Download Modeling a RLC Circuit`s Current with Differential Equations

Modeling a RLC Circuit’s Current with
Differential Equations
Kenny Harwood
May 17, 2011
Abstract
The world of electricity and light have only within the past century been explained in mathematical terms yet still remain a mystery
to the human race. R. Buckminster Fuller said; ”Up to the twentieth century, ”reality” was everything humans could touch, smell, see,
and hear. Since the initial publication of the chart of the electromagnetic spectrum ... humans have learned that what they can touch,
smell, see, and hear is less than one-millionth of reality.”[4] This paper gives an abbreviated description of the photovaltaic effect (solar
power production process) and then a RLC circuit will be modeled
that is powered by a photovaltaic panel that has its output voltage
passed through an inverter to produce an AC output signal where
voltage becomes a sinusoidal function of time.
1
Contents
1 A Means to Produce Power
3
2 The S.R.H process
4
3 Applying Free Electricity
5
4 In Search of an ODE
7
5 Analyzing Circuit for Numerical Values of Circuit Components
8
6 Solving the Ordinary Differential Equation
10
7 Matlab Method
12
8 Curtain Call
14
9 Appendix
15
2
1
A Means to Produce Power
The Sun has showered this blue speck that has been called home for
billions of years, showering free energy upon Earth’s surface unconditionally.
However, only recently has solar power been given the spotlight across this
globe. Solar panels currently are being produced and marketed in mass to
counteract the dependency humans have on the less forgiving fossil fuels.
In 2007, 18.8 trillion kilowatthours of electricity were produced globally[1].In
comparison, the sunlight received on the Earth’s surface in one hour is enough
to power the entire world for a year[2]. The question is, how do those radiant
warm rays of light become electricity? The answer in short, recombination.
3
2
The S.R.H process
In the early 1900’s, a mathematical model was published expressing the process in which light photons hand off valence electrons to latices (such as
silicon photovaltaic, or PV cells) and create an electric current and therefore
electricity. It was titled ’Shockley-Read-Hall Recombination’[7]. There are
many complicated ways to descrive recombination and the photovaltaic effect, but with the help of Jessika Toothman and Scott Aldous’s How Stuff
Works article[3], this should go by quite painlessly.
The basic idea of recombination is that photons (light wave-particles)
have electrons in their valence shell, or the outer-most orbit of an atom that
has electrons occupying it, and when the photons come in contact with a crystalline structure of silicon, the valence electrons are magnetically captured
by ’electron holes’ in the silicon lattice. The electron holes are present in all
the atoms in the silicon lattice so the captured electrons are passed through
the lattice to make more room at the silicon’s surface for more electrons.
But without an electric field, the cell wouldn’t work; the field forms when
the N-type (neutrally charged) and P-type (positively charged) silicon come
into contact. Suddenly, the free electrons on the N side see all the openings
on the P side, and there’s a mad rush to fill them. Eventually, equilibrium
is reached, and an electric field is present separating the two sides.
This electric field acts as a diode, allowing (and even pushing) electrons
to flow from the P side to the N side, but not the other way around. It’s
like a hill – electrons can easily go down the hill (to the N side), but can’t
climb it to the P side. And if one were to go through the mathematics of this
process starting with six spatial partial derivative equations, a solution to
the differential equation for the current being made by recombination would
be found to be I = Is (eVD /nVt − 1) which is known as the Shockley Diode
equation where;
•
•
•
•
•
I is the diode current,
Is is the reverse bias saturation current (or scale current),
VD is the voltage across the diode,
Vt is the thermal voltage, and
n is the ideality factor
4
When light, in the form of photons, hits a solar cell, its energy breaks
apart electron-hole pairs. Each photon with enough energy will normally
free exactly one electron, resulting in a free hole as well. If this happens
close enough to the electric field, or if free electron and free hole happen to
wander into its range of influence, the field will send the electron to the N
side and the hole to the P side. This causes further disruption of electrical
neutrality, and if an external current path is provided, electrons will flow
through the path to the P side to unite with holes that the electric field sent
there. The electron flow provides the current, and the cell’s electric field
causes a voltage. With both current and voltage, there is power, which is
the product of the two.
Yet, Silicon happens to be a very shiny material, which can send photons
bouncing away before they’ve done their job, so an antireflective coating is
applied to reduce those losses. The final step is to install something that will
protect the cell from the elements – often a glass cover plate. PV modules
are generally made by connecting several individual cells together to achieve
useful levels of voltage and current, and putting them in a sturdy frame
complete with positive and negative terminals.
