| Home | Articles | Forum | Glossary | Books |
|
AMAZON multi-meters discounts AMAZON oscilloscope discounts Gate commutated inverters (DC/AC converters) Most modern AC variable speed drives in the 1 kW to 500 kW range are based on gate commutated devices such as the GTO, MOSFET, BJT and IGBT, which can be turned ON and OFF by low power control circuits connected to their control gates. The difficulties experienced with thyristor commutation in the early days of PWM inverters have largely been overcome by new developments in power electronic technology. Diodes and thyristors are still used extensively in line-commutated rectifiers. Starting with a DC supply and using these semiconductor power electronic switches, it’s not possible to obtain a pure sinusoidal voltage at the load. On the other hand, it may be possible to generate a near-sinusoidal current. Consequently, the objective is to control these switches in such a way that the current through the inductive circuit should approximate a sinusoidal current as closely as possible. Single-phase square wave inverter To establish the principles of gate-controlled inverter circuits, four semiconductor power switches feeding an inductive load from a single-phase supply. ++++ Single-phase DC to AC inverter This circuit can be considered to be an electronic reversing switch, which allows the input DC voltage VD to be connected to the inductive load in any one of the following ways: (1) S1 = on, S4 = on, giving +VD at the load (2) S2 = on, S3 = on, giving -VD at the load (3) S1 = on, S2 = on, giving zero volts at the load S3 = on, S4 = on, giving zero volts at the load (4) S1 = on, S3 = on, giving a short circuit fault S2 = on, S4 = on, giving a short circuit fault However, these four switches can be controlled to give a square waveform across the inductive load. This makes use of switch configuration (1) and (2), but not switch configuration (3) or (4). Clearly, for continued safe operation, option (4) should always be avoided. In the case of a purely inductive load, the current waveform is a triangular waveform. In the first part of the cycle, the current is negative although only switches S1 and S4 are on. Since most power electronic devices cannot conduct negatively, to avoid damage to the switches, this negative current would have to be diverted around them. Consequently, diodes are usually provided in anti-parallel with the switches to allow the current flow to continue. These diodes are sometimes called reactive or free-wheeling diodes and conduct whenever the voltage and current polarities are opposite. This occurs whenever there is a reverse power flow back to the DC supply. ++++ Square wave modulation waveforms The frequency of the periodic square wave output is called the fundamental frequency. Using Fourier analysis, any repetitive waveform can be resolved into a number of sinusoidal waveforms, comprising one sinusoid at fundamental frequency plus a number of sinusoidal harmonics at higher frequencies, which are multiples of the fundamental frequency. The harmonic spectrum for a single-phase square wave output below. The amplitude of the higher order harmonics voltages falls off rapidly with increasing frequency. ++++ Square-wave harmonic spectrum The RMS value of the fundamental sinusoidal voltage component is: The RMS value of the nth harmonic voltage: This illustrates that the square wave output voltage has a lot of unwanted components of reasonably large magnitude at frequencies close to the fundamental. The current flowing in the load as a result of the output voltage is distorted, as demonstrated by the non-sinusoidal current wave-shape. In this example, the current has a triangular shape. If the square-wave voltage were presented to a single-phase induction motor, the motor would run at the frequency of the square-wave but, being a linear device (inductive/ resistive load), it would draw non-sinusoidal currents and would suffer additional heating due to the harmonic currents. These currents may also produce pulsating torques. To change the speed of the motor, the fundamental frequency of the inverter output can be changed by adjusting the speed of the switching. To increase frequency, switching speed can be increased and to decrease frequency, switching speed can be decreased. If it’s required to also control the magnitude of the output voltage, the average inverter output voltage can be reduced by inserting periods of zero voltage, using switch configuration (3) as shown. Each half cycle then consists of a square pulse which is only a portion of a half period as shown below. ++++ Square wave modulation with reduced voltage pulse width The process of changing the width of the pulse to reduce the average RMS value of a waveform is called pulse width modulation (PWM). In the single phase, pulse width modulation makes it possible to control the RMS value of the output voltage. The fundamental sinusoidal component of voltage is continuously variable in the following range: The harmonic spectrum of this modified waveform depends on the fraction that the pulse is of the full square wave, but is broadly similar to the waveform. Single-phase pulse width modulation (PWM) inverter The fact that the voltage supply to the stator of an AC induction motor is a square wave and is distorted is not in itself a problem for the motor. The main problem comes