 and through T 2 and T 4. Hence, if we consider the commutation from T 1 to T 3 and T 2 to T 4 and if we assume that the short- circuit current during commutation is positive through T 3, then el di dt ga c =− 3 (11.2-6) Figure 11.2-3. Single-phase, full-bridge converter operation for constant output current with l c and without fi ring delay. 438 SEMI-CONTROLLED BRIDGE CONVERTERS where i 3 is the current in thyristor T3. Substituting (11.2-1) into (11.2-6) and solving for i 3 yields i l Edt E l C c g gc g 3 1 2 2 =− =− + ∫ cos sin θ ω θ (11.2-7) At θ g = π /2, i 3 = 0, therefore C E l gc = 2 ω (11.2-8) whereupon i E l gc g3 2 1=− ω θ (sin) (11.2-9) At the end of commutation θ g = π /2 + γ and i 3 = I d , therefore I E l d gc =− 2 1 ω γ ( cos ) (11.2-10) where γ is the commutation angle (Fig. 11.2-3 ). The uppercase ( I d ) is used to denote constant or steady-state quantities. During commutation, the converter output voltage v d is zero. Once commutation is completed, the short-circuit paths are broken, and the output voltage jumps to the value of the source voltage since i d , and hence i ga , are assumed constant after commutation. Since i ga is constant, zero voltage is dropped across the inductance l c . It is recalled that V d 0 given by (11.2-5) is the average converter output voltage when l c is zero. When l c is considered, the output voltage is zero during commutation. Hence, the average output voltage decreases due to commutation. The average converter output voltage may be determined by VEd V dgg d = =+ −+ ∫ 1 2 2 1 2 2 0 π θθ γ πγ π cos ( cos ) / / (11.2-11) If (11.2-10) is solved for cos γ and the result substituted into (11.2-11) , the average converter output voltage with commutating inductance but without fi ring delay becomes VV l I dd gc d =− 0 ω π (11.2-12) SINGLE-PHASE LOAD COMMUTATED CONVERTER 439 It is interesting to note that commutation appears as a voltage drop as if the converter had an internal resistance of ω g l c / π . However, this is not a resistance in the sense that it does not dissipate energy. Operation without Commutating Inductance and with Firing Delay Thus far, we have considered the thyristor as a diode and hence have only considered rectifi er operation of the converter. However, the thyristor will conduct only if the anode voltage is positive and it has received a gating signal. Hence, the conduction of a thy- ristor may be delayed after the anode has become positive by delaying the gating signal (fi ring signal). Converter operation with fi ring delay but without commutating induc- tance is shown in Figure 11.2-4 . We can determine the average output by VEd E V dgg d = = = −+ + ∫ 1 2 22 2 2 0 π θθ π α α πα πα cos cos cos / / (11.2-13) where α is the fi ring delay angle (Fig. 11.2-4 ). If the current is maintained constant, the average output voltage will become negative for α greater than π /2. This is referred to as inverter operation, wherein average power is being transferred from the dc part of the circuit to the ac part of the circuit. Operation with Commutating Inductance and Firing Delay Converter operation with both commutating inductance and fi ring delay is shown in Figure 11.2-5 . The calculation of i 3 and V d are identical to that given by (11.2-6)– (11.2-12) , except that the intervals of evaluation are different. In particular, (11.2-7) applies, but it is at θ g = π /2 + α , where i 3 = 0, thus C E l gc = 2 ω α cos (11.2-14) Commutation