Appendix A
RMS Values of Commonly Observed Converter Waveforms
A.1
Results for some common waveforms
See text for results
A.2
General piecewise waveform
How to compute the rms value of a waveform that can be broken into
smaller piecewise segments
Example: transistor current waveform, including effects of short
turn-on current spike
Fundamentals of Power Electronics
1
Appendix A: RMS Values
General piecewise waveform
(rms value) =
Constant
segment
i(t)
Suppose the waveform can be
represented as a series of
segments, with the kth segment
having length DkTs and with Ts
equal to the switching period:
1
T
Triangular
segment
Basic expression for rms value of waveform
v(t) having period T:
Trapezoidal
segment
A.2
D1Ts
D2Ts D3Ts
T
v 2(t)dt
0
etc.
Ts
t
Then the rms value can be
expressed as:
n
rms =
Σ Dk u k
k=1
Fundamentals of Power Electronics
where uk is the contribution of the kth
segment — see following slides
2
Appendix A: RMS Values
Some basic segment shapes
i(t)
Sinusoidal segment
(half or full period)
i(t)
Constant segment
I1
Ipk
u k = I 21
u k = 1 I 2pk
2
t
t
i(t)
Triangular segment
I1
u k = 1 I 21
3
Sinusoidal segment
(partial period)
i(t)
Ipk
0
t
i(t)
I1
Trapezoidal segment
I2
ωt
θ1
θ2
u k = 1 I 21 + I 1 I 2 + I 22
3
sin θ 2 – θ 1 cos θ 2 + θ 1
u k = 1 I 2pk 1 –
2
θ2 – θ1
t
Fundamentals of Power Electronics
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Appendix A: RMS Values
Example
Transistor current waveform, including turn-on current spike induced by
diode reverse recovery
The turn-on current spike is short but of high magnitude. Does it
significantly increase the rms current?
The observed current waveform is approximated by piecewise linear
segments as shown below
Six segments:
i(t)
1-3 are from diode
reverse recovery
I1 = 20 A
1
2
3
4
5
6
4 is transistor on time
I2 = 2 A
0.2 µs
0.2 µs 0.1 µs
5 µs
0A
Ts t
0.2 µs
10 µs
Fundamentals of Power Electronics
4
5 is transistor turn-off
transition
6 is transistor off time
Appendix A: RMS Values
Calculation
i(t)
I1 = 20 A
1
2
3
4
5
6
I2 = 2 A
0.2 µs
0.2 µs 0.1 µs
5 µs
0A
0.2 µs
Ts t
10 µs
Result:
6
rms =
Σ Dk u k
k=1
= 3.76 A
Without the current spike, the rms
value is approximately 1.4 A. So in
this example, the diode reverse
recovery significantly increases the
transistor rms current.
Fundamentals of Power Electronics
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Appendix A: RMS Values