from the distortion of the current waveform, which results in extra copper losses and shaft torque pulsations. The ideal inverter output is one that results in a current waveform of low harmonic distortion. Since an AC induction motor is predominantly inductive, with a reactance that depends on the frequency (XL = j2pfL), it’s beneficial if the voltage harmonic distortion can be pushed into the high frequencies, where the motor impedance is high and not much distorted current will flow. One technique for achieving this is sine-coded pulse width modulation (sine-PWM). This requires the power devices to be switched at frequencies much greater than that of the fundamental frequency producing a number of pulses for each period of the desired output period. The frequency of the pulses is called the modulation frequency. The width of the pulses is varied throughout the cycle in a sinusoidal manner giving a voltage waveform. This also shows the current waveform for an inductive load showing the improvement in the waveform. ++++ Sine-coded pulse width modulated voltage and current The improvement in the current waveform can be explained by the harmonic spectrum shown. It can be seen that, although the voltage waveform still has many distortion components, they now occur at higher harmonic frequencies, where the high load impedance of the motor is effective in reducing these currents. ++++: Harmonic spectrum for a PWM inverter Increasing the modulation frequency will improve the current waveform, but at the expense of increased losses in the switching devices of the inverter. The choice of modulation frequency depends on the type of switching device and its frequency. With the force-commutated thyristor inverter (10 years ago), a modulation frequency of up to 1 kHz was possible. With the introduction of GTOs and BJTs, this could be pushed up to around 5 kHz. With IGBTs, the modulation frequency could be as high as 20 kHz. In practice, a maximum modulation frequency of up to 12 kHz is common with IGBT inverters up to about the 22 kW motor size and 8 kHz for motors up to about 500 kW. The choice of modulation frequency is a trade off between the losses in the motor and in the inverter. At low modulation frequencies, the losses in the inverter are low and those in the motor are high. At high modulation frequencies, the losses in the inverter increase, while those in the motor decrease. One of the most common techniques for achieving sine-coded PWM in practical inverters is the sine-triangle intersection method. A triangular saw-tooth waveform is produced in the control circuit at the desired inverter switching frequency. This is compared in a comparator with a sinusoidal reference signal, which is equal in frequency and proportional in magnitude to that of the desired sinusoidal output voltage. The voltage VAN is switched high whenever the reference waveform is greater than the triangle waveform. The voltage VBN is controlled by the same triangle waveform but with a reference waveform shifted by 180°. The actual phase-to-phase output voltage is then VAB, which is the difference between VAN and VBN, which consists of a series of pulses each of whose width is related to the value of the reference sine-wave at that time. The number of pulses in the output voltage VAB is double that in the inverter leg voltage VAN. E.g., an inverter switching at 5 kHz should produce switching distortion at 10 kHz in the output phase-to phase voltage. The polarity of the voltage is alternatively positive and negative at the desired output frequency. ++++ : Principle of triangle intersection PWM It can also be seen that the reference sine-wave is given a DC component so that the pulse produced by this technique has a positive width. This puts a DC bias on the voltage of each leg. However, each leg has the same DC offset which disappears from the load voltage. The technique using sine-triangle intersection is particularly suited for use with the older analogue control circuits, where the two reference waveforms were fed into a comparator and the output of the comparator was used to trigger the inverter switches. Modern digital techniques operate on the basis of a switching algorithm, E.g. by producing triggering pulses proportional to the area under a part of the sine wave. In recent times, manufacturers have developed a number of different algorithms that optimize the performance of the output waveforms for AC induction motors. These techniques result in PWM output waveforms which are similar to those shown. The sine-coded PWM voltage waveform is a composite of a high frequency square wave at the pulse frequency (the switching carrier) and the sinusoidal variation of its width (the modulating waveform). It has been found that, for lowest harmonic distortion, the modulating waveform should be synchronized with the carrier frequency, so that is it should contain an integral number of carrier periods. This requirement becomes less important with high carrier frequencies of more than about twenty times the modulating frequency. The voltage and frequency of a sinusoidal PWM waveform are varied by changing the reference waveform giving outputs as shown. • ++++ (a) shows a base case, with the rated V/f ratio • ++++ (b) shows the case where the voltage reference is halved, resulting in the halving of each pulse • ++++(c) shows the case where