High Power Factor and High Efficiency You Can Have Both


 Catherine Reed
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1 High ower Factor and High Efficiency You Can Have Both Isaac Cohen and Bg Lu Abstract Topic Although improvg the powersupply power factor (F) can offer significant and necessary reductions distribution losses, it is usually assumed that addg an active powerfactor correction (FC) stage will reduce the supply s ternal efficiency. It doesn t have to be this way. This topic shows that several new system architectures are possible that can mimize system losses, accommodate a wide range of AC le voltages, and meet powerquality requirements. I. Introduction Active powerfactor correction (FC) circuits are presently used virtually all power supplies tended for professional use and are startg to appear very costsensitive consumer applications, such as power supplies for personal computers. The widespread use of FC circuits is a result of regulatory activity that started Europe over a decade ago, culmatg the IEC standard EN63. By limitg the harmonic content jected to the AC utility le, EN63 directly imposes a high powerfactor (F) requirement on nonlear loads such as power supplies. This topic discusses the benefits (and some potentially negative effects) of FC and its effect on conversion losses, and vestigates ways to improve the overall efficiency. II. Review of the F Concept The F is the ratio of the real power (average power watts) absorbed by a load from a voltage or current source to the apparent power (the product of RMS voltage appearg across the load and the RMS current flowg it). F = T T vidt T T vdt T T idt () It must be emphasized that this defition is valid for any arbitrary loadandsource combation and cludes susoidalvoltage sources feedg lear loads. In the special case of lear loads (consistg of lear resistors, capacitors, and ductors) driven by a susoidal source, the F is numerically equal to the cose of the angle between the load voltage and the current phase: F = cos( Φ ) () The F resultg from an otherthanzero phase angle between the voltage and current is called displacement F. From Equation (), it follows that a lear resistor load will have unity F, and lear capacitors and ductors will have a F of zero. Nonlear loads will absorb nonsusoidal current from a susoidal voltage source, and because only the fundamental component of the current that is phase with the source voltage delivers power to the load, the product of the RMS voltage and current will always be larger than the average power absorbed, resultg a F of less than unity. III. Comparison of the Applicable Standards EN63 and Energy Star are most frequently mentioned standards that regulate the quality of the current drawn by loads connected to the utility les. However, these two standards focus on different aspects of the current quality. EN63 limits the harmonic content of current. resumably, these limitations are driven by the harmful effect of harmonics on various Energy Star is a registered trademark of the Environmental rotection Agency. 
2 Topic elements of the powerdistribution system (the build up of triplen harmonics the neutral wire beg one significant concern). On the other hand, Energy Star requires power supplies with greater than or equal to W put power must have a true power factor of.9 or greater at % of rated load when tested at 5 V at 6 Hz. Universalput power supplies will have to comply to both standards. It is terestg to exame the relationship between these two standards: Is one standard stricter than the other one? Does a power supply passg one standard automatically qualify for the other one? First we will calculate the power factor associated with an put current that has a harmonic content equal to the maximum permitted by EN63, class D (most power supplies of terest fall to this category). Accordg the powerfactor defition, the power factor is determed by both the displacement (cos(φ)) and distortions (ratio of the fundamental to RMS of the put current). Generally, displacement of the put current is caused by the net reactive current of the EMI filter. At fullload condition, the effect of the filter s net reactive current is negligible, so the power factor reduces to: I F = = 3 V In V + In n= 3, 3 n= 35, =. 76 (3) This value is significantly lower than the Energy Star.9 powerfactor requirement, so a circuit that just meets EN63 will fail Energy Star. Conversely, a power supply may be designed to absorb a square wave of current from the le to meet the Energy Star requirement by emulatg an ductiveput filter with a large ductance value (see Fig. ). The squarewave current shown yields a power factor >.9 and meets the Energy Star requirement. However, the spectral analysis shows that all harmonics above the th exceeds the EN63 limits. From the above it may appear that a potential problem is loomg, but, fortuitously, all the commonly used activefc circuits draw putcurrent waveforms that can easily comply to both standards. AC a. Rectifier with.9 power factor. b. Input current waveform Fig.. Current waveform with F =.9. IV. Benefits of Active FC The followg benefits can result from FC: Reduces RMS put current. For stance, a power circuit with a 5V/.5A ratg is limited to about 79 W of available power with a F of.5. Increasg the F to.9 will double the deliverable power to 94 W, allowg higherpower devices to be operated. Facilitates powersupply holdup. The active FC circuit matas a fixed termediate DC bus voltage that is dependent of the put voltage, so the energy stored the system does not decrease as the put voltage decreases. This allows use of smaller, less expensive bulk capacitors. Enables universal put voltage capability by providg a constant 4V bus voltage for the entire 85 to 65VAC voltage range. Improves efficiency of downstream converters. FC reduces the dynamic voltage range applied to the downstream DC/DC converters. As a result, voltage ratgs of rectifiers can be reduced, resultg lower forward drops; and operatg dutycycle/transformer turns ratios can be creased, resultg lower current switches and wdgs. 
