Chapter 2 : Power Factor Correction
1. Power Factor
1.1. Sinusoidal voltage and current in AC system
- Resistive load
-
Phase difference between voltage and current with phasor representation
I
V
Φ
-
Voltage as a time function
Current as a time function
Instantaneous power
𝑣𝑣(𝑡𝑡) = 𝑉𝑉𝑚𝑚 sin(𝜔𝜔𝜔𝜔)
𝑖𝑖(𝑡𝑡) = 𝐼𝐼𝑚𝑚 sin(𝜔𝜔𝜔𝜔 + 𝜙𝜙)
𝑝𝑝(𝑡𝑡) = 𝑣𝑣(𝑡𝑡)𝑖𝑖(𝑡𝑡)
= 𝑉𝑉𝑚𝑚 sin(𝜔𝜔𝜔𝜔) ∗ 𝐼𝐼𝑚𝑚 sin(𝜔𝜔𝜔𝜔 + 𝜙𝜙)
1
= 𝑉𝑉𝑚𝑚 𝐼𝐼𝑚𝑚 {cos(𝜙𝜙) − cos(2𝜔𝜔𝜔𝜔 + 𝜙𝜙)}
2
(1)
1.2. There are two components in the instantaneous power
1
o 2 𝑉𝑉𝑚𝑚 𝐼𝐼𝑚𝑚 cos(𝜙𝜙)
this is a constant term independent of time
-
1
this is a sinusoidal time function varying
at twice the supply frequency
The average real power is the first term only as the average of any sinusoidal
function is zero
o
𝑉𝑉 𝐼𝐼 cos(2𝜔𝜔𝜔𝜔 + 𝜙𝜙)
2 𝑚𝑚 𝑚𝑚
1
𝑉𝑉𝑚𝑚 𝐼𝐼𝑚𝑚
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑉𝑉𝑚𝑚 𝐼𝐼𝑚𝑚 cos(𝜙𝜙) =
cos(𝜙𝜙) = 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟 cos(𝜙𝜙)
2
√2 √2
1
-
-
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
Define the power factor 𝑃𝑃𝑃𝑃 = 𝑉𝑉
𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
= 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
When I & V are both sinusoidal
(2)
𝑃𝑃𝑃𝑃 = cos(𝜙𝜙)
1.3. Practical Importance of Power Factor
- In an AC system the voltage is fixed according to the standard. The maximum
current through a circuit is limited by the conductor size and temperature rise.
- The PF plays an important role. If PF is 1 which is the highest value, maximum
real power can be delivered. If PF is low, real power delivery is also low. In
other words bigger conductor is needed to carry the same amount of real
power with low PF.
Exercise :
A transmission line supplies power at power factor PF=0.7. If you want to
increase the power delivery by 30% without building more power lines, how
far should the power factor be improved?
2. Passive Power Factor Correction method
2.1. In AC system with sinusoidal current, the power factor can be corrected by
parallel inductor or capacitors depending on whether the current leads or lags
the voltage.
2.2. Current leading voltage in a capacitive load circuit
2.2.1. Adding a parallel inductor may cancel the reactive current
I
V
I
𝐼𝐼𝐿𝐿
I𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
Φ
I𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝑉𝑉
= 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
𝑗𝑗𝑗𝑗𝑗𝑗
𝑉𝑉
𝐿𝐿 = �
�
𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗𝑗
𝐼𝐼𝐿𝐿 =
V
𝐼𝐼𝐿𝐿
(3)
2
2.3. Current lagging voltage in an inductive load circuit
2.3.1. Adding a parallel capacitor may cancel the reactive current
I
V
V
I
I𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝐼𝐼𝑐𝑐
Φ
𝐼𝐼𝑐𝑐 = 𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉 = 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
𝐶𝐶 = �
𝐼𝐼𝑐𝑐
I𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼
�
𝑉𝑉𝑉𝑉𝑉𝑉
(4)
3. Non sinusoidal load current
3.1. Some loads draw non sinusoidal current. A typical example is Diode Bridge
feeding a capacitor for AC/DC conversion. This circuit is often found in
electronic appliances which draw power from the AC mains.
3
3.2. The non-sinusoidal current carries a lot of harmonics in addition to the
fundamental.
3.3. The non-sinusoidal current bring down the power factor and only the
fundamental current carries real power.