the reference frequency is halved, resulting in the extension of the modulation over twice as many pulses ++++ 31: Variation of frequency and voltage with sinusoidal PWM The largest voltage with sine-coded PWM occurs when the pulses in the middle are widest, giving an output with a peak voltage equal to the supply. The modulation index is defined as the ratio of the peak AC output to the DC supply. Thus the largest output voltage occurs when the modulation index is 1. It’s possible to achieve larger voltages than the DC supply by abandoning strict sine-PWM by adding some distortion to the sinusoidal reference voltage. This results in the removal of some of the pulses near the centre of the positive and negative parts of the waveform, a process called pulse dropping. In the limit, a square wave voltage waveform can be achieved with a peak value which is up to 127% of what can be achieved by strict sine-PWM. hree-phase inverter A three-phase inverter could be constructed from three inverters of the type. However, it’s more economical to use a 6-pulse (three-leg) bridge inverter. ++++ Three-phase inverter using gate controlled switches In its simplest form, a square output voltage waveform can be obtained by switching each leg high for one half-period and low for the next half-period, at the same time ensuring that each phase is shifted one third of a period (120 deg). The resulting phase-to-phase voltage waveform comprises a series of square pulses whose widths are two thirds of the period of the switch in each phase. The resulting voltage waveform is called a quasi-square wave (QSW) voltage. This simple technique was used in early voltage source inverters (VSI) which used forced commutated thyristors in the inverter bridge. To maintain a constant V/f ratio, the magnitude of the DC bus voltage was controlled by the rectifier bridge to keep a fixed ratio to the output frequency, which was controlled by the inverter bridge. This technique was sometimes also called pulse amplitude modulation (PAM). The output voltage of a three-phase converter has a harmonic spectrum very similar to the single-phase square wave, except that the triplen harmonics (harmonics whose frequency is a multiple of three times the fundamental frequency) have been eliminated. In an inverter with a 3-phase output, so the 3rd, 9th, 15th, 21st, etc. harmonics are eliminated. To develop a 3-phase variable voltage AC output of a particular frequency, the voltages VAN, VBN, VCN on the 3 output terminals a, b, and c can be modulated on and off to control both the voltage and the frequency. The pulse-width ratio over the period can be changed according to a sine-coded PWM algorithm. When the phase-phase voltage VAB is formed, the present modulation strategy gives only positive pulses for a half period followed by negative pulses for a half period, a condition known as consistent pulse polarity. It can be shown that consistent pulse polarity guarantees lowest harmonic distortion with most of the distortion being at twice the inverter chopping frequency. The presence of both positive and negative pulses throughout the whole period of the phase-phase voltage (inconsistent pulse polarity) gives distortion at the inverter chopping frequency, where it will have more effect on current distortion and is a sign of a poor modulation scheme. ++++ Quasi square wave modulation output waveforms ++++ Output voltage waveform of a 3-phase sine coded PWM Manufacturers of AC frequency converters continue to work on the development of more efficient PWM algorithms in an attempt to improve the current waveform. The ultimate objective is a completely sinusoidal current, which produces no harmonic losses in the motor. These more advanced PWM algorithms have become possible as a result of the increased speed and power of microprocessors. Most reputable PWM inverters can operate at modulation frequencies between 2 kHz and 16 kHz and produce a current waveform, which is sufficiently sinusoidal to overcome the problem of motor de-rating for harmonic losses. However, as a result of the high PWM frequencies, a new problem has emerged, the high frequency leakage current due to the motor cable capacitance. In practical inverters, there are two conflicting requirements which need to be met when it’s required to accelerate a motor from standstill to rated speed with constant V/f ratio. • The need to operate the inverter at its highest possible switching frequency to achieve low current distortion • The importance of maintaining synchronization A common strategy to achieve both, particularly for older PWM inverters, is to begin with the inverter switching frequency at about half the maximum value. As the speed is increased, the saw-tooth carrier frequency is increased in proportion to maintain synchronism. When the carrier frequency reaches its maximum, it’s then switched to half its value for further increase in the output frequency. Thus the inverter exhibits a continual ramp increase in frequency followed by a sudden reduction at the changeover point. If the inverter is operating in the audible range then a change in pitch will be heard similar to the sound of a car engine as the car accelerates through the gears, hence the term 'gear-changing'. ++++ Changing the modulation frequency in steps (gear-changing) When the motor speed is reduced from maximum to zero, there