Transcript MatLAB program - Virginia Tech

MatLAB program
Experiment 15 A Series RC Circuit
Capacitors
Energy Storage Devices
Capacitors
• Composed of two conductive plates separated
by an insulator (or dielectric).
– Commonly illustrated as two parallel metal plates
separated by a distance, d.
C = e A/d
where e = er eo
er is the relative dielectric constant
eo is the vacuum permittivity
Types of Capacitors
• Fixed Capacitors
– Nonpolarized
• May be connected into circuit with either terminal of
capacitor connected to the high voltage side of the
circuit.
– Insulator: Paper, Mica, Ceramic, Polymer
– Electrolytic
• The negative terminal must always be at a lower
voltage than the positive terminal
– Plates or Electrodes: Aluminum, Tantalum
Nonpolarized
• Difficult to make nonpolarized capacitors that
store a large amount of charge or operate at
high voltages.
– Tolerance on capacitance values is very large
• +50%/-25% is not unusual
PSpice Symbol
http://www.marvac.com/fun/ceramic_capacitor_codes.aspx
Electrolytic
Pspice Symbols
Fabrication
http://www.digitivity.com/articles/2008/11/choosing-the-right-capacitor.html
Types of Capacitors
PSpice Symbol
• Variable
http://www.tpub.com/neets/book2/3f.htm
• Electric double layer capacitor
– Also known as a
supercapacitor or ultracapacitor
http://en.wikipedia.org/wiki/File:Supercapacitor_diagram.svg
Electrical Properties of a Capacitor
• Acts like an open circuit at steady state when
connected to a d.c. voltage or current source.
• Voltage on a capacitor must be continuous
– There are no abrupt changes to the voltage
• An ideal capacitor does not dissipate energy, it
takes power when storing energy and returns
it when discharging.
Energy Storage
• Charge is stored on the plates of the capacitor.
Equation:
Q = CV
Units:
Coulomb = Farad.Voltage
C=FV
Adding Charge to Capacitor
• The ability to add charge to a capacitor
depends on:
– the amount of charge already on the plates of the
capacitor
and
– the force (voltage) driving the charge towards the
plates (i.e., current)
Charging a Capacitor
• At first, it is easy to store charge in the capacitor.
• As more charge is stored on the plates of the
capacitor, it becomes increasingly difficult to
place additional charge on the plates.
– Coulombic repulsion from the charge already on the
plates creates an opposing force to limit the addition
of more charge on the plates.
• Voltage across a capacitor increases rapidly as charge is
moved onto the plates when the initial amount of charge on
the capacitor is small.
• Voltage across the capacitor increases more slowly as it
becomes difficult to add extra charge to the plates.
Discharging a Capacitor
• At first, it is easy to remove charge in the capacitor.
– Coulombic repulsion from the charge already on the plates
creates a force that pushes some of the charge out of the
capacitor once the force (voltage) that placed the charge in the
capacitor is removed (or decreased).
• As more charge is removed from the plates of the capacitor,
it becomes increasingly difficult to get rid of the small
amount of charge remaining on the plates.
– Coulombic repulsion decreases as the charge spreads out on the
plates. As the amount of charge decreases, the force needed to
drive the charge off of the plates decreases.
• Voltage across a capacitor decreases rapidly as charge is removed
from the plates when the initial amount of charge on the capacitor is
small.
• Voltage across the capacitor decreases more slowly as it becomes
difficult to force the remaining charge out of the capacitor.
Capacitor Voltage vs. Time
d.c. voltage, Vc, is applied at t = 0s
d.c. voltage, Vc, is removed at t = 0s
Time constant, t
• The rate at which charge can be added to or
removed from the plates of a capacitor as a
function of time can be fit to an exponential
function.
Charging
Discharging
Transition to steady state
• We approximate that the exponential function
reaches its final value when the charging or
discharging time is equal to 5t.
Vc is not a d.c. voltage source
• If the time at which a voltage source applied
to the capacitor circuit alternates between Vc
and 0V is more than 5t, the capacitor fully
charges or discharges before the value of the
voltage source changes.
Period, T, is the time required to cycle the voltage
source on and off. Duty cycle, D, is the fraction (or
percentage) of time that the voltage source is on
divided by T.