3 Increases the efficiency of the powerdistribution system. Lower RMS current reduces distribution wirg losses. See Appendix for details. Reduces the VA ratg of standby power generators. Reduces stresses on neutral conductors. Reducg harmonics elimates the risk of triplen harmonics (the third and multiples thereof) that can add up to dangerous levels the neutral conductor of Yconnected 3phase systems. V. Effect of Active FC on owersupply Efficiency It follows from the above benefits that havg active FC at power levels above 75 W are selfevident. At the same time, we should remember that the FC stage is an additional powerprocessg stage that evitably adds losses to the system. It is structive to quantify these losses order to develop ways to mimize them. Because the typical FC stage is a boost converter, the converter has to boost the stantaneous leput voltage to a value equal to the output voltage order to transfer power from the leput voltage to the output. Note that if the put voltage is equal to the output voltage, the converter does not have to process any power sce power can pass directly from the put to the output. Under this condition, the losses are only caused by the DC drops on the output rectifier. These losses will add to less than % of the output power. Consequently, the power processed by the boost converter at any given moment is a function of the difference between the put and output voltages; the higher the difference, the more power the converter has to process and when more power is processed, more power is dissipated. This means that the difference between the put and output voltages of the FC stage affects both the efficiency of the stage and the ratgs of its components. The followg analysis quantifies this effect. To facilitate the analysis, we could visualize a converter model (referred to as the V converter ) that is fed from the rectified le voltage and has a floatg output connected series with the rectified le voltage. By controllg the output voltage of the V converter so it is equal to the difference between the momentary le voltage and the required output voltage of the FC, we obta a system that is equivalent to the boost converter. The V converter processes only the power transferred directly to the output, so the model allows separation between the power transmitted directly to the output (with nearly % efficiency) and the power processed by the FC converter. By assumg a practically achievable efficiency for the V converter, we can estimate the efficiency of an equivalent boost FC stage under various operatg conditions. The proposed model is not just a theoretical abstraction. As shown Fig., connectg the output of a flyback converter series with its put yields a system equivalent to a boost converter. The DC transfer function of the V flyback converter can be calculated as follows: D V= V (4) D V = V + V out (5) D V out = V + V D V = out V D (6) (7) Equation 7 is exactly the DC transfer function of a boost converter. For the purpose of the analysis, we will defe a boost factor (BF) as the ratio of the boostconverter output voltage to the peak value of the + Flyback In V Converter Flyback Out Fig.. Simplified block diagram of a flyback V converter. FC Out Topic 3
4 Topic put voltage. The BF is a measure of the amount of boostg that the FC stage has to perform as a function of the leput voltage. Vout BF = (8) V ( pk) Now we will exame how the variation of the BF affects the efficiency of the FC stage. Normalizg the average put power and RMS put voltage to unity, and the output voltage of the FC to the square root of (equal to the peak of the put voltage), we have and ( ave) = (9) V = out. () The stantaneous value of the put power is given by ( ωt) = s ( ωt), () where w is the angular velocity, pf, of the le voltage, and t is the time. The stantaneous value of the put voltage is given by V = V out s( ω t). () BF For the terval < wt < p, the output voltage of the V converter will be V = V out V. (3) Substitutg Equations () and () to Equation (3) yields V = s( t). BF ω (4) Sce the converter efficiency is high, the output power is nearly equal to the put power: ( ) = ( ave) (5) out ave The output current of the V converter stage is also equal to the output current of the entire FC stage and is given by I out = V out (6) and s ( ω t) I out =. (7) The output power of the V converter is the product of Equations (4) and (7): BF s( t) s ( t) V = ω ω (8) Let s compare the peak and average power ratgs of the V converter for two practical cases: one with a sgle putvoltage range of 87 to 65 VAC, and one with a universal putvoltage range of 85 to 65 VAC. Both cases assume the FC output voltage is peak value of 65 VAC. In the first case, BF = In the second case, =. 