is a similar change in carrier frequency with output frequency. However, the changeover points must be different, otherwise an inverter sitting at one of the changeover frequencies might continually oscillate between the upper and lower carrier frequency. This is avoided by introducing hysteresis in the control scheme. Gate controlled power electronic devices A number of gate controlled devices have become available in the past decade, which are suitable for use as bi-stable switches on power inverters for AC Variable Speed Drives. These can be divided into two main groups of components: • Those based on thyristor technology such as gate turnoff thyristor (GTO) and field controlled thyristor (FCT) • Those based on transistor technology such as the bipolar junction transistor (BJT), field effect transistor (FET) and the insulated gate bipolar transistor (IGBT) 3.8.1 Gate turn-off thyristor (GTO) A GTO thyristor is another member of the thyristor family and is very similar in appearance and performance to a normal thyristor, with the important additional feature that it can be turned off by applying a negative current pulse to the gate. GTO thyristors have high current and voltage capability and are commonly used for larger converters, especially when self commutation is required. SYMBOL; IDEAL
The performance of a GTO is similar to a normal thyristor. Forward conduction is blocked until a positive pulse is applied to the gate terminal. When the GTO has been turned on, it behaves like a thyristor and continues to conduct even after the gate pulse is removed, provided that the current is higher than the holding current. The GTO has a higher forward voltage drop of typically 3 V to 5 V. Latching and holding currents are also slightly higher. The important difference is that the GTO may be turned off by a negative current pulse applied to the gate terminal. This important feature permits the GTO to be used in self commutated inverter circuits. The magnitude of the off pulse is large and depends on the magnitude of the current in the power circuit. Typically, the gate current must be 20% of the anode current. Consequently, the triggering circuit must be quite large and this results in additional commutation losses. Like a thyristor, conduction is blocked in the reverse biased direction or if the holding current falls below a certain level. Since the GTO is a special type of thyristor, most of the other characteristics of a thyristor covered above also apply to the GTO and won’t be repeated here. The mechanical construction of a GTO is very similar to a normal thyristor with stud types common for smaller units and disc types common for larger units. GTO thyristors are usually used for high voltage and current applications and are more robust and tolerant to over-current and over-voltages than power transistors. GTOs are available for ratings up to 2500 amps and 4500 volts. The main disadvantages are the high gate current required to turn the GTO off and the high forward volt drop. Power electronic converters of all types are usually controlled by an electronic control circuit which controls the on/off state of the power electronic devices and provides the interface for the external controls. Until recently, all control circuits were of the analog type using operational amplifiers (Op-Amps). Modern control circuits are usually of the digital type using microprocessors. Field controlled thyristors (FCT) Although the GTO is likely to maintain its dominance for the high power, self commutated converter applications for some time, new types of thyristor are under development in which the gate is voltage controlled. Turn on is controlled by applying a positive voltage signal to the gate and turn off by a negative voltage. Such a device is called a field controlled thyristor (FCT) and the name highlights the similarity to the field effect transistor (FET). The FCT is expected to eventually supersede the GTO because it has a much simpler control circuit in which both the cost and the losses may be substantially reduced. Small FCTs have become available and it’s expected that larger devices will come into use in the next few years. Development of a practical cost effective device has been a bit slower than expected. Power bipolar junction transistors (BJT) Transistors have traditionally been used as amplification devices, where control of the base current is used to make the transistor conductive to a greater or lesser degree. Until recently, they were not widely used for power electronic applications. The main reasons were that the control and protective circuits were considerably more complicated and ex pensive and transistors were not available for high power applications. They also lacked the overload capacity of a thyristor and it’s not feasible to protect transistors with fuses. In the mid-1980s, the NPN transistor known as a bipolar junction transistor (BJT) has become a cost effective device for use in power electronic converters. Modern BJTs are usually supplied in an encapsulated module and each BJT has two power terminals, called the collector (C) and emitter (E), and a third control terminal called the base (B). SYMBOL; IDEAL:
A transistor is not inherently a bi-stable (on/off) device. To make a transistor suitable for the conditions in a power electronic circuit where it’s required to switch from the blocking state (high voltage, low current) to the conducting state (low voltage, high current) it must be used in its extreme conditions, fully off to fully on. This potentially stresses the transistor and the trigger and protective circuits must be coordinated to ensure the transistor is not permitted to operate outside its safe operating area. Suitable control and protective circuits have been developed to protect the transistor against over-current when it’s turned on and against over-voltage when it’s turned off. When turned on, the control circuit must ensure that the transistor does not come out of saturation otherwise it will be required to dissipate high power. In practice, the control system has proved to be cost effective, efficient, and reliable in service. ++++ Switching locus of a power BJT with an inductive load Transistors don’t tolerate reverse voltages. When BJTs are used in inverter bridges, they must be protected against high reverse voltages by means of a reverse diode in series or in parallel. For the same reason, transistors are not used in rectifier bridges, which have to be able to withstand reverse voltages. In general, transistors were considered to be less robust and less tolerant of overloads and 'spikes' than thyristors. GTO thyristors were often preferred for converters. In spite of the earlier problems experienced with transistors, AC converters have used power transistors at power ratings up to about 150 kW at 415 V. The main advantage of transistors is that they can be turned on and off from the base terminal, which makes them suitable for self commutated inverter circuits. This results in power and control circuits which are simpler than those required for thyristors. Unfortunately, the base amplification factor of a transistor is fairly low (usually 5 to 10 times) so the trigger circuit of the transistor must be driven by an auxiliary transistor to reduce the magnitude of the base trigger current required from the control circuit. The emitter current from the auxiliary transistor drives the base of the main transistor using the Darlington connection. ++++ a double Darlington connection, but for high power applications, two auxiliary transistors (triple Darlington) may be used in cascade to achieve the required amplification factor. The overall amplification factor is approximately the product of the amplification factors of the two (or three) transistors. ++++ Power Darlington transistor Transistors, used in VSD applications, are usually manufactured as an integrated circuit and encapsulated into a 3 terminal module, complete with the other necessary components, such as the resistors and anti-parallel protection diode. The module has an insulated base suitable for direct mounting onto the heat-sink. This type of module is sometimes called a power Darlington transistor module. The non-parallel diode protects the transistors from reverse biasing. In practice, this diode in the integrated construction is slow and may not be fast enough for inverter applications. Consequently, converter manufacturers sometimes use an external fast diode to protect the transistors. The following shows the saturation characteristic of Toshiba MG160 S1UK1 triple Darlington power transistor rated at 1400 V, 160 amp with a built-in free-wheeling diode. ++++ Characteristics of a 160 amp bipolar junction transistor (BJT) Although the control circuits are completely different, the power circuit performance of a BJT is similar to a GTO thyristor. Forward conduction is blocked until a positive current is applied to the gate terminal and will conduct as long as the voltage is applied. During forward conduction, it also exhibits a forward voltage drop which causes losses in the power circuit. The BJT may be turned off by applying a negative current to the gate. The main advantages of the bipolar junction transistor (BJT) are:
The main disadvantages of BJTs are:
Power bipolar junction transistors are available for ratings up to a maximum of about 300 amps and 1400 volts. For VSDs requiring a higher power rating, GTOs are usually used in the inverter circuit. Field effect transistor (FET) A field effect transistor (FET) is a special type of transistor that is particularly suitable for high speed switching applications. Its main advantage is that its gate is voltage controlled rather than current controlled. It behaves like a voltage controlled resistance with the capacity for high frequency performance. FETs are available in a special construction known as the MOSFET. MOS stands for metal oxide silicon. The MOSFET is a three terminal device with terminals called the source (S), drain (D), and the gate (G), corresponding to the emitter, collector, and gate of the NPN transistor. SYMBOL; IDEAL: Forward conduction: Resistance (less) Forward blocking: Loss (less) (no leakage current) Reverse blocking: Loss (less) (no leakage current) Switch on/off time: Instantaneous The overall performance of an FET is similar to a power transistor, except that the gate is voltage controlled. Forward conduction is blocked if the gate voltage is low, typically less than 2 volts. When a positive voltage Vgs is applied to the gate terminal, the FET conducts and the current will quickly rise in the FET to a level dependent on the gate voltage. The FET will conduct as long as gate voltage is applied. The FET may be turned off