Power Electronic Handbook P7

Modulation Strategies PWM Signals with DC Average • PWM Signals for AC Output7.4 Third Harmonic Injection for Voltage Boost of SPWM Signals and DSPs 7.7 Hysteresis Feedback Control Intr

Modulation Strategies

PWM Signals with DC Average • PWM Signals for

AC Output7.4 Third Harmonic Injection for Voltage Boost

of SPWM Signals

and DSPs

7.7 Hysteresis Feedback Control

Introduction • Principles of the Hysteresis Feedback Control Circuits • Design Procedure • Experimental

Results • Conclusions

How the SVPWM Works • Implementation • Switching Signals

7.1 Introduction

Michael Giesselmann

In this chapter, modulation techniques for power electronics circuits are discussed Modulation niques are strategies to control the state of switches in these circuits Switch mode is preferred to linearoperation since switches ideally do not dissipate any power in either the ON or OFF state Depending

tech-on the switches that are being used, it may tech-only be possible to ctech-ontrol the turn tech-on instants However,most modern power semiconductors such as IGBTs can be turned on and off tens of thousands of timesper second on command In parallel with the development of these modern power semiconductors, newmodulation techniques have emerged In the following sections, a number of modulation techniquesalong with their advantages and disadvantages will be discussed Most figures have been generated usingMathCAD® 2000 [1] The examples for the digital modulation techniques and the voltage source–basedcurrent control techniques have been generated using PSpice®[2]

7.2 Six-Step Modulation

Michael Giesselmann

Six-step modulation represents an early technique to control a three-phase inverter Six-step modulationuses a sequence of six switching patterns for the three phase legs of a full-bridge inverter to generate afull cycle of three-phase voltages A switch pair connected between the positive DC bus and the negative

DC bus represents a phase leg The output terminal is the midpoint of the two switches Only one switch

of a phase leg may be turned on at any given time to prevent a short circuit between the DC buses Onestate of the inverter leg represents the case when the upper switch is turned on whereas the opposite state

is represented by the lower switch being turned on If each phase leg has these two states, the inverter has

23= 8 possible switching states Six of these states are active states, whereas the two states in which eitherall of the upper or all of the lower switches are turned on are called zero states, because the line-to-lineoutput voltage is zero in these cases The six discrete switching patterns for six-step modulation are shown

in Fig 7.1a to f For clarity, free-wheeling diodes have been omitted After the switching pattern shown

in Fig 7.1f, the cycle begins anew with the switching pattern shown in Fig 7.1a Note that in subsequentpatterns, only a single inverter leg changes states The switching patterns shown in Fig 7.1a to f representthe following inverter states in the following order:

• Positive peak of Phase A

• Negative peak of Phase C

• Positive peak of Phase B

• Negative peak of Phase A

• Positive peak of Phase C

• Negative peak of Phase BThe aforementioned inverter states are equally spaced in a circle with 60° of phase shift between them.This is illustrated in Fig 7.2 The hexagon in Fig 7.2 represents the trace of a voltage vector around acircle for six-step modulation This scheme could be extended to space vector modulation, if the voltagevector would not make discrete 60° steps, but would alternate at high speed between two adjacent states.The switching control would be such that the average time spend in the previous state is graduallydecreasing, whereas the average time spent in the next state is gradually increasing Also by inserting zerostates, the magnitude of the output voltage could be controlled

Figure 7.3 shows the phase to neutral waveform of one inverter leg for six-step operation if the neutral point

is considered the midpoint between the positive and negative bus The resulting line-to-line output voltage

is shown in Fig 7.4 This waveform is closer to a sinusoid than the phase to neutral voltage but it still has aconsiderable amount of harmonics Figure 7.5 shows the spectrum of the line-to-line voltage for six-stepoperation normalized to the fundamental frequency The lowest harmonic component is the 5th harmonic The advantages of six-step modulation are the simplicity of the procedure and the ability to use slow-switching, high-power devices like GTOs However, the harmonic content of the output voltage and theinability to control the magnitude of the output voltage are serious drawbacks Because of these drawbacksand due to the recent advances in high-power IGBT technology, this modulation scheme is today seldomconsidered for new designs

