Nov. 22, 2009
ENEL585 - Short Review of Nearly Everything
ENEL 585 – A Short Review of Nearly Everything
Page 1 of 2
Th5: If average inductor current is constant, then the
inductor voltage displays Volt·second balance, or, as
an equation:
1. Principle
P1: The key principle of ENEL 585, Introduction to
Power Electronics, is that when you are using an
equation, or making an approximation, ask yourself:
Under what conditions is this equation valid?
2. Theory
The Basics
Th1: Average value means dc value.
Th2: RMS value means the effective value, or the
equivalent value, as if the waveform were dc in
nature. Why? Because instantaneous power for a
2
2
pure resistor is given by v /R or i R, so, given a
2
2
periodic waveform, if we average out v or i over
one period, we get the average power delivered by
that waveform. As an aside, note that in wind
turbines, power is proportional to the cube of wind
speed, therefore the effective wind speed over, say
one year, is given by the root mean cube value.
Th3: Orthogonal means, neither aligned, nor
opposite, but sideways. Consider two pure ac
voltages stacked up in series, they are aligned (if the
negative terminal of the top voltage is connected to
the positive terminal of the bottom voltage) and so
the total voltage is simply the addition of the two
voltages (this is true for any instant of time, or for
rms values). On the other hand, if those two ac
o
voltages are out of phase by 90 , then they are
orthogonal and the total rms value is given by the
square-root of the sum of the squared rms values.
There are many ways two periodic waveforms can
be orthogonal; the test is that the integral of their
product, over one period, is zero.
Key Equation of ENEL585:
v dt = L di → v dθ = ω L di (periodic case)
Note: for Theory Concepts Th4, Th5, Th6, v means
voltage across an inductor and i is the current
through that inductor, while for Th7, v is the voltage
across an ideal capacitor and i is the current through
that capacitor.
Th4: If average inductor current is constant, then
average inductor voltage is zero, or, as an equation:
<i> = constant → ∫period di = 0 → ∫period v dt = 0 = <v>
Theory Concept 4 is very useful when doing dc
circuit analysis. It means let that inductor voltage be
zero and results in much easier math.
<i>=const → ∫period di=0 → ∫ vpositive dt + ∫ vnegative dt = 0
Volt·second balance is a useful check that you have
correctly understood the operation of a power
converter circuit. It also has a more direct application
in finding the dc transfer function of switch mode
converters (SMCs).
Th6: Near constant voltages on the inductor mean
ramped currents, or what goes up must come down,
or in equation form:
<i> = constant → ∫upward di + ∫downward di = 0
or, from v dt = L di :
vpositive ∆ t = L ∆ iup or vnegative ∆ t = L ∆ idown
The Th6 theory concept helps us to find the ripple
current in an inductor (e.g. in a dc chopper or SMC),
in words, the positive (or negative) Volt·second area
divided by L is the inductor peak-to-peak current.
Th7: Capacitor Analogy means that the positive (or
negative) Amp·second area (in units of Coulombs)
divided by C is the capacitor peak-to-peak voltage,
or in equation form, using i dt = C dv we have:
ipositive ∆ t = C ∆ vup or inegative ∆ t = C ∆ vdown
3. Analysis Methods
When you are performing circuit analysis of a power
converter circuit, ask yourself, which method do I
need to use? Here are some examples.
AM1: DC Analysis is the application of KVL, KCL
and Ohm’s Law, noting which average voltages (or
currents) are zero.
e.g. <iL> = <iT> + <iD> for a SMC
AM2: AC Analysis is the application of KVL, KCL
and Ohm’s Law (with Z in place of R), noting which
impedances are negligible (and setting those to zero
when in series, or infinity when in parallel).
e.g. ILpp ≈ VBridge pp / (2π fripple)L for a diode rectifier
AM3: Transient Analysis - An example in 585 refers
to Th6 or Th7, ie, we are looking at a current or
voltage during a time frame less than one complete
period. In industry there is an important type of
transient that we have not had much time to study in
this course, and that is the power-up transient, which
typically lasts a few or tens of dc periods or ac
cycles. Components of a power converter need to
be designed to survive repeated transient cycles.
e.g. ILpp = vpositive ∆ t / L
for a SMC
Nov. 22, 2009
ENEL585 - Short Review of Nearly Everything
AM4: Fourier Analysis for Quarter Wave Symmetry
e.g v(θ) = ∑ vh cos(hθ), where vh = 4/π ∫0
π/2
v(θ) d θ
AM5: Energy Analysis tells us that the input power is
converted, on average, into the output power plus
the power losses of a given converter. This analysis
method is sometimes used as a check after
employing the above analysis methods (AM1 to
AM4), or it can be used to perform quick calculations
for design purposes.
