Unwind di Power Electronics - 4 di 5

Power Factor and Active Power

The power factor (PF) is equal to the cosine of the phase shift between voltage and current. If the voltage and the current are considered as phasors, they are parallel: P_F = cos(θ)

The projection of the phasor of the current on the phasor of the voltage represents the active current. The product between the active current and the active voltage represents the instantaneous active power. If the angle between the current and the active voltage is 0, then the active current corresponds to the total current and the power factor corresponds to 1.

Let's consider now a different scenario in which the voltage is sinusoidal but the current is distorted:

V = V₀sin(ωt)

I = I₀sin(ωt + φ)

Usually, the current injected in the system is not purely sinusoidal but presents other harmonics that introduce distortion. Let's evaluate now the active power:

P = V₀I₀cos(φ)

Let's evaluate the square of Irms: I²_rms = I₀²

This term represents the square of the root mean square current, which includes both the fundamental and harmonic components.

Total Harmonic Distortion (THD):
Total harmonic distortion of the current.
Note that the summation starts from 2. The square roots of the summation corresponds to the rms value of the signal obtained by removing the fundamental component from the current signal. The THD measures the relative weight of a rms value due to harmonics only and rms due to fundamental component.
Using the previous definition of the THD, the square value of Irms is:
ITitTHD Thisi iIr.ms Ir.ms Ii i t
The THD is 0 only if there are no current harmonics after the fundamental one. In this case, the current becomes purely sinusoidal. With more harmonics after the fundamental one, the waveform cannot be purely sinusoidal.
The PF is so: VenistiP Fa Cos Cos aaTHI THIi ivrmsI.it ii9I Cos iTHI i1 PF with a sinusoidal9P voltage and noF 1 Cos iTHI assumption on thei1 current
This term is called displacement factor, while this term is called Distortion Factor (DF). Both terms are comprised between 0 and 1.

The distortion factor accounts for the harmonic content of the current while the DF accounts for the phase shift between the fundamental component of the voltage and the fundamental component of the current.

The PF is limited by 2 factors and can be equal to 1 if and only if both are equal to 1, so there is no distortion in the current shape and there is no phase shift between voltage and current: Ò4I Reactive compensation. CosPF L THI i Harmonic compensation. DDI i

If the PF is equal to 1, both the reactive and harmonic compensation are present. Usually, in this last scenario the current is purely sinusoidal, so the constant that relates voltage and current is a resistance and this is the objective of the reactive compensation. In fact, the purpose of the reactive compensation is to modify the system in order to make sure the load appears like a purely resistive load.

A first example of rectifier in which the PF is limited principally by the THD is the Graetz rectifier with capacitive load:

isDi D C voRiHtt Da84There are some hypothesis made on this circuit.

Firstly, the voltage v(t) is purely sinusoidal:V times it wot

The 4 diodes are ideal.

The last hypothesis and the most important one is that the product: R*C, is much more larger than half of the fundamental period: ire

The behaviour of the Graetz rectifier is the following: when the input voltage is greater then the voltage on the capacitor, the current flows through a couple of diodes, charges the capacitor goes in the resistive load. Due to the topology of the circuit, this happens during the first semi-period (where the diodes D1 and D2 conducts) and during the second semi-period (where the diodes D3 and D4 conducts).

The small portion of time in which che current flows is called conduction angle or conduction time and it’s a very small quantity of time.

The waveforms are the following:

UtVm tto too 2Vm IV It iv Vm ti t

During this small interval, the current flows from the input voltage generator, recharges the

< p >capacitor and flows in the resistor. Outside of the conduction interval this current is 0 and the capacitor discharges on the capacitor but not completely because the time constant of the equivalent RC circuit is much larger then half the fundamental period. The discharge of the capacitor is only partial. Note that the output voltage is approximately constant and equal to the amplitude of the voltage sinusoid but this is true only in this example because for hypothesis the diodes are ideal but in a more realistic behaviour they presents a voltage drop so the maximum output voltage should be equal to Vm but at this value it should be subtracted 2 times the voltage drop of a diode. In terms of the current, the current that flows in the sinusoidal voltage generator can be positive or negative. In particular it presents large spikes of currents, both positive and negative due to the topology of the circuit. The output current is approximately constant because the output voltage is approximately< /p >

constant too. Moreover the current that flows through the diodes is approximately impulsive and the amplitude of these impulses is very high because the current has to recharge the capacitor that usually is presents a large value of capacitance.