7.3 Pulse Width Modulation

Michael Giesselmann

Pulse width modulation (PWM) is the method of choice to control modern power electronics circuits Thebasic idea is to control the duty cycle of a switch such that a load sees a controllable average voltage Toachieve this, the switching frequency (repetition frequency for the PWM signal) is chosen high enoughthat the load cannot follow the individual switching events Switching, rather than linear operation of the

FIGURE 7.1 (a) GTO inverter indicating conducting switches for step 1 in six step sequence (b) GTO inverter indicating conducting switches for step 2 in six-step sequence (c) GTO inverter indicating conducting switches for step 3 in six-step sequence (d) GTO inverter indicating conducting switches for step 4 in six-step sequence (e) GTO inverter indicating conducting switches for step 5 in six-step sequence (Continued)

FIGURE 7.1 (Continued.) (f) GTO inverter indicating conducting switches for step 6 in six-step sequence.

FIGURE 7.2 Graphical representation of the vector positions of the inverter states in a circle for six-step modulation.

FIGURE 7.3 Phase to neutral waveform of the inverter for six-step operation.

power semiconductors, is of course done to maximize the efficiency because the power dissipation in aswitch is ideally zero in both states In a typical case, the switching events are just a “blur” to the load,which reacts only to the average state of the switch.

PWM Signals with DC Average

There are a number of different methods to generate periodic rectangular waveforms with varying dutycycle A standard method is the so-called carrier-based PWM technique, which compares a control signalwith a triangular (or sawtooth shaped) waveform Figure 7.6 shows an example of a triangular waveformwith 10-kHz repetition (switching) frequency By comparing this signal with a reference level, which canvary between 0 and 1 V, a PWM signal with a duty cycle between 0 and 100% is generated Because ofthe triangular carrier, the relation between the reference level and the resulting duty cycle is linear

Figure 7.7 shows an example where a PWM signal with 80% duty cycle is created This method worksvery well for duty cycles in the range from 5% up to 95% as shown in Figs 7.8 and 7.9 However, if thereference signal exceeds 100% or falls below 0%, the resulting PWM signal would be always on or alwaysoff, respectively This is called overmodulation This regime must be avoided by proper conditioning ofthe control signal In addition, for control signals resulting in PWM signals with duty cycle values ashigh as 99% or as low as 1%, the switch may never fully reach the opposite state and spend an undueamount of time in transitions Therefore, it is typically recommended to limit the control signal to arange, which avoids overmodulation as well as extremely narrow pulses

FIGURE 7.4 Line-to-line waveform of the inverter for six-step operation.

FIGURE 7.5 Spectrum of the line-to-line voltage for six-step operation normalized to the fundamental frequency.

FIGURE 7.6 Triangular carrier wave for PWM modulation with a duty cycle between 0 and 100%.

FIGURE 7.7 Triangular carrier wave and PWM signal for 80% duty cycle.

FIGURE 7.8 Triangular carrier wave and PWM signal for 95% duty cycle.

The spectrum of a typical PWM signal with 25% duty cycle with a switching frequency of 10 kHz isshown in Fig 7.10 The DC magnitude of 25% is clearly visible The harmonics are multiples of thecarrier frequency The lowest harmonic is located at 10 kHz This spectrum might look dramatic,especially in comparison with Fig 7.5, but the reader should be reminded that, due to the switchingspeed of modern power semiconductors, the carrier frequency can be chosen sufficiently high that theharmonics can be easily filtered with capacitors and inductors of small size.

PWM Signals for AC Output

In addition to a DC reference signal, any other waveform could be used as the modulation signal as long

as the highest frequency of its AC components are at least an order of magnitude less than the frequency

of the carrier signal Figure 7.11 shows an example of a carrier waveform, which is symmetrical withrespect to the zero level To generate a sinusoidal output voltage for an inverter, which is often desired,this carrier can be modulated with a sinusoidal reference signal An example is shown in Fig 7.12 Notethat for clarity, the ratio between the carrier frequency and the frequency of the modulation signal islower than recommended for actual implementation The resulting sinusoidal PWM (SPWM) voltagedrives one phase leg of an inverter If the voltage level is +1, the upper switch is on, and vice versa Afterfiltering out the switching frequency components, the resulting output voltage has the shape and fre-quency of the modulation signal For the remaining phase legs, the same technique, with reference signals

FIGURE 7.9 Triangular carrier wave and PWM signal for 5% duty cycle.