47. BF = 65 = The peak power (normalized) of the V converter occurs at ωt = π/ for BFs of 3.8 and.47, and at π/4 for a BF of (see Fig. 3): (9) π V ωt =, BF = = V V () π ωt =, BF = = π ωt =, BF =.93 4 = () The average power (normalized) that the V converter can deliver is a function of the BF and is obtaed by averagg Equation (8) over a half cycle of the leput voltage: ave = π V( ) π V dωt Vout ave = π Vout s( t) BF π ω V( ) s ( ωt) d( ωt) () (3) 4
5 .36 Normalized ower, V BF =.48 (V = 87 V) BF = (V = 65 V) BF =3.8 (V =85V) Topic t Fig. 3. ots where peak power occurs..73 Normalized Average ower, V(ave) Boost Factor (BF) Fig. 4. lot of average put power for a V converter. Figs. 3 and 4 illustrate the impact of the BF on the peak and average power ratgs of the V converter (the case of unity BF is also added for completeness). At a BF of unity, the V converter has to handle an average of only 5% and a peak of 9% of the output power, creasg to 4% and 59% for a BF of.47, or 73% and 36% for a BF of 3.8. These figures provide an sight to the effect that a high BF has on the size and cost of the FC converter. A converter designed for universal put voltage (BF = 3.8) will be rated to a peak power that is.3 times higher than the power of a converter designed for an put range of only 87 to 65VAC put (BF =.47). In the proposed model, all the losses are generated by the V converter. For a V converter s efficiency of η = 9% (a reasonable value for a flyback converter), the average losses will be loss V(ave) = η V(ave). (4) 5
6 Topic Sce a good portion of the power goes directly to the output, the FCstage efficiency will be η = loss. (5) Fig. 5 illustrates the effect of the BF on FC efficiency. An efficiency of 98.3% for a BF of unity decreases to 9.9% for a BF of 3.8 (with a universal putvoltage range). In addition, we assumed a constant 9% efficiency for the V converter, when reality the efficiency is even lower at higher BFs. The primary contrib u tors to a decrease efficiency are: the resistance of the wdgs, the R DS(on) of the MOSFETs. These factors all represent power losses that can crease with higher temperatures caused by the higher power dissipation. Efficiency, (%) This analysis suggests that a converter s ability to handle a wide putvoltage range (a high BF) has significant penalties, while a low BF improves not only the efficiency but also the cost and/or power density of the system (the components of the boost converter have to be rated for only a relatively small percentage of the output power). For these reasons, a boost fol lower concept has been proposed the literature. A boost follower is Boost Factor (BF) Fig. 5. FC efficiency versus BF for a V converter. a boost converter that produces a variable output voltage that is only slightly higher than the peak value of the put voltage. Consequently, the BF is near unity over the full range of the put voltage, and the efficiency of the FC stage is maximized. A boost follower can be easily implemented usg any standard FC control IC. An application circuit usg the Texas Instruments (TI) UCC387 FC controller [] is shown Fig. 6. The output voltage of the boost follower at different le Rectified Le Voltage R 998 k Vout (Boost Output Voltage) R3 998 k UCC387 FC Controller Voltage Amplifier C 6.6 µf R 4.3 k Q N R5 39. k R4 39. k + Internal Reference Fig. 6. Application circuit to implement boost follower. 6