3.3.1. Current waveform is defined by the load while the voltage waveform is
controlled by the generating source which is often rigid and produces
undistorted waveform. Here it is assumed the utility input voltage is
undistorted but the current consists of a lot of harmonics.
1 𝑇𝑇1
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑇𝑇1 ∫0 𝑣𝑣(𝑡𝑡). 𝑖𝑖(𝑡𝑡)𝑑𝑑𝑑𝑑
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 =
=
𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑟𝑟𝑟𝑟𝑟𝑟
4
1 2𝜋𝜋
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = � √2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠(𝜔𝜔𝜔𝜔). 𝑖𝑖(𝑡𝑡)𝑑𝑑𝑑𝑑𝑑𝑑
𝑇𝑇1 0
(5)
where 𝑖𝑖(𝑡𝑡) = √2𝐼𝐼𝑠𝑠1 sin(𝜔𝜔𝜔𝜔 − 𝜙𝜙) + ∑ 𝐼𝐼𝑠𝑠ℎ sin(𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜙𝜙ℎ )
hence 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑠𝑠1 . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 +
1 2𝜋𝜋
∫ √2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠(𝜔𝜔𝜔𝜔). ∑ 𝐼𝐼𝑠𝑠ℎ sin(𝑚𝑚𝑚𝑚𝑚𝑚 − 𝜙𝜙ℎ ) 𝑑𝑑𝑑𝑑𝑑𝑑
𝑇𝑇 0
1
1
The harmonic terms become
𝑇𝑇1
𝜙𝜙ℎ ) 𝑑𝑑𝑑𝑑𝑑𝑑
(6)
2𝜋𝜋
√2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 ∫0 𝑠𝑠𝑠𝑠𝑠𝑠(𝜔𝜔𝜔𝜔). ∑ 𝐼𝐼𝑠𝑠ℎ sin(𝑚𝑚𝑚𝑚𝑚𝑚 −
as
sin(𝐴𝐴 − 𝐵𝐵) = 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 − 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠
become
the harmonic terms
2𝜋𝜋
1
√2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 � � 𝐼𝐼𝑠𝑠ℎ (sin(𝑚𝑚𝑚𝑚𝑚𝑚). 𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙𝑚𝑚
𝑇𝑇1
0
− cos(𝑚𝑚𝑚𝑚𝑚𝑚) . 𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙𝑚𝑚 ) sin(𝜔𝜔𝜔𝜔). 𝑑𝑑𝑑𝑑𝑑𝑑
2𝜋𝜋
1
= √2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 � � 𝐼𝐼𝑠𝑠ℎ (sin(𝑚𝑚𝑚𝑚𝑚𝑚). sin(𝜔𝜔𝜔𝜔) . 𝑐𝑐𝑐𝑐𝑐𝑐𝜙𝜙𝑚𝑚
𝑇𝑇1
0
− cos(𝑚𝑚𝑚𝑚𝑚𝑚) . sin(𝜔𝜔𝜔𝜔) . 𝑠𝑠𝑠𝑠𝑠𝑠𝜙𝜙𝑚𝑚 ) 𝑑𝑑𝑑𝑑𝑑𝑑
but
and
∫ 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑 = ∫
cos(𝑚𝑚−𝑛𝑛)𝑥𝑥
∫ 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐. 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑑𝑑𝑑𝑑 = ∫
𝑑𝑑𝑑𝑑 − ∫
2
sin(𝑚𝑚−𝑛𝑛)𝑥𝑥
2
cos(𝑚𝑚+𝑛𝑛)𝑥𝑥
𝑑𝑑𝑑𝑑 − ∫
𝑑𝑑𝑑𝑑
2
sin(𝑚𝑚+𝑛𝑛)𝑥𝑥
2
𝑑𝑑𝑑𝑑
integration of sin and cos terms with limits 0 and 2π result in zero, therefore
all harmonic terms become zero
and
𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝑠𝑠1 . 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑉𝑉
𝐼𝐼 .𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
1
𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝑃𝑃𝑃𝑃 = 𝑟𝑟𝑟𝑟𝑟𝑟
𝑉𝑉
𝐼𝐼
𝑟𝑟𝑟𝑟𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟
𝐼𝐼
= 𝐼𝐼 𝑠𝑠1 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐
𝑟𝑟𝑟𝑟𝑟𝑟
(7)
3.3.2. Note that only the fundamental current 𝐼𝐼𝑠𝑠1 contributes to real power.
The harmonic currents carry no real power but produce resistive losses in
conductors.