3
Applying Free Electricity
Now that the process of S.R.H recombination and the workings of a PV cell
have been presented, this power source can be put into a circuit consisting
of a resistor, capacitor, and inductor. Using the software Matlab, and skills
learned in differential equations, the circuit will be modeled where as the
circuit tunes to the frequency of the Flathead Valley’s own Kool 105.1 FM
station. Prior to modeling the circuit some assumptions must be laid out,
the first of which being that the voltage source coming from the solar panel
is an alternating current signal. Solar panels produce direct current (DC)
electricity. However, the electricity used in homes for lighting and power
is 240 volt Alternating Current (AC) electricity. Therefore, an electronic
component called an inverter is used in the transformation of DC electricity
to AC electricity. An inverter achieves this by use of electronic switches to
alternate the flow of the DC signal produced from solar panels.That is, switch
one opens and switch 2 is closed and the current flows one way across the
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circuit. Then switch 1 closes and switch 2 opens and the current runs the
opposite way across a circuit. Thus the DC electricity is converted to AC
electricity. For the remainder of this paper, the following assumptions will
be made:
• The Solar panel has its DC signal transformed to AC through an inverter
before being connected to the RLC circuit.
• Voltage efficiency of the PV cells do not waver.
(constant voltage peak for AC signal function)
• Frequency of AC signal is set at 105.1 kHz
• All wires in setup are ideal, therefore offering negligible resistance to
current.
• Initial charge of capacitor in circuit is zero.
For a simple example of how solar power can be used, an RLC circuit will
be modeled with a driving voltage that is produced from PV cells(about 0.5
Volts per cell)[6] in a 12-celled solar panel. The RLC circuit being powered
must have values for its components that let the frequency resonate at 105.1
kHz, which will in turn tune the circuit to pickup and resonates the AC signal
that oscillates at the frequency of Kool 105.1 FM. In the AC signal equation
where VAC (t) = Vpeak sin(ωt), the angular frequency ω can be expressed as
ω = 2πf , and Vpeak can be found as the product of the amount of PV cells
and the voltage output per cell. From this information, the Voltage function
for the AC signal can be expressed as; VAC = 6 sin(105.1 × 103 t) where t is
time measured in seconds.
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4
In Search of an ODE
With a basic understanding of how light is transformed into electricity, a
mathematical model can be presented of the electric current in an RLC
parallel circuit, also known as a ”tuning” circuit or band-pass filter. To
reach the ordinary differential equation
needed to model the RLC circuit,
R
dI
V = L dt + RI(t) + 1/C ∗ ((Qo ) + I(t)dt[5] must be differentiated. Therefore, V has been been replaced with VAC found above, and Qo (initial charge
of capacitor), assumed to equal zero, will make the voltage equation for the
RLC circuit
Z
1
dI
Vpeak sin(ω ∗ t) = L + RI(t) + ∗ I(t)dt
dt
C
taking the derivative of both sides of the equation with respect to t, the
Second Order ordinary differential equation (ODE) is found to be;
L
dI
1
d2 I
+
R
+
I(t) = Vpeak ω cos(ω ∗ t)
dt2
dt C
Where;
• I is the circuit current,
• V is the Voltage output from a solar panel system which will be
assumed to be constant,
• L is the inductance of the inductor in the circuit measured in Henrys,
H,
• R is the resistance of the resistor in the circuit measured in ohms, Ω,
and
• ω is the angular frequency of the AC signal which is also expressed as
ω = 2πf
.
In order to have this equation set up for being evaluated by Matlab’s
ode45 solver, the second derivative must be isolated on the left hand side
like so
d2 I/dt2 = −R/L ∗ dI/dt − 1/(C ∗ L) ∗ I(t) +
6ω
cos(ω ∗ t)
L
(1)
However, before solving the ODE, values must be assigned for the circuit
components.
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5
Analyzing Circuit for Numerical Values of
Circuit Components
Figure 1. RLC parallel circuit
V - the voltage of the power source
I - the current in the circuit
R - the resistance of the resistor
L - the inductance of the inductor
C - the capacitance of the capacitor
The components of this circuit are needing to resonate a frequency of
105.1 kHz. In order to find the values needed in a tuning circuit for the
Kool 105 radio frequency, common inductance and resistance values found
in older tuning circuits will be used. With the values of resistance (R =
10Ω), inductance (L = 130pF ) and frequency (f = 105.1kHz), the value of
capacitance can be found. From the equation for forced damped harmonic
oscillation
x00 + 2cx0 + ωo2 x = A cos(ωt)
a comparison can be made with equation (1) where ωo2 is equivalent to 1/(C ∗
L). Since this circuit is needing to tune to 105.1 kHz, the natural angular
frequency ωo must be equal to the driving angular frequency ω and therefore
the capacitance needed for this model can be calculated by setting
ω2 =
8
1
CL
(2)
After doing some algebra and substitution, equation(2) becomes C =
1/(L ∗ (2π ∗ f )2 ) and capacitance is calculated to be 17.64 ∗ 10−6 F or 17.64
µF. To support this calculation, Multisim, a virtual circuit simulator has
been used to show that the magnitude of frequency peaks at 105.1 kHz with
the given and found values for this RLC circuit.