7 4 Output Voltage (Conventional) 3.5 Voltage (V) 3 Output Voltage (Boost Follower) Input Voltage Time (ms) a. Output voltage at different le voltages. Fig. 7. Boost follower operation over a range of le voltages. Boost Factor (BF) Conventional Boost FC BoostFollower FC 5 5 Input Voltage, V (V) b. BF at different le voltages. Topic voltages is shown Fig. 7a. With this data, BFs at different le voltages can be calculated and are shown Fig. 7b. Because the output voltage is much closer to the put voltage, the BF of the boost follower is only. at an 85V put. The reduced BF leads to efficiency improvement for all the le voltages. Unfortunately, the boost follower solution is beneficial only if the dynamic range of the AC put voltage is relatively low and there is little or no requirement for holdup. If the AC le voltage varies widely (universal put voltage), the voltage applied to the downstream converter will vary from approximately to 375 VDC. This large variation will force compromises the design of the converter that will result signifi cant degradation of the system efficiency. The larger putvoltage range will impose a lower transformer turns ratio and reflect a higher current to the primary circuit, creasg the RMS current the primary wdgs and switchg transis tors as well as generatg higher differentialmode EMI. The low transformer turns ratio causes an crease the voltageblockg require ments of the secondary rectifiers; and the higher voltage ratg will result higher conduction drop (or higher R DS(on) the case of synchronous rectifiers) and poorer reverserecovery performance. In addition, the creased secondary voltage will require a larger filter ductance, resultg higher losses the output ductor. Sce the energy stored the FC capacitor at low put voltages will be low, very large capaci tance will be required to support the output voltage for the customary  to ms holdup time required many applications. All all, the power dissipation crease the downstream converter may equal or exceed the power savgs provided by the creased efficiency of the boost follower. A. Universal Input: Is it a Must? Considerg the significant impact that a wide putvoltage range has on the cost and efficiency of the FC converter s front end, it is important to look at the benefits and penalties associated with universal putvoltage designs. While portable products must operate with universal put voltages, other products will always be operated at a localpower voltage of V, 5 V, or 3 VAC. For stance, TV receivers are designed for regional use and are operated throughout their entire service life from a sgle le voltage of V for Japan and the U.S., and of 3 V for the European Union and other countries. When these products are designed for universal put voltage, the efficiency will be several percentage pots lower when operated at the V put range. Consequently, addition to the loss of power, costs will be curred for more expensive components and/or extra thermal management that are not necessary if the equipment is operated at the V put range. As previously shown, separate designs for V and 3 V will 7
8 Topic yield nearly equal efficiencies when operated with nearly the same BFs. (The V design will have an termediate VDC bus, and the 3V design will have a 4VDC bus.) Although separate designs for local voltage will yield higher efficiency, better power density, and lower cost, they have one major disadvantage different bills of material not only for the FC stage but also for the downstream converters, which will have to be designed for different voltages. This disadvantage may overshadow the benefits and is probably the major reason why virtually all applications are designed for universal put voltage. With efficiency becomg an ever more critical parameter, a different approach may be considered: a FC converter whose ternal connections between the components are configured on the assembly le so that the BF is roughly the same for both high and low putvoltage ranges. B. Ideal Configurable FC Converter An ideal configurable FC converters is shown Fig. 8. If the devices are configured parallel as shown Fig. 8a, a low leput voltage will generate a V output. If these devices are reconfigured series as shown Fig. 8b, a high leput voltage will generate a 4V output. Fig. 9 compares the BF of these converters with that of a conventional boost FC converter. The BF is reduced to half at low leput voltages, which significantly improves efficiency. In the ideal configurable FC circuit, all the components are active, operatg at both high and low leput voltages, which maximizes their utilization and lowers design cost. However, because there are different output voltages at different put les, the downstream DC/ DC