𝐼𝐼
3.3.3. The PF comprises of two terms : the 𝐼𝐼 𝑠𝑠1 represents the effect of
𝑟𝑟𝑟𝑟𝑟𝑟
current distortion; the 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 represents the effect of phase shift.
5
3.3.4. For a rectangular current waveform with a certain spectrum where
𝐼𝐼𝑠𝑠1 = 0.9𝐼𝐼𝑠𝑠
PF =
I
real _ power
= s1 = 0.9
apparent _ power I s
(8)
3.4. Total Harmonic Distortion (THD)
3.4.1. This is a measure of the distortion from a pure sinusoidal wave.
3.4.2. Root Mean Square (RMS) current of a non-sinusoidal current consists
of the fundamental as well as the harmonics.
1/ 2


I s =  I s21 + ∑ I sh2 
(9)
h ≠1


3.4.3. Harmonic currents other than the fundamental current are regarded
as distorted current.
1/ 2


I dis =  ∑ I sh2 
 h ≠1 
(
= I s2 − I s21
)
1/ 2
3.4.4. Total harmonic Distortion (THD) is defined as a percentage of the
useful fundamental component.
(10)
6
I
%THD = 100 x dis
I s1
(I − I )
= 100 x
2
s
2 1/ 2
s1
(11)
I s1
Exercise :
You are an engineer and you measure that an electrical product draws current at
120% THD. Is there anything wrong with the apparatus?
3.5. Regulatory requirement
3.5.1. Low power factor and high current distortion is most undesirable in the
power system.
3.5.2. In order to ensure power quality countries have set up regulations to
demand products to draw current with reasonable quality.
3.5.3. The EN61000-3-2 is an European standard which states limits for
harmonic current emissions to the power system.
4. Active Power Factor correction
4.1. The passive power factor correction method is not effective for distorted
current.
4.2. Active power factor correction is most effective. Essentially it is a power
converter that forces the input AC current to be sinusoidal through high
frequency switching. The PFC output is a DC which feeds a DC/DC in most
configurations.
7
4.3. Theoretically the PFC can be any high frequency converter but it has been
shown that a boost converter is a most suitable configuration.
5. Critical Conduction Mode (or Boundary Conduction Mode) operation of the boost
converter is a simple method
5.1. The MOSFET switch is ON for an on time 𝑡𝑡𝑂𝑂𝑂𝑂 which is constant over over the
entire cycle. After the switch is turned off, the next turn-on event is initiated
when the inductor current reaches 0A. This can be detected by observing for
a drop in MOSFET Drain voltage when the inductor current becomes zero.
5.2. This operation achieves PFC naturally without complicated control. Note that
8
the inductor current is the sum of the MOSFET and Diode currents, which is
triangular in shape with the base touching the zero line. Examine the average
current within one triangle
𝐼𝐼𝐿𝐿(𝐴𝐴𝐴𝐴𝐴𝐴) (𝑡𝑡) =
𝑉𝑉𝐼𝐼𝐼𝐼 (𝑡𝑡)
𝑡𝑡
2𝐿𝐿 𝑂𝑂𝑂𝑂
(12)
Since 𝑉𝑉𝐼𝐼𝐼𝐼 (𝑡𝑡) is sinusoidal, 𝐼𝐼𝐿𝐿(𝐴𝐴𝐴𝐴𝐴𝐴) (𝑡𝑡) is also sinusoidal. Thus giving PF=1.
5.3. As the input voltage varies within a sinusoidal cycle the MOSFET switching
frequency varies as well. The switching frequency variation over a half cycle
for a particular choice of L and 𝑡𝑡𝑂𝑂𝑂𝑂 is shown.
5.4. The on time 𝑡𝑡𝑂𝑂𝑂𝑂 is the control parameter which is generated by the control
signal 𝑉𝑉𝑐𝑐 (𝑡𝑡) in the following circuit. The output voltage 𝑉𝑉𝐵𝐵𝐵𝐵𝐵𝐵 (𝑡𝑡) is
monitored, and the signal is fed to an error amplifier which attempts to keep
𝑉𝑉𝐵𝐵𝐵𝐵𝐵𝐵 (𝑡𝑡) constant. Note that 𝑉𝑉𝑐𝑐 (𝑡𝑡) must be slow with respect to the sinusoidal
AC input line cycle as 𝑡𝑡𝑂𝑂𝑂𝑂 needs to be constant over the line cycle.