FIGURE 7.10 Spectrum of a PWM signal with 25% duty cycle.

that are phase shifted by 120 and 240°, is used The amplitude of the output voltage can be controlled

by varying the ratio between the peak of the modulation signal and the peak of the carrier wave If theamplitude of the modulation signal exceeds the amplitude of the carrier, overmodulation occurs and theshape of the fundamental of the output voltage deviates from the modulation signal

To appreciate the spectral content of sinusoidal PWM signals, a 20-kHz triangular carrier has been ulated with a 500-Hz sinusoid with an amplitude of 80% of the carrier signal The resulting SPWM signal isshown in Fig 7.13 The spectrum of this PWM signal is shown in Fig 7.14 The fundamental with an amplitude

mod-of 0.8 is located at 500 Hz The harmonics are grouped around multiples mod-of the carrier frequency [1]

It should be pointed out that this modulation scheme is far superior to the six-step technique describedearlier, because the difference between the switching frequency and the fundamental is much larger.Therefore, the carrier frequency components can be easily removed with LC filters of small size [2] Inaddition, the amplitude of the output voltage can be controlled simply by varying the amplitude ratiobetween the modulation signal and the carrier If six-step modulation is used, the DC bus voltage wouldhave to be controlled in order to control the amplitude of the output voltage

FIGURE 7.11 Triangular carrier wave AC modulation.

FIGURE 7.12 Illustration of the generation of sinusoidal PWM (SPWM) signals.

1 Mohan, N., Undeland, T., and Robbins, W., Power Electronics: Converters, Applications, and Design,

2nd ed., John Wiley & Sons, New York, 1995

2 Von Jouanne, A., Rendusara, D., Enjeti, P., and Gray, W., Filtering techniques to minimize the effect

of long motor leads on PWM inverter fed AC motor drive systems, IEEE Trans Ind Appl., July/Aug.,919–926, 1996

7.4 Third Harmonic Injection for Voltage Boost of SPWM Signals

Michael Giesselmann

It can be shown (Mohan et al.[1], p 105) that if a three-phase input voltage is rectified using a standardthree-phase rectifier, the resulting DC voltage is equal to 1.35 times the rms value of the AC line–lineinput voltage If this DC voltage is used to feed a three-phase inverter using the SPWM modulationtechnique described above, the theoretical maximum AC line–line output voltage is only 82.7% of the

AC line–line input voltage feeding the rectifier (Mohan et al.[1], p 228) To boost the output voltagewithout resorting to overmodulation, the third harmonic of the fundamental frequency can be added tothe modulation signal Figure 7.15 shows an example, where a third harmonic with an amplitude of21.1% has been added to the fundamental modulation signal

FIGURE 7.13 20-kHz carrier modulated with 500 Hz.

FIGURE 7.14 Spectrum of the SPWM signal shown in Fig 7.13

1 0.5

SPWM(t)

Sin(t)

20-kHz carrier modulated with 500 Hz

The amplitude of the fundamental has been increased to 112% in this example It can be seen, thatthe peak amplitude of the resulting signal does not exceed the amplitude of the pure sinusoid with100% amplitude By inspection of Fig 7.15 it is easy to see that the voltage–time integral will be higher

if a 3rd harmonic is added to the reference signal for the phase to neutral voltage This voltage boostbeyond the previously mentioned value of 82.7% is very desirable, to retrofit induction motors withadjustable speed drives in existing installations The 3rd harmonic components exactly cancel each other

in the line-to-line voltages of the inverter This is because the phase shift of the fundamental signals is

120° and therefore the phase shift of the 3rd harmonic is 3 × 120 = 360° Therefore, the 3rd harmonicvoltages precisely cancel and result in a pure sinusoidal output voltage being applied to the motor This

is shown in Fig 7.16, which illustrates the voltage boost that is obtained

FIGURE 7.15 Sinusoidal modulation signal with and without added 3rd harmonic.

FIGURE 7.16 Line-to-line signal showing the voltage boost obtained by 3rd harmonic injection.