The fundamental period of the line current is the same of voltage so the line current is in periodic steady-state but it's not sinusoidal so there is a large amount of distortion. In the DC side the fundamental period is half the period of the supply because the current iD is equal to the line current but all the pulses are positive. The fundamental frequency of the DC side is so 2 times the fundamental frequency of the AC side.

An important thing to note is that the couple of diodes that conducts are turned ON or OFF due to the voltage presented by the supply. So it's the line voltage that turns ON or OFF the diodes. This means that the switching noise containers the fundamental frequency of the AC side and it's multiplies.

There are

some approximated expression about the Graetz rectifier. The first one is the expression of the conduction angle: I_2TEWoP / (2π iCUORE)

Another important parameter is the relative peak-to-peak output voltage ripple: pIo / wo re

This last expression can be used to redefine the conduction angle: IEP Wo / (2π 2cuore)

Another important parameter is the peak like current: E Io / (2π Ire Io i P)

All these inequalities are valid as long as: Wore iIo

The initial hypothesis about the product RC and the fundamental period of the DC side yields a small conduction angle, to a small relative peak-to-peak output voltage and to a large peak line current. Obviously if the line current reaches high values, then it's highly distorted. LESSON #24 LESSON #25

Let's derive a design equation for the capacitive filter of the output side of Graetz rectifier: LESSON #26

ilPHI e RvPNEIsin PP Pcos Wot 2WhWotP Pft tPecos2

The power-controlled power source generates the power p(t) but only it's average value P is absorbed

by the resistive load. The instantaneous power exchanged by the capacitor with the rest of the circuit is the fluctuating power, indicated with pf(t). The instantaneous power absorbed by the load is constant because the output voltage is approximately constant, so the output current is constant too.

The energy stored and dissipated by the capacitor is named deltaE and is represented as follows:

In the first half of the period, the capacitor stores energy that is successively dissipated in the rest of the circuit during the second half of the period. This phenomenon is responsible for the peak-to-peak output voltage.

Let's evaluate deltaE that is the integral of p(t) over the time:

deltaE = ∫ p(t) dt = 1/2 ∫ P sin^2(ωt) dt = 1/2 P T

Clearly, the larger the power that must reach the load, the larger the energy stored inside the capacitor. The last expression represents the energy exchanged by the

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capacitor in period, but there is another expression toexpression this quantity that is the following:

E VèniaCUOIAEevqnax Eevqn.in ehchi Io IoVit voVo Vovo_VoIII IIIe ioeroE P ero ioW

This last expression is a design equation that solved for C gives:

p voUovo

This result represents the value of capacitance that must be used in a Graetz rectifier modelled with Loss FreeResistor (LFR)

The value of C is directly proportional by the power that the capacitor have to exchange with the rest of thecircuit. Moreover note that at the denominator there is the line frequency and this happens because this is not a acapacitor that must filter out some switching harmonics but a capacitor that must filter out the flactuating power.

Let’s now compare this last result with the Graetz rectifier with a capacitive filter:

i f néÈ rew.ie

Putting together these last two expressions gives: ietw.vnè vc

If:

• wo is the same;
• P is the same;
• Delta_vo is the same;

Vo=V_m (in the Graetz rectifier, the output voltage is approximately equal to the line voltage and it cannot be different); Then: i_z.I_T ElfiCGRAE The reason that the Graetz rectifier capacitance must be larger is that the current is impulsive so has a lot of spectral harmonics and the capacitor must be larger to filter out all the harmonics, while in the LFR Graetz rectifier the output current is made by the superposition of a DC value and only one harmonic (centered at two times the line frequency) so the capacitor can be smaller to filter the only one harmonic present at the output. Let's introduce now the power factor correction: i_g i_D i_D i C I RRemi pit voda piuDa84 control VVOLTAGECOMPENSATOR Voref In this representation, the output voltage is used to be compared with a reference signal V_o,ref and avoltagecompensator uses this difference to generate the control signal Vcontrol that adjusts the emulated re.