powerelectronics_AppendixA

8/9/2019 powerelectronics_AppendixA

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Appendix A

RMS Values of Commonly ObservedConverter Waveforms

The waveforms encountered in power electronics converters can be quite complex, containing modula-

tion at the switching frequency and often also at the ac line frequency. During converter design, it is often

necessary to compute the rms values of such waveforms. In this appendix, several useful formulas and

tables are developed which allow these rms values to be quickly determined.

RMS values of the doubly-modulated waveforms encountered in PWM rectifier circuits are dis-cussed in Section 18.5.

A.1 SOME COMMON WAVEFORMS

DC, Fig. A.1:

(A.1)rms= I

i(t)

t

I

Fig. A.1


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RMS Values of Commonly Observed Converter Waveforms

DC plus linear ripple, Fig. A.2:

(A.2)

Square wave, Fig. A.3:

(A.3)

Sine wave, Fig. A.4:

(A.4)

Pulsating waveform, Fig. A.5:

(A.5)

rms= I 1 + 13

iI

2

i(t)

t

I

Ts0

i

Fig. A.2

rms= Ipk

i(t)

t

Ipk

IpkFig. A.3

rms=Ipk

2

i(t)

t

Ipk

TsDTs0

0

Fig. A.4

rms= Ipk D


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4


Triangular waveform, no dc component, Fig. A.9:

(A.9)

Center-tapped bridge winding waveform, Fig. A.10:

(A.10)

General stepped waveform, Fig. A.11:

(A.11)

i(t)

t

Ipk

TsD1Ts0

0

Fig. A.8

rms= i3

i(t)

t0

i

Fig. A.9

rms= 12Ipk 1 +D

i(t)

t

Ipk

TsDTs0

0

(1+D)Ts 2Ts

1

2Ipk

1

2Ipk

Fig. A.10

rms= D1I12

+D2I22

+

i(t)

t

I2

Ts

D1Ts

0

I1

D2Ts

Fig. A.11


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A.2 General Piecewise Waveform

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A.2 GENERAL PIECEWISE WAVEFORM

For a periodic waveform composed of n

piecewise segments as in Fig. A.12, the rms value is

(A.12)

whereD

k

is the duty cycle of segment k

, and u

k

is the contribution of segment k

. The u

k

s depend on the

shape of the segmentsseveral common segment shapes are listed below:

Constant segment, Fig. A.13:

(A.13)

Triangular segment, Fig. A.14:

(A.14)

i(t)

tTsD1Ts D2Ts D3Ts

Trian

gular

segm

ent

Constant

seg

ment

Trapezoidal

segm

ent

etc.

Fig. A.12 General piecewise waveform.

rms= Dkukk= 1n

uk= I12

i(t)

t

I1

Fig. A.13

uk= 1

3 I

1

2

i(t)

t

I1

0

Fig. A.14


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Trapezoidal segment, Fig. A.15:

(A.15)

Sinusoidal segment, half or full period, Fig. A.16:

(A.16)

Sinusoidal segment, partial period: as in Fig. A.17, a sinusoidal segment of less than one half-period,

which begins at angle

1

and ends at angle

2

. The angles

1

and

2

are expressed in radians:

(A.17)

uk= 13 I12

+I1I2 + I22

i(t)

t

I1 I2

Fig. A.15

uk= 1

2Ipk

2

i(t)

t

Ipk

Fig. A.16

uk= 1

2Ipk

21

sin 2 1 cos 2 + 1

2 1

i(t)

t

Ipk

12Fig. A.17


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A.2 General Piecewise Waveform

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Example

A transistor current waveform contains a current spike due to the stored charge of a freewheel-

ing diode. The observed waveform can be approximated as shown in Fig. A1.18. Estimate the rms cur-

rent.

The waveform can be divided into six approximately linear segments, as shown. TheD

k

and u

k

for each segment are

1. Triangular segment:

D

1

= (0.2 s)/(10 s) = 0.02

u

1

= = (20 A)

2

/3 = 133 A

2

2. Constant segment:

D2 = (0.2 s)/(10 s) = 0.02

u2= = (20 A)2

= 400 A2

3. Trapezoidal segment:

D3= (0.1 s)/(10 s) = 0.01

u3= = 148 A2

4. Constant segment:

D4= (5 s)/(10 s) = 0.5

u4= = (2 A)2= 4 A2

5. Triangular segment:

D5= (0.2 s)/(10 s) = 0.02

u5= = (2 A)2/3 = 1.3 A2

6. Zero segment:

u6= 0

i(t)

tTs0.2 s

I1= 20 A

0.2 s

I2= 2 A

0.1 s

10 s

0.2 s5 s

0 A

1 2 3 4 5 6

Fig. A.18 Example: an approximate transistor current waveform, including estimated current spike due to diode

stored charge.

I12

3

I12

I12

I22

I32

+ +( ) 3

I22

I22

3


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8 RMS Values of Commonly Observed Converter Waveforms

The rms value is

(A.18)

Even though its duration is very short, the current spike has a significant impact on the rms value of the

currentwithout the current spike, the rms current is approximately 2.0 A.

rms= Dkukk= 16

= 3.76 A

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