1 Mohan, N., Undeland, T., and Robbins, W., Power Electronics: Converters, Applications, and Design,

2nd ed., John Wiley & Sons, New York, 1995

7.5 Generation of PWM Signals Using Microcontrollers

and DSPs

Michael Giesselmann

Modern power electronics controllers are rapidly moving toward digital implementation Typical tions consist of microcontrollers or DSPs In addition, coprocessors, such as the ADMC200/201 fromAnalog Devices, are available that are specifically designed to support inverter control Most of theprocessors, such as the 68HC12B32 from Motorola, that are commonly used to control power electronicshave built-in hardware support for PWM generation Figure 7.17 shows the basic principle of their digitalPWM generation

solu-For clarity, the circuit shown in Fig 7.17 has only 4-bit resolution for the duty cycle of the generatedPWM signals, resulting in only 16 discrete duty cycles In actual applications, 8 to 12 bits of resolution

is typical In Fig 7.17, a digital counter (74163) counts from zero to its maximum value and repeats thecycle afterward The count is continuously compared with a digital value representing the duty cycleusing a hardware comparator (7485) The PWM signal is available on the output of the comparator

Figure 7.18 shows the simulation results from the example circuit shown in Fig 7.17 The duty cycle inthis example is 3/16

If more than one channel is present, the PWM signals can be left, right, or center aligned To be centeraligned, up–down counters are used, which count up to their maximum count and then back to zerobefore starting the next cycle The maximum count (2bits − 1) is determined by the number of stages(bits) the digital counter has In a digital PWM modulator each counter has an associated period register.The content of this register determines the maximum count at which the counter resets If this number

is less than the maximum count (2bits − 1), the repetition (switching) frequency is increased and the

FIGURE 7.17 Principle of digital PWM signal generation.

FIGURE 7.18 Simulation results from the circuit shown in Fig 7.17

resolution of the duty cycle is decreased for a given clock speed It is often important to make the correcttrade-off between the switching frequency and the resolution.

The advantage of hardware support for PWM generation is that the processor typically only needs toaccess any registers if the duty cycle is to be changed, since the period is typically only initialized onceupon program start-up It should also be mentioned that the duty cycle registers are typically “double-buffered,” meaning that an update of a duty cycle does not need to be synchronized with the currentstate of the counter In double-buffered systems, the new duty cycle will only be chosen once the previousperiod is completed to avoid truncated PWM signals If necessary, a software override can disable thisfeature

7.6 Voltage Source–Based Current Regulation

Michael Giesselmann

In motor drive applications, it is often desired to control directly the input current of the motor to controlthe torque DC control also limits dynamics resulting from the electrical characteristics of the machine.Controlling the torque provides direct control over the angular acceleration, which is essential for precisemotion control Current control is typically performed in the innermost loop of a cascaded feedbackcontrol loop arrangement [1] However, most power electronics converters are circuits with controllablevoltage output To achieve current control, the voltage of the power electronics converter can be controlled

in such a way, that the desired current is obtained Several methods can be used to achieve this:

• A feedback control loop, typically using a PI controller can be used control the current

• The necessary voltage can be calculated in real time and applied to the motor

• The necessary voltage for fast transients can be calculated in real time and applied to the motorand the residual error can be corrected by a PI controller

Examples illustrating each of the schemes are described in the following Figure 7.19 shows an example

of a DC motor in which the current is controlled by adjusting the applied voltage using a PI controllersuch that the current follows the desired trajectory The result is presented in Fig 7.20, which shows thatthe current indeed follows the desired value at all times

Sometimes even better results and higher loop bandwidth can be obtained if known information aboutthe motor and the load is used to calculate the required voltage in real time Figure 7.21 shows somefundamental equations of a permanent magnet (PM) DC motor Here a capacitor is used to representthe kinetic energy stored in the machine Therefore, the second voltage loop equation in Fig 7.21

represents the voltage across the motor at all times To test this theory, the “compensator” in the circuitshown in Fig 7.22 calculates this voltage and applies the result to the DC motor The subcircuit of the

FIGURE 7.19 Voltage source–based current control using a PI feedback loop.

compensator is shown in Fig 7.23 The result is identical to that shown in Fig 7.20 However, for thecorrect implementation of this scheme, the load and the inertia of the system needs to be known precisely,which is not realistic Therefore, the best strategy is to implement the first two terms on the right side

of the second voltage loop equation in Fig 7.21 using a compensator and to use a PI controller to eliminatethe residual error

