Convert the time functions v(t) = 24
cos(377t-45o)V and i(t) = 12 sin(377t +
120o)A to phasors. Irwin. page 405 example 9.6
Phasors are represented in the frequency domain by a complex
number written as a magnitude and an angle in degrees. This is
possible because the frequency of an input signal is not changed
within a circuit. Phasors are written from time functions following
the formulas.
Time Domain
Frequency Domain
For Safe keeping, let's add these conversions to the text
file.
On the TI-92, complex numbers in the form of a magnitude and an
angle are expressed in
notation. Where r is the magnitude, i is the imaginary
operator [2nd] [i] on the TI-92 keyboard, and theta is
the angle in radians. LEAVE THE CALCULATOR IN RADIAN MODE! To
work around this we can force degrees by using the degree operator
[2nd] [D]. In order to get the calculator to return an answer
that we recognize, we can use a function that is a part of the
ThevNor program discussed on the Network
Theorems page. The function is named DegAng. Degang takes a
complex number for an input parameter and returns a string in the
recognizable form of a magnitude and an angle. Since we will be
using this function a lot, lets put it in the custom menu for easy
access.
Find the impedance of a 20-mH inductor when a
sinusoidal voltage source with a frequency of 60 Hz is
applied. Irwin. page 413 example
9.11
The impedance for an inductor is found by applying the formula:
This will make a handy function
From the home screen press [APPS] [7] [3] to open a new
program editing session.
Select function as the type, enter the variable name as
ZL, and press [ENTER] twice.
Use obscure variables to avoid circular definition
errors.
The formula will be added to the text file and added to the
custom menu as well.
Using the function that we just wrote, pass in the parameters.
L equals the inductance.
omega equals 2 * Pi * frequency.
Find the impedance of a 100-microF capacitor
when a sinusoidal voltage source with a frequency of 50 Hz is
applied. Irwin. page 411 example
9.10
The impedance for a capacitor is found by applying the formula:
For completeness let's make this a function as
well
From the home screen press [APPS] [7] [3] to open a new
program editing session.
Select function as the type, enter the variable name as
ZCap, and press [ENTER] twice.
Use obscure variables to avoid circular definition
errors.
The formula will be added to the text file and added to the
custom menu as well.
Using the function that we just wrote, pass in the parameters.
C equals the capacitance.
omega equals 2 * Pi * frequency.
Find and graph all of the voltages and currents
of the circuit. Irwin. page 426 example
9.17
We will solve this entire circuit using nothing more than ohm's
law, voltage division and the parallel function. The only difference
between this circuit and one containing only resistors, is that we
get to use complex numbers.
Using the parallel function we can change this circuit into a
simple circuit containing only one loop. We combine the 8 ohm
resistor and the -j4 ohm capacitor with the j6 ohm
inductor.
We are then left with the resulting circuit.
I1 can now be found easily using Ohm's law. Store in the
variable i1 for graphing later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
V1 can now be found using the VoltDiv function. Store the
answer in the variable v1 for graphing later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Knowing the voltage at V1, I2 can now be found using the Ohm's
law. Store the answer in the variable i2 for graphing
later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Knowing the voltage at V1, I3 can now be found using the Ohm's
law. Store the answer in the variable i3 for graphing
later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Knowing the current through I3, V2 can now be found using the
Ohm's law. Store the answer in the variable v2 for graphing
later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Now that we know all of the voltages and currents, all we have
to do is graph them. Even though we refer to these as being in
phasor form, they are really the same as complex numbers. I have
written a program named VecGraph for graphing complex numbers
as a magnitude and an angle that is available in circuits.92g.
The input parameter for this program is a list of complex numbers.
For easy access, this program has been placed in the custom tool
menu under Prgms. Since we stored the currents under their variable
names, we can enter them as the list.
The graph shows which currents are leading and lagging. Also
this graph is an example of how KCL is still applicable even in
complex circuits. At the node V1: I1 = I2 + I3 Try it with the
program.
The voltages can be graphed the same way. From the home screen:
Select the VecGraph program from the custom menu.
Enter {v1,v2,24e^(i60o)} as
the list of complex numbers.
Press [ENTER].
The graph shows which voltages
are leading and lagging. Also this graph is an example of how KVL is
still applicable even in complex circuits. Around the outside loop:
Vs = V1 + V2 Try it with the program.
Find and graph all the currents of the
circuit. Irwin. page 446 prob
9.28
The first step on this type of problem is to transform the
voltage to phasor form and find the complex impedances of the
elements within the circuit. The voltage source can be transformed
by inspection as having a magnitude of 120 at an angle of
0o. The value of omega (2*Pi*f) from the source is 5000.
Use the ZL(L,w) function to find the impedance of the
inductor with 16E-3 as the value of L and
5000 as the value of w.
Use the ZCap(L,w) function to find the impedance of the
capacitor with 2.5E-6 as the value of C and
5000 as the value of w.
Now the circuit can be redrawn with the voltage source in phasor
form and the impedances in complex form.
Using the parallel function we can transform this circuit into
a circuit with one voltage source and one complex
impedance.
Using Ohm's law, calculate the current for I1. Store this to
the variable i1 for graphing later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Using the CurrDiv function and the complex impedances, find
the current I2 through the capacitor. Store this to the variable
i2 for graphing later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Using the CurrDiv function and the complex impedances, find
the current I3 through the capacitor. Store this to the variable
i3 for graphing later.
With the custom menu in place press [F3] and
[ENTER] twice to use the DegAng function to convert the
complex number to phasor form.
Use the VecGraph program to graph the
currents.
The graph of the currents show which currents are leading and
lagging. Also notice that the current through I2 is purely
imaginary. Once again the graph shows how KCL still holds true even
for complex circuits.
notation. Where r is the magnitude, i is the imaginary
operator [2nd] [i] on the TI-92 keyboard, and theta is
the angle in radians. LEAVE THE CALCULATOR IN RADIAN MODE! To
work around this we can force degrees by using the degree operator
[2nd] [D]. In order to get the calculator to return an answer
that we recognize, we can use a function that is a part of the
ThevNor program discussed on the 