9
5.5. Assume that the PFC works well and produces PF=1, the power drawn from
the input can be expressed as
𝑃𝑃𝐷𝐷 (𝑡𝑡) =
2
2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟
𝑠𝑠𝑠𝑠𝑠𝑠2 (𝜔𝜔𝜔𝜔)
𝑅𝑅𝑒𝑒 (𝑡𝑡)
where 𝑅𝑅𝑒𝑒 (𝑡𝑡) is a resistive load defined by 𝑉𝑉𝑐𝑐 (𝑡𝑡).
Since 𝑅𝑅𝑒𝑒 (𝑡𝑡) is fictitous it is better to rewrite the equation in terms of
maximum power
𝑃𝑃𝐷𝐷 (𝑡𝑡) = 2𝑠𝑠𝑠𝑠𝑠𝑠2 (𝜔𝜔𝜔𝜔)
Using the trigonometric identity
𝑉𝑉𝑐𝑐 (𝑡𝑡)
𝑃𝑃
𝑉𝑉𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀) 𝑀𝑀𝑀𝑀𝑀𝑀
𝑠𝑠𝑠𝑠𝑠𝑠2 (𝜔𝜔𝜔𝜔) =
𝑃𝑃𝐷𝐷 (𝑡𝑡) = {1 − cos(2𝜔𝜔𝜔𝜔)}
1−cos(2𝜔𝜔𝜔𝜔)
(13)
(14)
2
𝑉𝑉𝑐𝑐 (𝑡𝑡)
𝑃𝑃
𝑉𝑉𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀) 𝑀𝑀𝑀𝑀𝑀𝑀
Note that there are two terms in the power equation. The average power
𝑉𝑉 (𝑡𝑡)
delivered is
𝑃𝑃𝐷𝐷_𝐴𝐴𝐴𝐴𝐴𝐴 (𝑡𝑡) = 𝑉𝑉 𝑐𝑐
𝑃𝑃𝑀𝑀𝑀𝑀𝑀𝑀
(15)
𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀)
and there is an AC ripple at twice the line frequency
𝑉𝑉𝑐𝑐 (𝑡𝑡)
𝑃𝑃
𝑃𝑃𝐷𝐷_𝐴𝐴𝐴𝐴 (𝑡𝑡) = −cos(2𝜔𝜔𝜔𝜔)
𝑉𝑉𝑐𝑐(𝑀𝑀𝑀𝑀𝑀𝑀) 𝑀𝑀𝑀𝑀𝑀𝑀
5.6. Examples
5.6.1. In a 110 Vrms 60Hz system, a display panel draws an average power of
200W. It has a front end CrCM PFC with input waveforms as shown. What
is the maximum input average current? What is the corresponding peak
inductor current?
10
Solution
Eqtn (15) is rewritten as
𝑃𝑃𝐷𝐷 (𝑡𝑡) = {1 − cos(2𝜔𝜔𝜔𝜔)}𝑃𝑃𝐼𝐼𝐼𝐼
𝑃𝑃𝐷𝐷 (𝑡𝑡) is maximum when cos(2𝜔𝜔𝜔𝜔) = −1 , which is the instantaneous power
So the maximum input average current can be estimated
𝑃𝑃𝐷𝐷𝑀𝑀𝑀𝑀𝑀𝑀 (𝑡𝑡) = 2𝑃𝑃𝐼𝐼𝐼𝐼 = √2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀
2 ∗ 200 = √2 ∗ 110 ∗ 𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀
𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀 = 2.571𝐴𝐴
Since the input current waveform is triangular, the average value is related to
the peak value as
𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 = 2 ∗ 𝐼𝐼𝐴𝐴𝑉𝑉𝑀𝑀𝑀𝑀𝑀𝑀 = 2 ∗ 2.571𝐴𝐴 = 5.14𝐴𝐴
5.6.2. Refer to the same CrCM PFC converter. If the CrCM PFC turn on time
𝑡𝑡𝑂𝑂𝑂𝑂 is set to 3𝜇𝜇𝜇𝜇, what is the recommended inductor value?
