Dk4141 Appa

Appendix ARMS Values of Waveforms

A.1 Definitions

The RMS (root-mean-square) values are denoted with uppercase letters with-out index. We use currents in the examples given here.

Physical Meaning of the RMS Value

The RMS value (often called the effective value or DC-equivalent value) of acurrent is an equivalent of a DC current, which has the same heat dissipationas the real current on any resistor.

RMS Value in the Frequency Domain

When a given wave form (a current in the case) comprises components withdifferent frequency, i.e., different harmonics, then its RMS value is given bythe sum:

(A.1)

where Ik is the RMS value of the kth harmonic.This sum can also be separated in the frequency domain into two components:

• The DC component: IDC = I0

• The AC component: IAC =

The basic (the first) harmonic is I1. The RMS value of the higher harmonics is

(A.2)

I Ikk

== ∞

∞

∑ 2

0...

Ikk

2

1=

∞

∑

I Ih kk

==

∞

∑ 2

2

DK4141_AppA.fm Page 409 Wednesday, January 19, 2005 2:51 PM

Copyright 2005 by Taylor & Francis Group, LLC

410 Inductors and Transformers for Power Electronics

Using Equation (A.2) we can write

(A.3)

RMS Value in the Time Domain

The RMS value of a current, defined as i(t) in the general case, is

(A.4)

where the period of the repeating signal is T and t0 is an arbitrary time instant.

A.2 RMS Values of Some Basic Waveforms

A.2.1 Discontinuous Waveforms

The current flows during some time interval DT and it is zero for the restof the period T. For this case we can write

(A.5)

whereD is the duty ratioID is the RMS value corresponding to the nonzero part of the waveform in

the frames of one period (see Fig. A.1)

FIGURE A.1Waveforms: (a) D < 1; (b) D = 1.

I I I I I IDC AC DC h= + = + +2 2 212 2

IT

i t dtt

t T

= ⋅+

∫1 2

0

0

( )

I I DD=

i(t) i(t )

(a) (b)

0 DT T 2Tt t

0 T 2T 3T



RMS Values of Waveforms 411

A.2.2 Repeating Line Waveforms

Repeating line wave forms are shown in Fig. A.2. The start value of thecurrent is I1, the end value is I2, the period is T. For 0 < t < T the current is

(A.6)

After calculating the integral corresponding to RMS value we obtain thefollowing result:

(A.7)

We can also write this result as

(A.8)

Equation (A.8) describes the RMS value as a function of the average value(I1 + I2)/2 and the deviation (I1 − I2)/2. The current waveform is divided intoDC current component (I1 + I2)/2 and AC current component (I1 − I2)/2. ThenRMS value is calculated in a similar way to the equation (A.3).

A.2.3 Waveforms Consisting of Different Repeating Line Parts

The curve is composed from the line parts A, B, and C (see Fig. A.3), forwhich RMS values IA, IB, and IC are calculated by Equation (A.8). The RMSvalue of that waveform is

(A.9)

whereDA = TA/T, IA is the RMS value of the part with duration TA

DB = TB/T, IB is the RMS value of the part with duration TB

DC = TC/T, IC is the RMS value of the part with duration TC

FIGURE A.2Repeating line waveforms.

i(t)

0 T 2Tt

3T

I1

I2

i t It I I

T( )

( )= + −1

2 1

IT

i t dtI I I I

T

= =+ +∫1

32

0

12

22

1 2( )

II I I I

=+

+−

1 22

1 22

213 2

I I D I D I DA A B B C C= + +2 2 2




Note that the current may also be discontinuous.

A.3 RMS Values of Common Waveforms

A.3.1 Sawtooth Wave, Fig. A.4.

A.3.2 Clipped Sawtooth, Fig. A.5.

FIGURE A.3Different repeating line parts.

FIGURE A.4Sawtooth wave.

FIGURE A.5Clipped sawtooth.

TDA TDB TDC

T

i(t )

0 t

i(t)

t0 T 2T 3T

Ipk

II

rmspk=3

i(t)

0 DT T 2Tt

Ipk

I ID

rms pk=3




A.3.3 Triangular Waveform, No DC Component, Fig. A.6.

A.3.4 Triangular Waveform with DC Component, Fig. A.7.

A.3.5 Clipped Triangular Waveform, Fig. A.8.

FIGURE A.6Triangular waveform, no DC component.

FIGURE A.7Triangular waveform with DC component.

FIGURE A.8Clipped triangular waveform.

i(t)

0

Ipk

t

II

rmspk=3

i(t)

0t

IDC

∆IDC

I II

II

Irms DC

DC

DCDC

DC= +

= +113 3

2

22∆ ∆( )

i(t)

0 DT Tt

2T

Ipk

I ID

rms pk=3




A.3.6 Square Wave, Fig. A.9.

A.3.7 Rectangular Pulse Wave, Fig. A.10.

A.3.8 Sine Wave, Fig. A.11.

FIGURE A.9Square wave.

FIGURE A.10Rectangular pulse wave.

FIGURE A.11Sine wave.

i(t)

0 T 2Tt

Ipk

I Irms pk=

i(t)

0 DT 2TTt

Ipk

I I Drms pk=

i(t)

0 T 2Tt

Ipk

II

rmspk=2




A.3.9 Clipped Sinusoid, Full Wave, Fig. A.12.

A.3.10 Clipped Sinusoid, Half Wave, Fig. A.13.

A.3.11 Trapezoidal Pulse Wave, Fig. A.14.

FIGURE A.12Clipped sinusoid, full wave.

FIGURE A.13Clipped sinusoid, half wave.

FIGURE A.14Trapezoidal pulse wave.

i(t)

0 TDT 2Tt

Ipk

I I Drms pk=

i(t)

0 TDT 2Tt

Ipk

I ID

rms pk=2

i(t)

0 TDT 2Tt

∆IDCIDC

I I DI

Irms DCDC

DC

= +

1

13

2∆



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Dk4141 Appa