ends at θ g = π /2 + α + γ , whereupon i 3 = I d , thus I E l d gc =−+ 2 ω ααγ [cos cos( )] (11.2-15) From (11.2-13) 440 SEMI-CONTROLLED BRIDGE CONVERTERS Figure 11.2-4. Single-phase, full-bridge converter operation for constant output current without l c and with fi ring delay. VEd V dgg d = =++ −++ + ∫ 1 2 2 2 2 0 π θθ ααγ παγ πα cos [cos cos( )] / / (11.2-16) Solving (11.2-15) for cos ( α + γ ) and substituting the results into (11.2-16) yields the following expression for the average output voltage with commutating inductance and fi ring delay. SINGLE-PHASE LOAD COMMUTATED CONVERTER 441 Figure 11.2-5. Single-phase, full-bridge converter operation for constant output current with l c and fi ring delay. VV l I dd gc d =− 0 cos α ω π (11.2-17) The equivalent circuit suggested by (11.2-17) is shown in Figure 11.2-6 . The average-value relations and corresponding equivalent circuit depicted in Figure 11.2-6 were developed based upon the assumptions that (1) the rms amplitude of the ac source voltage, E , is constant, and (2) the dc load current i d is constant and hence denoted I d . This equivalent circuit provides a reasonable approximation of the 442 SEMI-CONTROLLED BRIDGE CONVERTERS Figure 11.2-6. Average-value equivalent circuit for a single-phase full-bridge converter. V d0 cos a w gc l p d V d I + + - – Figure 11.2-7. Single-phase, full-bridge converter operation with RL load. (a) α = 0°; (b) α = 45°; (c) α ≈ 90°, discontinuous operation. average dc voltage even if E and i d vary with respect to time provided that the varia- tions from one conduction interval to the next are small. Modes of Operation Various modes of operation of a single-phase, full-bridge converter are illustrated by simulation results in Figure 11.2-7 , Figure 11.2-8 , and Figure 11.2-9 . The source voltage is 280 V (rms) and the commutating inductance in 1.4 mH. In each case, e ga , i ga , i 1 , i 3 , v d , and i d are plotted, where i 1 and i 3 are the currents through thyristors T 1 and T 3, respectively. In Figure 11.2-7 , the converter is operating with a series RL load con- nected across the output terminals, where R = 3 Ω and L = 40 mH. In Figure 11.2-7 a, SINGLE-PHASE LOAD COMMUTATED CONVERTER 443 Figure 11.2-8. Single-phase, full-bridge converter operation with RL and an opposing dc source connected in series across the converter terminals. (a) α = 0°; (b) α = 60°. [...]... ⎟ + cos ⎜ α + γ + ⎟ ⎥ ⎝ ⎝ π 6⎠ 6 ⎠⎦ ⎣ (11. 3-50) 456 SEMI- CONTROLLED BRIDGE CONVERTERS ˆa vqg ˆa vdg a* (11. 3-10) qga ˆ i ˆ i (11. 3-23) (11. 3-37) (11. 3-9) a qg a dg E ˆg iqg (11. 3-51) ˆ i g dg ˆ (11. 3-28) id a g (11. 3-46) (11. 3-47) (11. 3-49) (11. 3-50) (11. 3-40) (11. 3-41) ˆ id a ˆ id a ˆ id ˆ ed ˆ pid 1 p ˆ id Figure 11. 3-5 Average-value model of three-phase full -bridge converter At this point, the total... 3 (11. 3-20) Substituting (11. 3-18)– (11. 3-20) into (11. 3-17) and simplifying yields ˆ vd = ˆ di 3 6 3 ˆ E cos α − lcω g id − 2lc d dt π π (11. 3-21) ˆ ˆ From Figure 11. 3-1, vd can be related to id and ed using ˆ di ˆ ˆ vd = rdc id + Ldc d + ed dt (11. 3-22) Combining (11. 3-21) and (11. 3-22) yields 3 6 3 E cos α − ⎛ rdc + lcω g ⎞ id − ed ⎜ ⎟ˆ ˆ ⎝ ⎠ did π π = dt Ldc + 2lc (11. 3-23) 452 SEMI- CONTROLLED BRIDGE. .. id ⎤ ⎦ (11. 3-24) and vas = vbs = 0 (11. 3-25) Algebraically manipulating (11. 2-2), (11. 3-1), (11. 3-2), (11. 3-13), (11. 3-14), (11. 3-24), and (11. 3-25), it is possible to show that diag 6E 5π = cos ⎛ θ g − ⎞ ⎝ dt 2lcω g 6⎠ (11. 