converter has to be configurable as well. Other wise, the configurable FC converter only moves the design stress from the FC stage to the downstream DC/DC stage stead of really solvg the problem. One simple example of configurg a downstream DC/DC Boost Factor (BF) a. Configuration for low leput voltage. b. Configuration for high leput voltage. Fig. 8. Ideal configurable FC converter. converter is to use a full bridge at the V FC output and a half bridge at the 4V FC output. Because of the herent doublg effect that occurs when the topology changes from fullbridge to halfbridge, the transformer and secondaryside designs can stay the same. Nevertheless, usg a configurable converter both the FC and the DC/DC stage is complicated compared with the origal design and may not be attractive enough for commercial applications. Conventional Boost FC Converter Ideal Configurable FC Converter 5 5 Input Voltage, V (V) Fig. 9. BF for boost FC and ideal configurable FC converters. 8
9 S S Topic Fig.. Configurable threelevel boost converter. C. Configurable ThreeLevel FC Converter A more practical solution, a configurable threelevel FC converter (also know as a threelevel converter), uses the same FCstage components as the ideal configurable FC converter, but it uses jumpers to implement different topologies for high and low leput voltages. Because of the reconfigurable structure, the outputbus voltage does not change, and the DC/DC stage is not affected. A configurable threelevel boost converter is shown Fig., with two boost converters series to generate the output voltage. Because of the threelevel structure, the voltage stresses on the MOSFETs and diodes are only half of the output voltage, so lowvoltage devices can be used to achieve better performance []. At high leput voltages, the range switch, S, is open (see Fig. ). The whole converter operates as a sgle boost converter with an output voltage of 4 V. It has two MOSFETs series and two diodes series, all of which creases the conduction loss. However, at high leput voltages, the boost FC converter is highly efficient because of the low current stress and the lower BF. The creased conduction loss still results very high efficiency. At low leput voltages, the range switch, S, is closed to construct a voltage doubler. On each half cycle of the leput voltage, only one boost converter is operatg, as shown Fig., generatg V on one of the two seriesconnected output capacitors. The other boost converter operates durg the subsequent le half cycle, generatg V on the other output capacitor. The output voltage of the FC stage is the sum of the voltages on the two capacitors, i.e. 4 V. The reduced voltage delivered by each a. Chargg ductor. S b. Dischargg ductor. Fig.. Equivalent circuit with high leput voltage. S a. ositive half cycle. S b. Negative half cycle. Fig.. Equivalent circuit with low leput voltage. 9
10 Topic converter greatly reduces the duty cycle and also reduces conduction loss, switchg loss, and diode reverserecovery loss. Sce each MOSFET and diode see only half of the output voltage, lowvoltage devices can be used to easily achieve better conduction loss and switchg loss and further crease the overall system efficiency. Sce the configurable threelevel FC con verter has a V output voltage for each converter at low leput voltages and 4 V at high leput voltages, it has the same BF as the ideal configurable FC circuit. The reduced BF helps to improve the FCstage efficiency at low leput voltages. Another major source of efficiency loss is the put diode bridge. The voltage drop on the diode bridge stays relatively constant regardless of the level of current flowg through it. For a conventional boost FC converter, there are two diodes the diode bridge conductg current at any time. The total voltage drop on those two diodes is about V. At a high leput voltage of V, this V drop causes a less than % drop efficiency. However, at a low leput voltage of V, the same V drop causes a.