# The Three Dimensional Surface on A Two Dimensional Plane Question

Homework-111.What are response contours? Illustrate for a k-factor problem with k=2,3. (5075 words)

2. A response that is a smooth function over the levels of two factors may or

may not be similar when viewed over different levels of a third factor. Can you

explain this statement? Provide example.

3.Suppose a second-order model in three factors had been fitted. Using

canonical analysis, how could you detect and analyze a sloping ridge system?

Describe and illustrate two different kinds of limiting models for a sloping ridge

and their corresponding canonical form?

4.

1. In the MMC of Figure 1,

1.1) If the dc-side voltage is 500 kV, using IGBTs with a voltage rating of 6.5 kV, explain the way you

size the number of submodules N in each arm and write it down with enough supporting

justification. Determine the number of submodules.

1.2) Repeat part 1.2) if the submodules are replaced with full-bridge submodules.

1.3) Write the mathematical equations to prove that if a sine-triangular PWM with 3rd order

harmonic injection employed, 4th order harmonic will appear in the circulating current.

Figure 1: The circuit diagram of the MMC.

2- The components specify for a resonant buck converter, as shown in the following figure are:

πΏπ = 4ππ», πΆπ = 47ππΉ and πΏ = 1ππ», and πΆ = 150ππΉ. The input voltage is 48V and output voltage is

12V. The load resistance is 2β¦. Answer the following questions when the converter is operating in

steady state.

2.1) Find the switching frequency

2.2) Find the duration of resonant mode

2.3) Find the peak voltage across the resonant capacitor.

2.4) Determine the minimum duty cycle of the gate signal, in which the ZVS is guaranteed.

2.5) Determine the rating values (current and voltage) of the power MOSFET.

2.6) Assuming a rise time and fall time of π‘π = π‘π = 100 ππ , calculate the MOSFET turn-on and turnoff

losses.

2.7) What happens if the load varies in the range of 1β¦ to 10 β¦.

3. In the power circuit of the following figure, IGBTs can be turned off either with a zero voltage applied

to the gate-emitter (unipolar turn off) or with a negative voltage (bipolar turn-off). In high-power

applications, the bipolar turn off is the common practice. Using the parasitic inductances πΏπ1 and πΏπ2

shown in the figure, explain the risk of using unipolar turn off and justify the reason behind using the

bipolar turn off.

ECE6331-A: Power

Electronic Circuits

Losses in Hard Switching Converters

2

2

Soft Switching

Semiconductor devices are switched on or off at the zero

crossing of their voltage or current waveforms:

β’Zero-current switching (ZCS): Switching transition occurs

at zero current.

β’Zero-voltage switching (ZVS): Switching transition occurs

at zero voltage.

3

3

ZVS and ZCS

Illustration of ZVS and ZCS for a power switch: (a) ZVS turn-on and turn-off

transitions; (b) ZCS turn-on and turn-off transitions.

4

4

Trajectories of current and voltage

5

5

Undamped Series-Resonant Circuit

6

6

Series-Resonant Circuit with CapacitorParallel Load

7

7

Resonant Switches

Configurations of the zero-current switch

Configurations of the zero-voltage switch

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8

Quasi-resonant Converters

In a conventional PWM converter, replace

the PWM switch network with a switch

network containing resonant elements.

9

9

Quasi-Resonant Switch Cells

10

10

The ZCS Quasi-Resonant Switch Cell

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11

Waveforms of the Half-wave ZCS QuasiResonant Switch Cell

Waveforms:

Waveforms:

Each switching period contains four

subintervals

12

12

Subinterval 1

β’

13

13

Subinterval 2

β’

14

14

Subinterval 2

15

15

Boundary of ZCS

β’

16

16

Subinterval 3

β’ All semiconductor devices are off and the equivalent

circuit is:

17

17

Subinterval 4

β’

18

18

Maximum Switching Frequency

β’

19

The length of the fourth subinterval cannot be negative, and the switching period

must be at least long enough for the tank current and voltage to return to zero by

the end of the switching period. The angular length of the switching period is

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The Average Output Voltage

The average switch input current is given by:

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20

The Average Output Voltage (Conβed)

β’ During subinterval 2, we have:

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21

The Average Output Voltage (Conβed)

β’ First term: integral of the capacitor current over

subinterval 2. This can be related to the change in

capacitor voltage:

Substitute results for the two integrals:

Substitute into expression for average

switch input current:

22

22

β’

23

23

The Average Output Voltage (Conβed)

This is of the form:

24

24

ZCS Boundary

ZCS boundary

1

0.9

0.8

0.8

0.7

0.6

0.6

Js

0.5

0.4

0.4

0.3

0.2

0.2

F = 0.1

0

0

0.2

0.4

0.6

0.8

1

ο

25

25

Properties of ZCS Resonant DC-DC

Converter

β’ Peak current of switch

– High when compared to the output current

– Conduction losses are higher than the hardswitching

26

26

ZCS Switch Cell

27

27

Analysis of Full-Wave ZCS

β’ Analysis in the full-wave case is nearly the same as in the half-wave

case. The second subinterval ends at the second zero crossing of the

tank inductor current waveform. The following quantities differ:

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28

Analysis of Full-Wave ZCS

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29

Resonant Switches for ZVS

Configurations of the zero-voltage switch

30

30

The ZVS Quasi-resonant Switch Cell

β’ A half-wave version based on the PWM buck

converter

31

31

Half-Wave ZVS Resonant-Switch

Converter

32

32

Full-wave ZVS

When switch T is implemented by a MOSFET and an anti-parallel diode, D, as shown

in Fig. (b), the voltage across capacitor Cr is clamped by D to positive values, and the

resonant switch is operating in a half-wave mode.

On the other hand, when switch T is implemented by a MOSFET in series with D, as

shown in Fig. (C), and the voltage across Cr can oscillate freely, then the resonant

switch is operating in a full-wave mode.

Notice that in a current-mode resonant switch, the resonant interaction between Lr

and Cr occurs during the major portion of the on-time; while in a

voltage-mode resonant switch, the resonant interaction occurs during the major

portion of the off-time.

33

33

Full-Wave ZVS Resonant-Switch

Converter

In the full-wave version, the voltage across Cr can

oscillate freely and become negative too as shown

below. ZVS can be implemented in the second zero

voltage crossing point of VCr.

34

34

ZVS Resonant-Switch Converter

35

35

Comparison of ZCS and ZVS

β’

36

36

Resonant Pole Converter

37

37

Resonant Pole Converter

β’ Switch turn on and off with zero voltage

β Maximum voltage is clamped to input voltage

β’ Lf is small when compared to hard switching

β Its current is both positive and negative

β’ T+ conduct current and it is turned off

β Voltage over it is zero because of C+

38

38

Resonant Pole Converter

39

39

Operation (1/2)

40

40

Operation (2/2)

41

41

Control of output voltage

42

42

Modular Multilevel Converters

Half-Bridge and Full-Bridge Modules

2

Cascaded Full-Bridge Modules

β Provides scalability and

modularity

β More modules:

β’ More voltage levels

β’ Higher voltages

β’ More redundancy

3

Cascaded Half-Bridge Modules

β Provides scalability and

modularity

β More modules:

β’ More voltage levels

β’ Higher voltages

β’ More redundancy

4

Cascaded Half-Bridge Modules

π£ = ππ·πΆ + πΰ·‘π cos ππ‘

π = πΌπ·πΆ + πΌΰ·‘π cos ππ‘ β π

2ππ·πΆ πΌπ·πΆ + πΰ·‘π πΌΰ·‘π cosπ = 0

5

Limits of Cascaded Modules

Assuming the same amount for the total capacitor voltages Vοcap, the

operating region with cascaded full-bridge modules is wider.

Half-bridge cascaded modules

Full-bridge cascaded modules

6

The Modular Multilevel Converter (MMC)

The MMC is built based upon cascaded

half-bridge submodules (SMs).

7

Basics of Operation of the MMC

ππ·πΆ

π£

2π

β’ On the AC side, sinusoidal

voltages are provided.

β’ Cascaded SMs

produce AC and DC

voltage components.

β

ππ·πΆ

2

ππ·πΆ

2

ππ·πΆ

2

0

ππ·πΆ

ππ·πΆ

2

ππ·πΆ

ππ·πΆ

2

ππ·πΆ

2

0

ππ·πΆ

2

8

Steady-State Currents

9

Steady-State Operation of the MMC

π£π’π

ππ·πΆ

=

β πΰ·‘π sin ππ‘

2

π£π = πΰ·‘π sin ππ‘

π£πππ€

ππ·πΆ

=

+ πΰ·‘π sin ππ‘

2

Power balance for three-phase converter:

ππ’π =

πΌπ·πΆ 1

+ πΌΰ·‘π sin ππ‘ β π

3

2

ππ = πΌΰ·‘π sin ππ‘ β π

ππππ€ =

πΌπ·πΆ 1

β πΌΰ·‘π sin ππ‘ β π

3

2

3

πΰ·‘ πΌΰ·‘ cos π = ππ·πΆ πΌπ·πΆ

2 π π

SM Capacitor Voltage Balancing

Similar to any other

multilevel converter

topology, the MMC

needs an active

voltage balancing

strategy to balance and

maintain the SM

capacitor voltages at

VDC/N

11

SM Capacitor Voltage Balancing

12

SM Capacitor Voltage Balancing for a 5Level MMC

N=4

13

Circulating Current

ππ’π,π =

ππ

2

+

ππ·πΆ

3

ππ

ππππ€,π = β +

2

+ππ§,π

ππ·πΆ

3

ππ§,π =

+ππ§,π

14

ππ’π,π + ππππ€,π ππ·πΆ

β

2

3

Circulating Currents

β’

Circulating currents contain negative sequence components with the

frequencies twice the fundamental one (predominantly a 2ndβorder

harmonic component)

β’

If not properly controlled/suppressed, they:

β

β’

increase the arm rms and peak currents, which consequently increase the converter

power losses, rating values of the semiconductor devices and the ripple magnitude

of the SM capacitor voltages.