Solution
Note that current rise in an inductor
𝑑𝑑𝑑𝑑
𝑣𝑣 = 𝐿𝐿 𝑑𝑑𝑑𝑑
Take the approximation that the input voltage during 𝑡𝑡𝑂𝑂𝑂𝑂 is constant,
√2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 = 𝐿𝐿
𝐼𝐼𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝
𝑡𝑡𝑂𝑂𝑂𝑂
11
√2 ∗ 110 = 𝐿𝐿
5.14
3𝑢𝑢𝑢𝑢
𝐿𝐿 = 90.8𝜇𝜇𝜇𝜇
5.7. Comments on the CrCM PFC
5.7.1. The control configuration is simple and does not need current sensing.
5.7.2. The MOSFET turns on with zero current and the diode turns off with
zero current, which can reduce corresponding losses.
5.7.3. The inductor and MOSFET current is high at twice the input average
value, high switching loss turn off loss may occur.
5.7.4. Applications limited to low power applicances below 300W, such as
TVs, PCs and power supplies.
5.7.5. High ripple input current may produce Electromagnetic Interference
problems and filtering is needed.
5.7.6. Voltage ripple at twice the line frequency will appear at the DC output
because of the slow voltage control loop. This has to be compensated by
sophisticated methods.
6. Continuous Conduction Mode (CCM) PFC
6.1. In order to solve the high input current ripple problem in the CrCM PFC a
solution is the CCM PFC. It has the same circuit configuration except that
bigger inductor is used such that the inductor current is not allowed to drop
to zero in every switching cycle.
6.2. The ripple current is greatly reduced compared to the input current average
value and the boost converter can operate a fixed frequency which is much
preferred to variable frequency.
6.3. A current sensor is needed to sense the input current. The duty cycle within a
switching cycle varies to control the input peak current with respect to a
sinusoidal reference.
12
6.4. The sinusoidal reference for the input current waveform is taken from the
sinusoidal input voltage waveform
𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡) =
𝑝𝑝(𝑡𝑡)√2
|sin(𝜔𝜔𝜔𝜔)|
𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟
where 𝑝𝑝(𝑡𝑡) is the output power given by a feedback circuit. It is in
general much slower than the sinusoidal frequency 𝜔𝜔.
(16)
If we want to use the input voltage as a template for shaping the input
current peak, the amplitude must include an additional denominator 𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟
𝑖𝑖𝑖𝑖𝑖𝑖 (𝑡𝑡) =
where 𝑣𝑣𝑖𝑖𝑖𝑖 (𝑡𝑡) = √2𝑉𝑉𝑖𝑖𝑖𝑖 |sin(𝜔𝜔𝜔𝜔)|
𝑝𝑝(𝑡𝑡)
𝑣𝑣𝑖𝑖𝑖𝑖 (𝑡𝑡)
2
𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟
(17)
6.5. The power 𝑝𝑝(𝑡𝑡) is a feedback variable derived from an error amplifier which
attempts to produce a constant output voltage. As said the power should vary
much slower than the line frequency.
6.6. The control scheme is illustrated in the following diagram. Notice the fast
current loop and the slow voltage loop.
13
6.7. CCM PFC has successfully solved the current ripple problem and brings PFC to
higher power well beyond 300W. However in the presence of practical issues
it is not perfect.
6.7.1. The continuous conduction mode makes the diode turns off at non
zero current. This brings in diode reverse recovery effect which may affect
efficiency. The diode should have excellent reverse recovery effect and
modern SiC devices have greatly improved characteristics.
6.7.2. At zero crossings of the sinusoidal waveform the input voltage would
be too low to produce the designated high output voltage. Thus current
distortion may lower THD performance.
6.8. Examples
6.8.1. In a CCM PFC for a 110 Vrms 60Hz system, the output voltage is set to
200V dc. The load draws a power of 400W. The switching frequency of
the boost converter is 200kHz. Given that a CCM boost converter has the
following voltage equation
1
𝑉𝑉𝑜𝑜𝑜𝑜𝑜𝑜
=
𝑉𝑉𝑖𝑖𝑖𝑖
1 − 𝐷𝐷
calculate the minimum duty cycle.
Solution
The minimum duty cycle occurs when the input voltage is at its peak.
Follow the boost converter equation
1
200
=
110 ∗ √2 1 − 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚
110 ∗ √2
𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 = 1 −
= 0.222
200
6.8.2. Refer to the same CCM PFC configuration as above, if it is desired that
the inductor current rise at 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 is one-tenth of the peak average input
current, what is the inductor value?