The advantage of the mixed (compensator and PI residual controller) approach is that the fast dynamicsare covered by the feedforward path through the compensator, whereas the effect of the slower integralterm is taken care of by the PI controller The compensator will immediately apply the correct voltage toovercome the ohmic resistance of the winding and to establish the correct current slope in the rotor induc-tance As the machine accelerates, the PI controller adds the appropriate voltage to offset the back-emf

FIGURE 7.20 Simulation result for the circuit shown in Fig 7.19

FIGURE 7.21 Fundamental equations for the PM DC machine.

FIGURE 7.22 Voltage source–based current control using a feedforward approach.

Voltage loop equation

Equivalent Capacitance

Voltage loop equation

Vs = Rrot⋅Irot + L rot⋅ − Irotd + K m⋅ω

of the motor An example for this approach is shown in Fig 7.24 Again, the results are identical to theones shown in Fig 7.20.

Reference

1 Mohan, N., Electric Drives, An Integrative Approach, MNPERE, Minneapolis, MN, 2001

7.7 Hysteresis Feedback Control

Hossein Salehfar

This section presents a hysteresis feedback control technique for DC-DC buck converters operating inboth continuous and discontinuous conduction modes A dead band with a high boundary above thevoltage reference and a low boundary below the voltage reference are used to avoid chattering in theswitch The output voltage is regulated by comparing it to a reference voltage and the difference betweenthe two voltages (error) is used to turn ON or OFF the switch Regardless of where the voltage starts,switching takes place as soon as a boundary is encountered

Initially, the switch turns ON because the output voltage of the converter is below the turn-on boundary.The output voltage then rises at a rate limited only by the inductor, the capacitor, and the load Theswitch then turns OFF when the output voltage crosses the upper boundary and it remains OFF until

FIGURE 7.23 Subcircuit for the compensator shown in Fig 7.22

FIGURE 7.24 Voltage source–based current control using a feedforward approach for the fast transients and a PI controller for the residual error.

the output voltage falls below and crosses the lower boundary where the switch is turned ON again Oncethe voltage is between the upper and lower boundaries, switching actions will keep it in that vicinityunder all conditions This type of operation becomes independent of line, load, inductor, and capacitorvalues Hysteresis control in principle eliminates the output variations other than the ripples The systemwill stay close to the desired output voltage even if the input voltage, the load, or the component valueschange Hysteresis control also provides an immediate response to dynamic disturbances The controlcircuit is very simple and relatively straightforward to design and physically implement

Introduction

Because of their high efficiency, compact size, and low cost, switching power supplies continue to gainpopularity Switching power supplies could be as high as three times more efficient than linear powersupplies and in some cases eight times smaller in size The heart of a switching power supply is its switchcontrol circuit Generally, the control circuit is a negative-feedback control loop connected to the switchthrough a comparator and a pulse width modulator (PWM) This control circuit regulates the outputvoltage against changes in the load and the input voltage A feedforward loop may also be used tocompensate for changes in the input voltage Several topologies of switching power supplies have beendeveloped and used DC-DC buck converters are one of the widely used topologies

The PWM-based voltage and current mode feedback controllers used in buck converters are widelyused to improve line regulation [1] Apparently, however, PWM voltage mode controllers have disad-vantages Since the input voltage is a significant parameter in the loop gain, any changes in the inputvoltage will alter the gain and will change the dynamics of the system The central issue is that a voltagemode controller alone cannot correct any disturbances or changes until they are detected at the output

In the voltage-based controllers the compensation loop is difficult to implement A limitation of thecurrent mode controllers is the limit on the duty ratio If the duty ratio exceeds 50%, then instabilityoccurs [1] In the current mode controllers a sensing resistor is used in the current loop, which increasesthe power losses in the converter Most feedback controllers in buck converters use both the PWM voltageand current mode controllers to produce a better steady-state response and to reduce the voltage overshootsduring start-ups