Increasing the arm inductor would reduce the magnitude of

circulating currents. However, this would adversely impact the

magnitude of the maximum attainable voltage on the AC-side of the

converter

Dynamics of Circulating Currents

ππ’π,π + ππππ€,π ππ·πΆ

ππ§,π =

β

2

3

πππ’π,π

ππ·πΆ

β π£π’π,π = πΏπ

+ π
π ππ’π,π + π£π

2

ππ‘

πππππ€,π

ππ·πΆ

β π£πππ€,π = πΏπ

+ π
π ππππ€,π + π£π

2

ππ‘

β’

Circulating current dynamics:

πΏπ

Lo

πππ§,π

ππ·πΆ π£π’π,π + π£πππ€,π

ππ·πΆ

+ π
π ππ§,π =

β

β π
π

ππ‘

2

2

3

diz , j

dt

+ Roiz , j = vz , j

Circulating Current Control: PR

Controller

β’

Circulating Current Dynamics:

πΏπ

Lo

β’

πππ§,π

ππ·πΆ π£π’π,π + π£πππ€,π

ππ·πΆ

+ π
π ππ§,π =

β

β π
π

ππ‘

2

2

3

diz , j

dt

+ Roiz , j = vz , j

PR Controller:

K p1 +

K i1s

K s

+ 2 i2 2

2

s + ο·n1 s + ο·n 2

2

Where Οn1 and Οn2 are tuned to 2nd and 4thorder harmonics.

Circulating Current Control: PI

Controllers

ππ§,π = πΌ2π sin 2π0 π‘ + π0

2π

+ π0

3

2π

= πΌ2π sin 2π0 π‘ + π0 +

3

ππ§,π = πΌ2π sin 2 π0 π‘ β

2π

+ π0

3

2π

= πΌ2π sin 2π0 π‘ + π0 β

3

ππ§,π = πΌ2π sin 2 π0 π‘ +

πΌ2π is the peak value of the double line-frequency

circulating current.

π0 is the fundamental frequency and π0 is its initial

phase angle.

Circulating Current Control: PI

Controllers

ππππ/ππ =

2

3

πππ π

βπ πππ

where π = β2π0 π‘

2π

3

2π

βπ ππ π β

3

πππ π β

2π

3

2π

βπ ππ π +

3

πππ π +

Circulating Current Control: PI

Controllers

By rewriting

Lo

diz , j

dt

+ Roiz , j = vz , j in abc form

π

ππ§,π

π£π§,π

π π§,π

π£π§,π = πΏ0

ππ§,π + π
0 ππ§,π

ππ‘

π£π§,π

ππ§,π

ππ§,π

Substituting and multiplying the transformation matrix ππππ/ππ yields

π’π§,π

π π2π,π

0

=

πΏ

+

0

π’π§,π

2π0 πΏ0

ππ‘ π2π,π

π2π,π

π2π,π

β2π0 πΏ0

β

+ π
0

π2π,π

π2π,π

0

Circulating Current Control: PI

Controllers

π2π,π = 1.5 πΌ2π sin π0

π2π,π = 1.5 πΌ2π cos π0

Pulse Width Modulation

22

Direct Modulation

πππ

ππ’π,π

ππ·πΆ

πππ

β π£π

=π 2

ππ·πΆ

πππ

ππππ€,π

ππ·πΆ

πππ

+ π£π

=π 2

ππ·πΆ

Indirect Modulation

πππ

ππ’π,π

ππ·πΆ

β

πππ

π§

β π£π β π£πππ,π β π£πππ,π

=π 2

βπ

π=0 π£π,π’π,π,π

πππ

ππ·πΆ

β

πππ

π§

+ π£π β π£πππ,π β π£πππ,π

2

=π

βπ

π=0 π£π,πππ€,π,π

ππππ€,π

Control of a Grid-Connected MMC

25

Closed-Loop Control of the MMC

Closed-Loop Control of the MMC

HVDC Transmission System

P =

VDC. IDC

MMC-Based HVDC Transmission

Half-Bridge SM

Full-Bridge SM

The clamp-double SM

29

DC-Side Fault

Half-Bridge SM

Full-Bridge SM

The clamp-double SM

30

Alternative SM Circuit: Full-Bridge SM

Full-Bridge SM

Half-Bridge SM

31

Full-Bridge SM

Full-Bridge MMC During DC-Side Fault

Current path in a Full-Bridge SM during a DCside short circuit fault

32

Alternative SM Circuit: Clamp-Double SM

Clamp-Double

Half-Bridge

SM

33

SM

Clamp-Double MMC During DC-Side Fault

Half-Bridge

SM path

Current

in aFull-Bridge

Clamp-Double

SM

SM

during a DC-side short circuit fault

34