Solution
Eqtn (15) is rewritten as
𝑃𝑃𝐷𝐷 (𝑡𝑡) = {1 − cos(2𝜔𝜔𝜔𝜔)}𝑃𝑃𝐼𝐼𝐼𝐼
𝑃𝑃𝐷𝐷 (𝑡𝑡) is maximum when cos(2𝜔𝜔𝜔𝜔) = −1.
So the maximum input average current can be estimated
𝑃𝑃𝐷𝐷𝑀𝑀𝑀𝑀𝑀𝑀 (𝑡𝑡) = 2𝑃𝑃𝐼𝐼𝐼𝐼 = √2𝑉𝑉𝑟𝑟𝑟𝑟𝑟𝑟 𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀
2 ∗ 400 = √2 ∗ 110 ∗ 𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀
𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀 = 5.1426𝐴𝐴
14
Inductor current rise ∆𝑖𝑖 at 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 is
0.1 ∗ 𝐼𝐼𝐴𝐴𝐴𝐴_𝑀𝑀𝑀𝑀𝑀𝑀 = 0.5143𝐴𝐴
Note that for a boost converter during the ON state
∆𝑖𝑖
𝑣𝑣𝐿𝐿 = 𝐿𝐿
∆𝑡𝑡
0.5143
110 ∗ √2 = 𝐿𝐿
0.222/200𝑘𝑘
𝐿𝐿 = 0.336 𝑚𝑚𝑚𝑚
7.
Bridgeless PFC
7.1. The input rectifiers have considerable forward voltage drop (> 0.7V) which
produces a lot of power loss. MOSFETs are known to have low on resistance
which can cut down power loss. So it is a nice idea to use MOSFET to replace
the diode and implement PFC as well.
7.2. Here two diodes are replaced by MOSFETs in a single phase bridgeless PFC
7.3. The current paths in the positive and negative half cycles are shown. In the
positive half cycle Q1 is switching on and off while Q2 is turned on. Together
with the inductors Q1 acts as the switch for a boost converter. In the negative
half cycle Q2 acts as the switch while Q1 is turned on.
7.4. The boost converter can work in the Critical Mode or Continuous Mode as
described.
7.5. Actually to further reduce losses the remaining two diodes can also be
replaced by MOSFETs with synchronized control.
15
7.6. Furthermore, this bridge configuration can be extended to three phase PFC
system. Actually this is the well known AC/DC inverter configuration.
References
[1] Brent McDonald and Ben Lough, “Power Factor Correction (PFC) Circuit Basics”,
Texas Instruments Training,
https://training.ti.com/sites/default/files/docs/power_factor_correction_circuit_basi
cs_-_paper.pdf
[2] “Power Factor Correction (PFC) Circuits”, Toshiba Application Notes, 2019
https://www.google.com/search?q=+%E2%80%9CPower+Factor+Correction+%28PFC
%29+Circuits%E2%80%9D%2C+Toshiba+Application+Notes%2C+2019+&sxsrf=ALeKk0
38dc1pYWJiuxAFbQKxa9ofZDEWUg%3A1628055693224&source=hp&ei=jSgKYdbhCs
WSr7wPy_60kA4&iflsig=AINFCbYAAAAAYQo2nY7yAPO8jsDXwgFqFHHHWZ6qjWg&oq=+%E2%80%9CPower+Factor+Correction+%28PFC%29+Circu
its%E2%80%9D%2C+Toshiba+Application+Notes%2C+2019+&gs_lcp=Cgdnd3Mtd2l6E
AM6BwgjEOoCECdQ9w9Y9w9gxBloAXAAeACAAUeIAUeSAQExmAEAoAECoAEBsAEK&s
client=gws-wiz&ved=0ahUKEwiW3JCK1JbyAhVFyYsBHUs_DeIQ4dUDCAc&uact=5
[3] Siu-Chung Wong; C. K. Tse; M. Orabi; T. Ninomiya, “The method of double
averaging: an approach for modeling power-factor-correction switching converters”,
16
IEEE Transactions on Circuits and Systems I, 2006, Volume: 53, Issue: 2. Pages: 454 –
462
[4] Johann W. Kolar, and Thomas Friedli, “The Essence of Three-Phase PFC Rectifier
Systems—Part I”, IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 1,
JANUARY 2013
17