3-26) ˆ From (11. 3-26) and noting that at θg = α + π/3, we have that iag = −id, we conclude that ˆ iag (θ g ) = −id + π⎞⎤ 6 ⎛ ⎡ E ⎢cos(α ) − cos ⎜ θ g − ⎟ ⎥ ⎝ 2lcω g ⎣ 3⎠⎦ (11. 3-27)... egb + lc dibg dt (11. 3-14) vcs = egc + lc dicg dt (11. 3-15) Substituting (11. 3-13)– (11. 3-15) into (11. 3 -11) yields 451 THREE-PHASE LOAD COMMUTATED CONVERTER ˆ vd = 3 π 2π +α 3 ∫ π +α 3 2π +α 3 3 (egb − egc )dθ g + lcω g (ibg − icg ) π π +α (11. 3-16) 3 Substituting (11. 3-2) and (11. 3-3) into (11. 3-16) and simplifying yields 2π +α 3 3 6 3 ˆ vd = E cos α + lcω g (ibg − icg ) π π π +α (11. 3-17) 3 Further... ⎡cos θ ga Kg = ⎢ ⎣ sin θ ga − sin θ ga ⎤ cos θ ga ⎥ ⎦ (11. 3-8) 450 SEMI- CONTROLLED BRIDGE CONVERTERS and where f = [fq fd]T can be a voltage v or current i, and θga = θg − θa, where θa is the position of the arbitrary reference frame Manipulating (11. 3-5) through (11. 3-8), a a θ ga = − angle(vqg − jvdg ) E= 1 a a (vqg )2 + (vdg )2 2 (11. 3-9) (11. 3-10) a a where vqg and vdg are the q- and d-axis voltages... and/or id Thus, (11. 3 -11) may be interpreted as the short-term average of vd Likewise, the short-term average of ˆ id (average of id over a π/3 interval) will be denoted as id The firing delay angle α in (11. 3 -11) is defined such that T3 fires when θg = π +α 3 (11. 3-12) The average dc voltage indicated in (11. 3 -11) may be evaluated by noting from Figure 11. 3-1 that vas = ega + lc diag dt (11. 3-13) vbs =...444 SEMI- CONTROLLED BRIDGE CONVERTERS Figure 11. 2-9 Single-phase, full -bridge converter operation with RL and an aiding dc source connected in series across the converter terminals (a) α = 108°; (b) α = 126° THREE-PHASE LOAD COMMUTATED CONVERTER 445 the converter is operating without firing delay In Figure 11. 2-7b, the firing delay angle is 45° In Figure 11. 2-7c, the firing delay... represent the back emf of a dc machine or the capacitor voltage in a dc filter 446 SEMI- CONTROLLED BRIDGE CONVERTERS ega lc iag egb vas + ibg vbs + lc T1 e f3 T3 e f5 T5 ef 4 T4 e f6 T6 e f 2 T2 + icg vcs ef 1 + id + - lc rdc Ldc + - egc ed – + - Figure 11. 3-1 Three-phase full -bridge converter Figure 11. 3-2 Three-phase, full -bridge converter operation with RL load (a) α = 0°; (b) α = 45°; (c) α = 90° Modes... connected across the output terminals of the converter In Figure 11. 3-2a (2-3 mode), the converter is 448 SEMI- CONTROLLED BRIDGE CONVERTERS Figure 11. 3-4 Three-phase, full -bridge converter operation with RL and an aiding dc source connected in series across the converter terminals (a) α = 140°; (b) α = 160° operating without firing delay In Figure 11. 3-2b, the firing delay angle is 35°, and the output current... with the average-values plotted in Figure 11. 3-2 3 Assume that the ac source voltage applied to the three-phase load commutated converter have an acb phase sequence Indicate the sequence in which the thyristors should be fired 459 PROBLEMS 4 5 6 7 8 9 10 Derive (11. 3-9) and (11. 3-10) Starting with (11. 3-26) obtain (11. 3-27) Using (11. 3-26) and (11. 3-32), infer (11. 3-33) Perform the detailed mathematical . d T iiii=−− ⎡ ⎣ ⎤ ⎦ ˆˆ (11. 3-24) and vv as bs ==0 (11. 3-25) Algebraically manipulating (11. 2-2) , (11. 3-1) , (11. 3-2) , (11. 3-13) , (11. 3-14) , (11. 3-24) , and (11. 3-25). I E l d gc =−+ 2 ω ααγ [cos cos( )] (11. 2-15) From (11. 2-13) 440 SEMI- CONTROLLED BRIDGE CONVERTERS Figure 11. 2-4. Single-phase, full -bridge converter operation