% drop efficiency. Efficiency can drop even further when the put voltage is lower. In a configurable threelevel FC converter at low leput voltages, the voltagedoubler structure has only one diode of the diode bridge the conduction path. This removes a diode voltage drop of V, which can be easily translated to a % crease efficiency with a V put. An efficiency plot combg all the benefits of the configurable threelevel FC converter is shown Fig. 3. At a high leput voltage, the efficiency is about the same as with a conventional boost FC converter; but at a low leput voltage, the efficiency improves dramatically. Although this configurable threelevel FC converter improves overall system efficiency, it also presents many challenges. The threelevel structure requires a highside driver. if a drive transformer is to be used, the maximum duty cycle may have to be limited to perhaps 8%. An tegrated halfbridge driver IC Efficiency, (%) Input Voltage, V Fig. 3. Efficiency of configurable threelevel FC converter. could simplify the driver design but would crease the overall system cost as compared to the simple driver of a conventional FC circuit. Besides, to simplify the drivg scheme, the control ground would have to be set at the source of the bottom switch. Because of the boost diode, the output voltage is not directly connected to the control ground, and isolation would be required for the voltagefeedback loop. Voltage balancg on the output capacitors is another concern. For series capacitors, voltage balancg is determed by the capacitance tolerances and the leakage difference. However, a configurable threelevel FC converter at a high leput voltage, both MOSFETs turn on and off at the same time. A slight difference the switchg speed will cause current to flow through the middle pot, where two capacitors are series together. Voltage balancg can be realized by placg an equalizg resistor parallel to each capacitor, which causes extra loss, or by addg a dedicated control loop to slightly adjust the ON time of each MOSFET. The designer usg a configurable threelevel FC converter also faces the difficulty of implementg current sensg. Although usg four current transformers can reconstruct the ductor current, the system implementation becomes much more complicated. Furthermore, at low leput voltages, only one of the boost converters is operatg durg (V) 
11 each le half cycle, which means that component utilization is not optimal at this operation mode. As the demand for higher efficiency creases, a pot may be reached where the potential efficiency improvement may justify the effort and expense associated with the solution of these problems. D. The Buck FC It is terestg to note that a buck converter can also be used to shape the put current and provide powerfactor correction. Sce the output voltage of a buck converter can only be equal to or lower than its put voltage, the buck FC, as shown Fig. 4, will draw current from the put durg only a fraction of levoltage cycle, leadg to the buck s capa bility to yield very low harmonic distortions or correct the F to close to unity. Nevertheless, as it will be shown below, the buck FC can be a valuable circuit some applications. A quantitative analysis of the buckfc efficiency is outside the scope of this topic, but it is tuitively clear that the efficiency will be higher at lower put voltages where the switchg and ductor losses will be lower. Similarly to the boostfc case, a buck factor can be defed as: Buck Factor = V V out (6) Decreasg the buck factor (i.e. creasg the output voltage of the buck FC for a given voltage) improves the efficiency, but, at the same time, decreases the putcurrent conduction angle that causes a deterioration of both the F and the harmonic distortion. Increasg the buck Factor improves FC and harmonic distortions, but reduces efficiency. Also, reducg the output voltage reduces the energy storage capability of the FC output capacitors. At the correct balance between these para meters, the buck FC becomes an effective solution for meetg the Energy Star efficiency goals when converters are tended for universal voltage. For stance, if the FC output voltage is set to 8 V, the buck factor at 5 VAC will be and the conduction angle will be approximately (see Fig. 5). The result is a F of.97 (well above the Energy Star mimum) and a THD of 4.7%. I + Control (Modified eak CMC) Fig. 4. Buck FC. I (A) load I (A) load Time (ms) a. Buck factor =. Time (ms) b. Buck factor = 4. Fig. 5. Buck FC normalized put current. Vboost Out As the put voltage creases to 6 V, the buck factor will crease to 4 and both the F and THD will improve, reachg.997 and 7.3% respectively. The spectral analysis of the current waveform shows that all harmonics are just a fraction of the maximum allowed by EN63. Although a buckfc output voltage has an adverse effect on energystorage and holdup capability, there are applications, such as laptop AC adapters, where the holdup requirements are mimal and an 8V bus voltage allows use of Topic 
