Electric Power analysis using Fouries series
If a sinusoidal voltage source generates a periodic, non-sinusoidal current (for example, in a light dimmer, a speed controller in a vacuum cleaner, rectifiers, etc.), then these currents cause reactive power not only due to the phase shift of the fundamental component of the current relative to the voltage, but also distortion reactive power caused by higher-order harmonics. An extended example can then be used to discuss analyze and calculate all related properties such as power factor, distortion factor, etc.
It is advised to visit the page about Fourier series theory before continuing
Triac control circuit
The triac control circuit, shown here below, is a widely used control system for regulating the power delivered to the load. The voltage source is a sinusoidal. With the so-called fire angle the power is controlled using the firing angle.
The firing angle is the angle at which the triac begins to conduct until the source voltage crosses zero. For the circuit we use firing angles of 135 and 315 degrees. A source voltage of 120 V RMS value with a frequency of 60 Hz. The load is a 9 Ohm resistor. The peak current is calculated to be 18.86 A. First we will use the program TRIACCT.8XP to analyze the current (RMS), the power and powerfactor. Following this we will delve deeper into the concepts distortion, reactive power. and Fourier analysis.
The triac circuit with resistor load with current @ firing angle of 135 and 315deg
The input parmeters for the program TRIACCT.8XP
The output of the program TRIACCT.8XP.
The results show that the power factor is very low (0.301). With the help of Fourier analysis we shall show that the low powerfactor is caused by two facts namely: the phase shift of the first harmonic and the presence of a significant amount of higher harmonics.
It is known that the real power Pw in a circuit with sinusoidal source voltage URMS is determined by the first harmonic of the current I1(RMS) and its phaseshift Ï•1 relative to the source voltage.
Pw=URMS*I1(RMS)*cos(Ï•1)
The reactive power caused by the first harmonic current is
PQ1=URMS*I1(RMS)*sin(Ï•1)
The apparent power of the first harmonic current is
PS1=URMS*I1(RMS)
The total apparent power of the system is
PS=URMS*I(RMS) I(RMS) is the RMS value of the total current
The powerfactor Pf= Pw/Ps=(Pw*PS1)/(PS* PS1)= (Pw/PS1)*( PS1/PS)=cos(Ï•1)*cos(δ)
cos(δ)is called the distortion facor and is related to the amount of higher harmonics.
For the reactive powet it yields
(PQ)^2=(PQ1)^2+(PQn)^2
To calculate real power, reactive power and the distortion factor we we need to analyze the current with Fourier to determine at least the first harmonic of the current. With the program Funcgen.8XP the current figure is generated to use it as input for the FOURIERL4.8XP program. The results are stored in the Lists: L1-L5. More info gives the website page Fourier for Ti84.
​ Important is the first harmonic of the current (L4) 3,4555 A andt its phase shift towards the source voltage (L5) -60.28 deg. As the calculated firt harmonic current is a peak value we have to calculate the RMS value which equals √(3.4555)=2.44305A.
The power Pw=URMS*I1(RMS)*cos(Ï•1)=120*2.44305*cos(-60.28)=145,34W which agrees with the results calulated with TRIACCT.8XP.
The reactive power caused by the first harmonic current:
PQ1=URMS*I1(RMS)*sin(Ï•1)=120*2.44305*sin(-60.28)=-254.603 Var
The apparent power PS1 of the first harmonic current
PS1=URMS*I1(RMS)=120*2.44305=293.17 VA
The total RMS current according to the program TRMAVE3.XP or program TRIACCT.8XP
I(RMS)=4.019 A
Total apperent power
PS=URMS*I1(RMS)=120*4.019=482.28VA​
Total reactive power
(PQ)^2=(Ps)^2-(Pw)^2P=(482.28)^2-(145.34)^2=211470,28
PQ=(211470)^(1/2)=459.86Var,
(PQn)^2=(PQ)^2-(PQ1)^2=(459.86)^2-(254.602)^2=146648.1044
PQn=√2(146648.1044)=382,95 Var reactive power caused by higher harmonic currents
(Pw/PS1)=cos(Ï•1)=pf = 145.34/293.17
Pw/PS1)=cos(Ï•1) = 145.34/293.17=0.4958
( PS1/PS)=293.17/482.28=cos(δ)=0.60788
Pf=Pw/Ps=145.34/482.28=0.30136=cos(Ï•1)*cos(δ)=0.4958*0.60788=0.30139
The low powerfactor of this system is both caused by the phase shift of the first harmonic and
and the reactive lossed caused by the higher harmonics​​​​​​​​​​​
This page is still in developement
The output of the program FOURIERL4.8XP.