Appendix A
RMS Values of Commonly Observed
Converter Waveforms
The waveforms encountered in power electronics converters can be quite complex, containing modulation 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 discussed in Section 18.5.
A.1
SOME COMMON WAVEFORMS
DC, Fig. A.1:
(A.1)
rms = I
i(t)
I
t
Fig. A.1
1
2
RMS Values of Commonly Observed Converter Waveforms
DC plus linear ripple, Fig. A.2:
rms = I
1 + 1 ∆i
3 I
2
(A.2)
i(t)
∆i
I
0
Fig. A.2
Ts
t
Square wave, Fig. A.3:
(A.3)
rms = I pk
i(t)
Ipk
t
– Ipk
Fig. A.3
Sine wave, Fig. A.4:
rms =
i(t)
I pk
2
(A.4)
Ipk
0
Fig. A.4
0
DTs
Ts
t
Pulsating waveform, Fig. A.5:
rms = I pk D
(A.5)
A.1
i(t)
Some Common Waveforms
3
Ipk
0
0
Fig. A.5
Ts
DTs
t
Pulsating waveform with linear ripple, Fig. A.6:
rms = I D
i(t)
I
1 + 1 ∆i
3 I
2
(A.6)
∆i
∆i
0
0
Fig. A.6
Ts
DTs
t
Triangular waveform, Fig. A.7:
rms = I pk
D1 + D2
3
(A.7)
i(t)
Ipk
0
Fig. A.7
0
D1Ts
D2Ts
Ts
t
Triangular waveform, Fig. A.8:
rms = I pk
D1
3
(A.8)
4
RMS Values of Commonly Observed Converter Waveforms
i(t)
Ipk
0
0
Fig. A.8
D1Ts
Ts
t
Triangular waveform, no dc component, Fig. A.9:
rms = ∆i
3
(A.9)
i(t)
∆i
0
t
Fig. A.9
Center-tapped bridge winding waveform, Fig. A.10:
rms = 1 I pk 1 + D
2
i(t)
Ipk
1
2
Fig. A.10
(A.10)
0
1
2
I pk
DTs Ts
I pk
0
(1+D)Ts 2Ts
t
General stepped waveform, Fig. A.11:
rms =
i(t)
(A.11)
D1 I 21 +D2 I 22 +
I2
I1
D1Ts
Fig. A.11
0
D2Ts
Ts
t
Trapezoidal
segment
Constant
segment
i(t)
General Piecewise Waveform
Triangular
segment
A.2
D1Ts
D2Ts D3Ts
5
etc.
Fig. A.12 General piecewise waveform.
A.2
Ts
t
GENERAL PIECEWISE WAVEFORM
For a periodic waveform composed of n piecewise segments as in Fig. A.12, the rms value is
Σ Dk u k
k=1
n
rms =
(A.12)
where Dk is the duty cycle of segment k, and uk is the contribution of segment k. The uks depend on the
shape of the segments—several common segment shapes are listed below:
Constant segment, Fig. A.13:
(A.13)
u k = I 21
i(t)
I1
Fig. A.13
t
Triangular segment, Fig. A.14:
u k = 1 I 21
3
(A.14)
i(t)
I1
0
Fig. A.14
t
6
RMS Values of Commonly Observed Converter Waveforms
Trapezoidal segment, Fig. A.15:
u k = 1 I 21 + I 1 I 2 + I 22
3
(A.15)
i(t)
I1
I2
Fig. A.15
t
Sinusoidal segment, half or full period, Fig. A.16:
u k = 1 I 2pk
2
(A.16)
i(t)
Ipk
t
Fig. A.16
Sinusoidal segment, partial period: as in Fig. A.17, a sinusoidal segment of less than one half-period,
which begins at angle θ1 and ends at angle θ2 . The angles θ1 and θ2 are expressed in radians:
sin θ 2 – θ 1 cos θ 2 + θ 1
u k = 1 I 2pk 1 –
2
θ2 – θ1
i(t)
Ipk
θ1
Fig. A.17
ωt
θ2
(A.17)
A.2
i(t)
General Piecewise Waveform
7
I1 = 20 A
1
2
3
4
5
I2 = 2 A
0.2 µs
0.2 µs 0.1 µs
5 µs
6
0A
0.2 µs
Ts t
10 µs
Fig. A.18 Example: an approximate transistor current waveform, including estimated current spike due to diode
stored charge.
Example
A transistor current waveform contains a current spike due to the stored charge of a freewheeling diode. The observed waveform can be approximated as shown in Fig. A1.18. Estimate the rms current.
The waveform can be divided into six approximately linear segments, as shown. The Dk and uk
for each segment are
1. Triangular segment:
D1 = (0.2 µs)/(10 µs) = 0.02
u1 = I 1 ⁄ 3 = (20 A)2/3 = 133 A2
2
2. Constant segment:
D2 = (0.2 µs)/(10 µs) = 0.02
2
u2 = I 1 = (20 A)2 = 400 A2
3. Trapezoidal segment:
D3 = (0.1 µs)/(10 µs) = 0.01
u3 = ( I 1 + I 2 + I 3 ) ⁄ 3 = 148 A2
2
2
2
4. Constant segment:
D4 = (5 µs)/(10 µs) = 0.5
2
u4 = I 2 = (2 A)2 = 4 A2
5. Triangular segment:
D5 = (0.2 µs)/(10 µs) = 0.02
u5 = I 2 ⁄ 3 = (2 A)2/3 = 1.3 A2
2
6. Zero segment:
u6 = 0
8
RMS Values of Commonly Observed Converter Waveforms
The rms value is
Σ Dk u k
k=1
6
rms =
= 3.76 A
(A.18)
Even though its duration is very short, the current spike has a significant impact on the rms value of the
current—without the current spike, the rms current is approximately 2.0 A.