Feedforward controllers may also be used to improve the line regulation in applications with a widerange of input voltages and loads [1, 2] An apparent disadvantage of these types of controllers, however,

is that the feedforward scheme with its direct sensing of the input quantities may have adverse effectswhen the converter is subjected to abrupt line transients The sensing of input voltage through thefeedforward loop may induce large-signal disturbances that could upset the normal duty cycle of thecontrol mode These concerns appear to be legitimate in light of the fact that such an effect is oftenobserved with other forms of feedforward controls as well [2]

Most buck converters are designed for a continuous-current mode operation [3] What if changes inthe load or input voltage cause the system to operate in a discontinuous-current mode? In these situations,

a voltage-based hysteresis feedback control circuit is a viable alternative [4–6] Hysteresis feedback trollers enable buck converters to operate in both continuous- and discontinuous-current modes Withthe ability of the system to operate in both modes, the size of the inductor and the capacitor are minimizedcompared with the other standard converters where a minimum size of the inductor is required to produce

con-a continuous-current mode [3] A minimum size of the ccon-apcon-acitor is con-also required to limit the outputvoltage ripples as well The output voltage of a hysteresis controller is stable and exhibits a robust behavior.The output is maintained even under extreme changes in the load, the line, or in the component values[4] In some cases, the hysteresis controller determines the output ripple The voltage-based hysteresisfeedback controller presented in this section combines the advantages of both the PWM voltage andcurrent-based controllers but it does not require a PWM circuit It needs only a comparator circuit,which make it simpler and cheaper to build Hysteresis controllers work well with DC-DC buck converters.But they do not work with other types of DC converters If they are used, for example, with a boostconverter, the results could be disastrous [4]

Principles of the Hysteresis Feedback Control Circuits

The main function of a DC-DC converter is to provide a good output voltage regulation The output

voltage must be maintained at the desired level against all changes in the input voltage or the load The

converter must act quickly to correct errors in the output voltage due to changes in the input voltage or

in the load

Figure 7.25 shows the basic circuit of the hysteresis controller for a buck converter The circuit consists

of a comparator and a switch (transistor) The comparator compares the output voltage Vout to a

reference voltage Vref If Vout>Vref, the switch is turned OFF If Vout < Vref, the switch is turned ON This

process repeats and Vout is maintained at a value close to Vref However, the circuit in Fig 7.25 leads to

chattering in the switch In an attempt to keep Vout equal to Vref, the switch chatters when it rapidly

turns ON and OFF as Vout moves back and forth across Vref In power converters, an excessively fast

switching action associated with chattering is destructive and it is therefore essential to avoid this

condition

Chattering is eliminated in hysteresis controllers by creating a dead band around Vref The dead band

is created by using an upper boundary above Vref and a lower boundary below Vref The space between

the two boundaries is the dead band Figure 7.26 shows the circuit of the hysteresis controller with a

dead band R1 and R2 resistors are added to the control circuit to provide the required dead band The

values of R1 and R2 determine the upper boundary and the lower boundary of the dead band

The comparator in Fig 7.26 is a Schmitt trigger The input voltage V+ of the positive terminal of the

op-amp is no longer a fixed reference voltage Vref as is the case in the basic comparator (without R1 and

R2) in Fig 7.25 V+ now depends on Vref, V o, R1, and R2 It switches from one boundary of the dead band

to another During the initial start-up of the buck converter, the input to the negative terminal (−) of

the op-amp (Vout) is a small positive value and is less than V+ The amplifier will be saturated so that

FIGURE 7.25 The basic circuit of a hysteresis controller for buck converters.

+ +

-V o =Vcc, thus the switch is turned ON V+ then becomes

(7.1)

As time progresses, Vout will increase gradually in the positive direction The output voltage, V o, remains

unchanged at Vcc until Vout is equal to V+ The op-amp will then enter its linear region and V o will decrease

and thus V+ will decrease as well This process will continue until Vo reaches zero and the op-amp will

again saturate The voltage V+ will no longer be given by Eq (7.1) but is given:

(7.2)Equations (7.1) and (7.2) represent the control boundaries of the control circuit Equation (7.1) defines

the upper boundary and Eq (7.2) gives the lower boundary of the dead band The dead band is computed

by subtracting Eq (7.2) from Eq (7.1) The dead band is given by Eq (7.3)

(7.3)

R1 and R2 may be chosen to give the required value of ∆D b/V o

FIGURE 7.26 The basic circuit of the hysteresis controller with a dead band.

+

-+ -