by removing the voltage applied to the gate terminal or making it negative. MOSFETs are majority carrier devices, so they don’t suffer from long switching times. With their very short switching times, the switching losses are low. Consequently, they are best suited to high frequency switching applications. A typical performance characteristic of a field effect transistor is shown below. ++++ Typical characteristic of a field effect transistor Initially, high speed switching was not an important requirement for AC converter applications. With the development of pulse width modulated (PWM) inverters, high frequency switching has become a desirable feature to provide a smooth output current waveform. Consequently, power FETs were not widely used until recently. At present, FETs are only used for small PWM frequency converters. Ratings are available from about 100 amp at 50 volt to 5 amp at 1000 volt, but for VSD applications MOSFETs need to be in the 300-600 volt range. The advantages and disadvantages of MOSFETs are almost exactly the opposite of BJTs. The main advantages of a power MOSFET are:
The main disadvantages of a power MOSFET are:
Insulated gate bipolar transistor (IGBT) The insulated gate bipolar transistor (IGBT) is an attempt to unite the best features of the bipolar junction transistor and the MOSFET technologies. The construction of the IGBT is similar to a MOSFET with an additional layer to provide conductivity modulation, which is the reason for the low conduction voltage of the power BJT. The IGBT construction avoids the MOSFET's reverse conducting body diode but introduces a parasitic thyristor, which could give spurious operation in early devices. The IGBT device has good forward blocking but very limited reverse blocking ability. It can operate at higher current densities than either the power BJT or MOSFET allowing a smaller chip size. The IGBT is a three terminal device. The power terminals are called the emitter (E) and collector (C), using the BJT terminology, while the control terminal is called the gate (G), using the MOSFET terminology. SYMBOL; IDEAL: Forward conduction: Resistance (less) Forward blocking: Loss (less) (no leakage current) Reverse blocking: Loss (less) (no leakage current) Switch on/off time: Instantaneous The electrical equivalent circuit of the IGBT, shows that the IGBT can be considered to be a hybrid device, similar to a darlington transistor configuration, with a MOSFET driver and a power bipolar PNP transistor. Although the circuit symbol above suggests that the device is related to a NPN transistor, this should not be taken too literally. ++++ 40: The equivalent circuit of an IGBT The gate input characteristics and gate drive requirements are very similar to those of a power MOSFET. The threshold voltage is typically 4 V. Turn-on requires 10 V to 15 V and takes about 1 µs. Turn-off takes about 2 µs and can be obtained by applying zero volts to the gate terminal. Turn-off time can be accelerated, when necessary, by using a negative drive voltage. IGBT devices can be produced with faster switching times at the expense of increased forward voltage drop. An example of a practical IGBT driver circuit is shown. This circuit can drive two IGBTs, connected to a 1000 V supply, at a switching frequency of 10kHz with propagation times of no more than 1µs. ++++ Circuit diagram of semikron SKHI 20 hybrid double IGBT or double MOSFET driver IGBTs are currently available in ratings from a few amps up to around 500 A at 1500 V, which are suitable for 3-phase AC VSDs rated up to about 500 kW at 380 V/415 V/480 V. They can be used at switching frequencies up to 100kHz. bipolar junction transistors (BJTs) have now largely been replaced by IGBTs for AC variable speed drives. The main advantages of the insulated gate bipolar transistor (IGBT) are: • Good power handling capabilities • Low forward conduction voltage drop of 2 V to 3 V, which is higher than for a BJT but lower than for a MOSFET of similar rating; • This voltage increases with temperature making the device easy to operate in parallel without danger of thermal instability; • High speed switching capability; • Relatively simple voltage controlled gate driver; • Low gate current Some other important features of the IGBT are: • There is no secondary breakdown with the IGBT, giving a good safe operating area and low switching losses • Only small snubbers are required • The inter-electrode capacitances are not as relatively important as in a MOSFET, thus reducing miller feedback • There is no body diode in the IGBT, as with the MOSFET, and a separate diode must be added in anti-parallel when reverse conduction is required-- E.g. in voltage source inverters Comparison of power ratings and switching speed of gate controlled power electronic devices ++++ Performance limits of gate controlled devices Other power converter circuit components Inductance SYMBOL; IDEAL: Inductance: constant (linear) Resistance: zero (no losses) EQUATIONS: L f j = X p 2 L Capacitance SYMBOL: IDEAL: Capacitance: constant (linear) Resistance: infinity (no losses) EQUATIONS: Resistance SYMBOL: IDEAL: Resistance: constant (linear) and free of inductance and capacitance EQUATIONS: I R = V Transformer SYMBOL: IDEAL: Magnetizing current negligible; Free of losses and capacitance |
| PREV: Power electronic converters (2) | NEXT: | Guide Index | HOME |