e.g. Pout = Pin – Ploss = η Pin
- these are averages
AM6: Power Factor Analysis
In equation form for a single-phase source or load:
PF = Pavg / Vrms Irms = Preal / Papparent = P / S
AM7: Commutation Analysis
u
Page 2 of 2
AT5: Orthogonality and RMS Calculation;
in equation form:
if f(θ) = f1(θ) ± f2(θ) and f1 ⊥ f2 → Frms2 = F1rms2 + F2rms2
AT6: Orthogonality applied to Fourier components;
in equations form, if fh are orthogonal to each other:
if f(θ) = f1(θ) + f2(θ) + f3(θ) + f4(θ) + …
2
2
2
2
2
then Frms = F1rms + F2rms + F3rms + F4rms + …
AT7: Orthogonality and RMS applied to THD;
In equation form (given a Fourier series as in AT6):
THD ≡ √ ( ∑h=2∞ Fh2 ) / F1 where Fh can be expressed in
one of several forms (e.g. the peak value of that
component, or the rms value of that component, as
long as the same form is used in all instances).
2
2
Therefore THD = √ ( Frms – F1rms ) / F1rms
5. Rules of Thumb
e.g. ∫0 vcom (θ) d θ = ω L ∫ di
RT1: 50% Low Ripple RMS Approximation
4. Analysis Tools
If f(θ) is a periodic waveform such that
Fpp ≤ ½ Favg → Frms ≈ | Favg | to within about 1% error
The above analysis methods are aided by using the
following tools.
AT1: KVL = Kirchhoff’s Voltage Law
Is derived from the conservation of energy or, as
expressed in Maxwell’s equation ∫ E · dℓ = 0. KVL is
applicable in both DC and AC analysis methods.
RT2: 50% Low Ripple Power Approximation
We would apply this rule of thumb in the case where
current and voltage both have a dc component.
Consider the voltage v across an element and a
current i through that element. If the current has less
than 50% ripple, then we can say:
Pavg = <vi> ≈ <v> Iavg = <v> <i>
AT2: KCL = Kirchhoff’s Current Law
Is derived from the conservation of charge or, in
words, charge can neither be created nor destroyed.
This is a bit of a stretch actually – what we’re saying
is that you can’t have charge indefinitely building up
at some spot in a circuit (a counter example of this is
a Van de Graff generator – one of those large metal
balls that keeps building up charge with a rubber belt
conveyor to the point that the voltage can reach
millions of volts and your hair sticks out real frizzy
style), and so charge keeps moving in circular paths;
hence the name: electrical circuit. KCL is applicable
in both DC and AC analysis methods.
In the extreme case where i(t) ≈ constant, we can
think of this as factoring out the constant I in <vi>.
The above rule of thumb also applies to the case
where the voltage has less than 50% ripple, which
we have not seen in ENEL585.
AT3: Ohm’s Law v=iR our lovely humble little law
For a 3-phase source or load that is balanced, we
can find the average power for one phase and
multiply by three. Consider a Y-connected source:
AT4: Average Power Pavg = 1/T ∫period v(t) i(t) dt = <vi>
One could say that this is the dc component of
power, where we recall that power is an
instantaneous quantity (= vi) that can change quite
quickly (just step on the accelerator – an exception
is a coal-fired electrical generation plant) but energy
(the integral of power over time, possibly integrated
over a few periods during a transient power-up
analysis) takes time to change (about 25 seconds
for my little car to go from zero to 60mph).
RT3: Balanced 3-phase (Real) Power Approximation
First noting that for a single-phase source or load,
power factor in general has two factors within it,
namely distortion factor (DF) and the displacement
power factor (DPF = cos φ ), which gives us:
Pavg = Vrms Irms PF = Vrms Irms DF · DPF
(single-phase)
Pavg = 3 VLN rms ILN rms PF = 3 VLN rms ILN rms DF · DPF
But since the line-to-neutral voltage cannot always
be measure, we divide the line-to-line voltage by √3:
Pavg = 3 (VLL rms /√3) ILN rms PF = √3 VLL rms ILN rms DF · DPF
e.g. 3-phase SCR rectifier: DF=3/π and DPF= cos α