Values of Components
L=130 nH for VHF (FM)
f =105.1 kHz
C=17.64 µF
R= 10 Ω
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6
Solving the Ordinary Differential Equation
In order to find a closed form solution to the RLC ODE, a general solution
to the homogeneous and particular equations must be found and then solved
for the initial conditions. At time t = 0 seconds, there is no current going
through the circuit and therefore no initial rate of change of current either,
also stated I(0) = 0 and dI/dt(0) = I 0 (0) = 0.
To solve for the homogeneous equation, the ODE(1) must have no driving
force, therefore
I 00 + R/L ∗ I 0 + 1/CL ∗ I(t) = 0
the characteristic roots are found;
q
R
±
λ=−
2L
(
L)2
R
−
4
CL
2
which equate to λ1 = −5.6694 × 103 and λ2 = −7.6917 × 107 . The homogeneous part to the general solution is now known as
Ih = C1 eλ1 ∗t + C2 eλ2 ∗t
Where C1 and C2 are constants.
However, before solving for C1 and C2 , a particular solution for I(t) = Ih +
Ip must be found. The complex method laid out on page 167 of ”Differential
Equations With Boundary Value Problems”[5] will be used in finding the
particular solution Ip where
I 00 + R/L ∗ I 0 + 1/CL ∗ I(t) = 6ω/L ∗ cos(ωt)
will be put in the complex form of
z 00 + R/L ∗ z 0 + 1/CL ∗ z(t) = 6ω/L ∗ eωit
√
Where i is the imaginary number −1.
In order to solve for z(t), a guess must be made first. Let z(t) = aeωit
and substitute that guess in for z(t). After taking a few derivatives and some
algebra,
6( 1 − ωL − iR)
a = 1Cω
( Cω − ωL)2 + R2
10
In the complex method of solving particular solutions, it is known that
the final particular solution to this initial problem will be equal to the real
part of the z(t) solution. Therefore,
Ip = Re(z(t)) = −8.4229 × 10−8 sin(ωt) − 0.6 cos(ωt)
The final step to solving for the explicit solution to equation1 is to let
I(t) = Ih + Ip and solve for the initial conditions C1 and C2 . By doing so,
the closed form solution to the second order ODE is equated as,
I(t) = C1 eλ1 t + C2 eλ2 t + A ∗ cos(ωt) + B ∗ sin(ωt)
(3)
C1 = 0.0051
C2 = −0.0051
λ1 = −5.6694 × 103
λ2 = −7.6917 × 107
A = −8.4229 × 10−8
B = 0.6000
ω = 2π105.1 × 103
Plotting the closed form solution in Matlab over a 0.1 millisecond time
interval produces the graph;
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7
Matlab Method
With the values for the RLC components having set values, the ODE is ready
to be solved by MATlab as well. By inputting equation [3] into the function
file cited in the appendix, Matlab’s ODE45 numerical solver estimated this
graph of I(t)
which is the solution to the current I equation in this RLC resonating circuit.
By graphing I with respect to its derivative I 0 , a phase plane plot can be
viewed.
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The phase plane graph shows a circular path with which the current amplitude remains constant (radius of the circle) as the change in current moves
from positive to negative repeatedly in a circular fashion. This confirms the
ODE45 system is solved for a resonating system. To show that both the current and the rate at which the current changes do not decay as time passes,
a composite graph has also been provided with Matlab’s tools.
Here also is the same composite graph yet with a longer time interval of
10 milliseconds to further back the previous statement.
As is evident, the graph of the closed form solution matches up nicely
to the graph that ODE45 produced for current versus time. Given all the
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assumptions and procedures made are in tandem with current scientific theories and laws, this paper has outlined the process of sunlight being harnessed
as electric current and has successfully provided and solved a second order
ordinary differential equation regarding the electric current within a RLC
circuit.