Example 1
• R = 500 W and C = 0.1mF
– t = 50ms
• Voltage source changes from 0V to 5V (square wave) with T
= 1ms (f = 1000 Hz) and D = 0.5 (50%)
– T > 5t
Example 2
• R = 500 W and C = 0.1mF
– t = 50ms
• Voltage source changes from 0V to 5V (square wave) with T = 100ms
and D = 0.5 (50%)
– T < 5t
Insufficient time to fully charge C
• A few cycles are required to reach steady state
conditions for charging and discharging C.
– More generic equations - assumes D = 50%
Charging
Discharging
MatLAB program
• User must enter values for R, C, and f of voltage
source
• Calculate t and T
• Perform piecewise calculation
– Calculate the voltage and current of the capacitor as it
charges (time, t = 0s to 0.5T)
– Then calculate the voltage and current of the
capacitor as it discharges (time, t = 0.5T to T)
• Plot the voltage and current through the
capacitor for 3 periods
Enter Data
• To obtain information from the program use
• A= input (‘Message to user’,);
– Where A is the name of the variable
Scalar vs. Vector
• Scalar number is a single quantity
–X=3
• X is a scalar quantity
• Vector is a string of numbers
– Y = [9 12 15]
• Y is a vector where the first element is 9, the second is
12, and the third element is 15
– Z=[3:-1:1]
• Z is a vector where the first element is 3, the second is
2, and the third element is 1
Mathematical Operations
– Scalar mathematical operations
• normal *, +, -, / signs
– The result of 5*X is 15
– The result of Y/X is 3 4 5
– Vector operations
• +, -, .*, and ./
– The result of Z.*Y is 27 24 15
» The multiplication of the first element of Z with the first
element of Y, then the multiplication of the second element of
Z with the second element of Y, and finally the multiplication of
the third element of Z with the third element of Y
• The lengths (or number of elements) of the two vectors
must be equal
If Function
• The if statement evaluates a logical expression
and executes a group of statements when the
expression is true.
– The optional elseif and else keywords provide for
the execution of alternate groups of statements
when the first statement is false.
– An end keyword, which matches the if, terminates
the last group of statements.
• The groups of statements are delineated by the four
keywords—no braces or brackets are involved.
FOR Function
• The for loop repeats a group of statements a
fixed, predetermined number of times.
– A matching end completes the for loop.
Example 3
if X >= 3
% if X is greater than or equal to 3,
Y = Y+Z % the next statement is performed
else Y = Y-Z % otherwise this statement is performed
end
Since X = 3, the result is Y = 12 14 16
Example 4
if X==2
Y=Y.*Z
% if X equals 2, the next statement is performed
% if not, the next statements are checked
elseif X > 3 % if X is >3, next statement is performed
Y=Y+Z
else Y = Y - Z % otherwise, this calculation is performed
end
Since X = 3, the result is Y = 6 10 14
Example 5
for i=1:2
% two loops through the next statement
Y=Y.*Z
% multiplication of the 2 vectors defined
End
The result is:
Y = 27 24 15
at end of the 1st time through the loop
Y = 81 48 15
at end of the 2nd time through the loop
Plotting Functions
plot(Y)
graphs the columns of Y versus the element
number (index)
plot(Y,Z);
title('Z vs Y')
xlabel('Z (time)');
ylabel('Y(volts)');
grid on
% Y is the x-axis & Z is the y-axis
%Inserts a title on the plot
%Inserts a x-axis label on the plot
%Inserts a label for the y-axis
% Causes a grid to appear on the plot
MatLAB Program Assignment
1.
Plot a graph of vC(t) given by Eqs (2) and (3) versus t
–
–
–
2.
3.
for the time constant calculated from the resistor and capacitor values
entered by the user*
And a voltage source cycles between 0V and 5V with a period of 1ms and
50% duty cycle
Also show the clock signal on the same graph.
Plot a graph of iC(t) versus t using Eqs (5) and (6), for the time constant
calculated in step 1.
Plot a graph of vC(t) using Eqs (7) and (8).
• Equations are listed in Experiment 15 of the lab manual
• All plots should show three full clock periods.
• All plots should also include a plot of the clock signal on the same graph.
* Note: This is different from the lab manual instructions.