12 Topic V MOSFETs the secondstage converter. These MOSFETs are expensive with extremely low dratosource capacitance and low R DS(on), which results a highlyefficient, costeffective power tra. The creased lowle efficiency of the buck converter balances the crease of losses the EMI filter and put rectifier. Fig. 6 shows the fullload efficiency of the 8V buck FC stage used a highperformance, 9W laptop adapter. In this design, the efficiency exceeds 96% over the full 9 to 7V putvoltage range. VI. Conclusion Active FC can improve the efficiency of ACtoDC conversion. The improvement may be curtailed and the FCstage size and cost will be adversely affected if the system is designed to accommodate a wide range of put voltages. These problems may be mitigated by the use of different FC circuits that reduce the BF and mimize the power processed by the FC stage. VII. References [] Michael O Loughl. Simple circuitry gets that old FC controller workg a boostfollower FC application. [Onle]. Available: [] M.T. Zhang, Yim Jiang, F.C. Lee, M.M. Jovanovic, "Sglephase threelevel boost power factor correction converter," Applied ower Electronics Conference and Exposition, AEC '95, pp , Vol., March 995. [3] High power factor boost rectifier apparatus, by Dragan Maksimovic and Robert W. Erickson. (995, Jan. 7). U.S. atent [Onle]. Available: =esmbaaaaebaj&dq=maksimovic Efficiency (%) AC Input (V) Fig. 6. Efficiency of a 8V, 9W buck FC stage as a function of the AC put voltage. [4] Office of Gas and Electricity Markets. (3, Jan.). Electricity distribution losses: A consultation document. [Onle]. Available: ElecDist/olicy/DistChrgs/Documents/ 363distlosses.pdf [5] Brian Fortenbery and Jonathan G. Koomey. (6, Mar. 3). Assessment of the impacts of power factor correction computer power supplies on commercial buildg le losses. California Energy Commission. CA. [Onle]. Available: powersupplies.org/pages/latest_rotocol/ ower_factor_report_cec_543.pdf 
13 Appendix: Effect of FC on owerdistribution Losses Because creasg the power factor (F) reduces the RMS put current drawn by a load, it should be expected that a low F will crease the conduction losses the transmission and distribution wirg that connects the load to the power station. This appendix presents a simplified correla tion to the F for loads, load currents, power losses, and system efficiency. We first assume that the total loadrelated transmission power loss at a F of unity is 5%. With the variables normalized such that load = V load = I load = at F =, and where d is the wirg power loss at F = and R w is the wirg resistance, the followg equations show the basic relationship between the F and the primary system variables: d =.5 d = Iload Rw (7) = R w d The load current as a function of the F is Iload = F. (8) The generalized wirg power loss, w, as a function of the F is d = w Iload and (9) = d d F. The system efficiency as a function of the F is ( ) = +. (3) η df d The losses the FC circuit will degrade the improvement the distri bu tion efficiency if the efficiency of the FC circuit is low, so correctg the F may actually decrease rather than crease the overall system efficiency. For stance, if we assume a F of.7 for a supply without FC, the system efficiency will be η d ( ) = (3) Fig. 7. Simplified equivalent circuit of a powerdistribution system. If the F is corrected to unity, the distribution efficiency will be η d ()=.95. (3) The overall system efficiency, η do, cludg the distribution and the FC efficiency, η FC, will be as follows: ηdo = ηd () ηfc (34) Note that if the FCstage efficiency, η FC, is equal to.953, correctg the power factor will not provide any efficiency enhancement. η do = =.97 Distribution Efficiency, Wirg R w =.5 Generator V gen ower Factor (F) Load R load = Fig. 8. Effect of the power factor on powerdistribution efficiency. Topic 3
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