8
Curtain Call
Though the mathematics of recombination are beyond the scope of ordinary differential equations, the smaller scale RLC system shows how little
an amount of PV cells can be used for electronics that are used day to day
in society, like the tuner circuit. The RLC second order differential equation
allowed for an easier access to the power of mathematical tools, built into
software as well as taught in textbooks abroad. As Man ventures into this
new era of scientific enlightenment, the principles of harmony and the logic
that math attests to must be made a prominent means to the motives of this
and future generations’ progress. The Sun will be shedding free energy upon
the Earth for billions of years so it is a reliable resource where as fossil fuels
are running low. Electricity is a realm that holds up to mathematical models
in the micro and macroscopic domains which is not common in mathematical
modeling of the known universe. With the combination of these; light and
electricity and the mathematics pertaining to them, there are guaranteed to
be massive leaps in technology and the human understanding of the universe.
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9
Appendix
Matlab mfile
function KennysolveRLCwithode45
[t,y]=ode45(@RLC,[0,0.0001],[0;0]);
%I vs t ODE45 graph
plot(t,y(:,1))
xlabel(’time (seconds)’)
ylabel(’Current (Amperes)’)
title(’ODE45 approximation of RLC current second order differential equation’)
shg
pause
%Phase plane graph
figure
plot(y(:,1),y(:,2))
xlabel(’I (current)’)
ylabel(’Iprime’)
title(’Phase Plane for RLC Current and its Derivative’)
shg
pause
%Composite graph
figure
plot3(y(:,1),y(:,2),t)
xlabel(’I (current)’)
ylabel(’Iprime’)
zlabel(’time (seconds)’)
title(’Composite plot for RLC Current and its Derivative’)
shg
pause
%calculating values for Capacitance, eigenvalues, particular constants and
%closed form constants with Initial conditions of I(0)=0 and I’(0)=0
%w=2*pi*105.1*10^3; R=10; L=130*10^-9; C=17.64*10^-6;
15
% lam1=-(R/(2*L))+((((R/L)^2)-(4/(C*L)))^(1/2))/2
%
lam1=-5.669352080553770e+003;
%
% lam2=-(R/(2*L))-((((R/L)^2)-(4/(C*L)))^(1/2))/2
%
lam2=-7.691740757099637e+007;
% a=((6/(C*w))-6*w*L)/((((1/(C*w))-w*L)^2)+R^2)
a=-8.422914076544442e-008;
%
% b=(6*R)/((((1/(C*w))-w*L)^2)+R^2)
b=0.599999999999988;
%
% c1=(-b*w-0.6*lam1)/(lam2-lam1)
c1=0.0051;
%
% c2=a-c1
c2=-0.0051;
%graph closed form
t=linspace(0,0.0001,1000);
W=2*pi*105.1*10^3;
I=(c1)*exp((lam1)*t)+(c2)*exp((lam2)*t)+a*cos(W*t)+b*sin(W*t);
plot(t,I)
xlabel(’time (seconds)’)
ylabel(’Current (Amperes)’)
title(’Closed form solution of RLC current second order differential equation’)
function yprime=RLC(t,y)
yprime=zeros(2,1);
w=2*pi*105.1*10^3; R=10; L=130*10^-9; C=1.7640e-005;
yprime(1)=y(2);
yprime(2)=(6*w/L)*cos(w*t)-(R/L)*y(2)-(1/(C*L))*y(1);
16
References
[1] U.S. Energy Information Administration, International Energy Outlook
2010. http://www.eia.doe.gov/oiaf/ieo/electricity.html, Mar., 2011.
[2] Cool
Earth
Solar,
FAQ:
Frequently
http://www.coolearthsolar.com/faq, Apr., 2011.
Asked
Questions.
[3] Jessika Toothman, Scott Aldous, How Solar Cells
http://science.howstuffworks.com/environmental/energy/solarcell3.htm, Apr., 2011.
Work.
[4] R. Buckminster Fuller, Cheap Thoughts.
http://www.angelo.edu/faculty/kboudrea/cheap/cheap1_f.htm
, Apr., 2011.
[5] John Polking, Albert Boggess, David Arnold, Differential Equations with
Boundary Value Problems. Pearson Education, Inc., Upper Saddle River,
NJ 07458, 2nd Edition, 2006.
[6] Colorado
Solar
Inc.,
Solar
http://www.solarpanelstore.com/, Apr., 2011.
Power
Store.
[7] Sallese J.-M., Krummenacher F., Fazan P., Derivation of Shockley-ReadHall recombination rates. Pearson Education, Inc., Solid-State Electronics, 48 (9) pp. 1539-1548, 2004.
17