Download Modeling an RLC circuit`s current with Differential Equations as well

Modeling an RLC circuit’s current with
Differential Equations as well as a general
overview of the mathematics that describe Solar
Power
Kenny Harwood
April 29, 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. There is still no way to categorize light as a particle or
wave so scientist have made its own niche as a matter-wave. And if that
property of light doesn’t make the brain squeamish, how about light’s
ability to be converted into electricity? 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.” This
paper will be giving an abbreviated description of the mathematics that
model the photovaltaic effect (solar power production process) and then
for the differential equations aspect, an RLC circuit will be modeled that
is powered by a photovaltaic panel consisting of 12 solar cells that has
its output voltage passed through an inverter to produce an AC output
signal where voltage becomes a sinusoidal function of time. The RLC
parallel will have it’s current expressed in mathematical termsby deriving
a second order differential equation from the equation
Z
V = Ldi/dt + Ri(t) + 1/C ∗ ((Qo ) + (i(t)dt)
that describes the circuit’s components as functions of time.
Also contained in this paper is some background on the mathematical
process in which sunlight is transformed to electricity by giving a basic
outline of Recombination and the Shockley Diode Equation
I = Is (eVD /nVt − 1)
that gives the electric current flow due to sunlight.
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A Means to Produce Power
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1
picture
The Sun has showered this blue speck that has been called home for billions
of years, showering free energy upon this planet’s surface unconditionally. However, only recently has solar power been given the spotlight across this globe
and solar panels 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 were produced globally[1].In comparison, the sunlight received on
the Earth’s surface in 1 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.
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The S.R.H process
In the mid 1900’s, a model for mathematically 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 published by the title ’Shockley-Read-Hall Recombination’. How stuff
works.com has a wonderful explanation of the recombination process and the
next bit of this section is taken from their Solar Cell page. 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.
Do all the free electrons fill all the free holes? No. If they did, then the whole
arrangement wouldn’t be very useful. However, right at the junction, they do
mix and form something of a barrier, making it harder and harder for electrons
on the N side to cross over to the P side. Eventually, equilibrium is reached,
and we have an electric field 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.
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.
There are a few more components left before the PV cell can be used. 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] *NOTE: equations from S.R.H article must be added to
the corresponding steps described above to be summated in the Shockley Diode
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equation
I = Is (eVD /nVt − 1)
Applying Free Electricity
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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 and modeled where as the circuit tunes
to the frequency of the Flathead Valley’s own Kool 105.1 FM station. First,
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 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.
Assumptions:
1.)The Solar panel has its DC signal transformed to AC through an inverter before
being connected to the RLC circuit.
2.)Voltage efficiency of the PV cells do not waver. (constant voltage peak for AC
signal function)
3.)Frequency of AC signal is set at 105.1 MHz
4.)All wires in setup are ideal, therefore offering negligible resistance to current.
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 .5 Volts per cell)
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 MHz, which will in turn tune the
circuit to pickup and amplify the AC signal that oscillates at the frequency of Kool
105.1 FM. In the AC signal equation where NOTE: will not texify with dollar signs so
this section has been commented out until help is available
In Search of an ODE
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To reach the ordinary differential
R equation needed to model the RLC circuit, V =
Ldi/dt + Ri(t) + 1/C ∗ ((Qo ) + (i(t)dt) must be differentiated. Therefore, V has
been been replaced with VAC found above, and Qo (initial charge of capacitor) being
assumed to equal zero will make the voltage equation for the RLC circuit
Z
Vpeak ∗ sin(ω ∗ t) = Ldi/dt + Ri(t) + 1/C ∗ (i(t))dt
taking the derivative of both sides of the equation with respect to t, the Second Order
differential equation is found to be;
L ∗ d2 i/dt2 + R ∗ di/dt + 1/C ∗ i(t) = ω ∗ cos(ω ∗ t)
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 = 2π ∗ f .
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Analyzing Circuit for Numerical Values of
Circuit Components
Now that there is a basic understanding of how light is transformed into electricity,
we can move on to the meat of this paper and model the electric current in an RLC
parallel circuit, also known as a ”tuning” circuit or band-pass filter. 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
Yet, the ODE cannot be solved until constant values are found for the components
in an RLC circuit. In this model, the components of this circuit are needing to resonate
the frequency of the AC signal to the Kool 105.1 FM radio station. Using Multilab??
I have calculated the values of the resistor and capacitor with the given inductor’s
inductance of 130 nH. I accomplished this by creating a pole-zero diagram for the
RLC circuit and found where the resistance and capacitance values resonate the Kool
105 frequency of 105,100,000 Hz.
the values of the components for this model are as follows: create table?
L=130 nH for VHF (FM)
f =105.1 MHz
C=?
R=?
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The derivation of my ODE to find the current equation for the LED circuit
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Now that the values for the RLC components have been found, the ODE is ready to
be solved by MATlab. By inputting
L ∗ d2 i/dt2 + R ∗ di/dt + 1/C ∗ i(t) = ω ∗ cos(ω ∗ t)
Matlab computed that
I(t) =?
which is the solution to the current equation in this RLC resonating circuit. By
graphing I with respect to t, *FIGURES and discuss and graph dfield of ODE with
x1=i(t) x2=i’(t)=x1’ and i”t=x2’, discuss behavior of current as initial conditions are
changed. also use P=V*I to discuss and possibly graph power of the signal as V and
I change with respect to time.
Curtain Call
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should this be at the beginning Don? probably...
Relevant equations
My ODE with AC driving signal
L ∗ d2 i/dt2 + R ∗ di/dt + 1/C ∗ i(t) = ω ∗ cos(ω ∗ t)
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 = 2π ∗ f .
Current related to Power and Voltage
I = P/V
Voltage related to Current and Resistance
V =I ∗R
Shockley Diode Equation
I = Is (eVD /nVt − 1)
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, also known as the quality factor or
sometimes emission coefficient. The ideality factor n varies from
1 to 2 depending on the fabrication process and semiconductor
material and in many cases is assumed to be approximately equal
to 1 (thus the notation n is omitted).
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Bibliography or output on photovaltaic cell http://www.solarpanelstore.com/
citation1http : //www.eia.doe.gov/oiaf /ieo/electricity.html citation2http : //www.coolearthsolar.com/f aq
citation3http : //science.howstuf f works.com/environmental/energy/solar−cell3.htm
citationf orR.BuckminsterF ullerquotehttp : //www.angelo.edu/f aculty/kboudrea/cheap/cheap1f .htm
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