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Transcript of Phasors and Kirchoff’s Current Law - Virginia Tech · Kirchhoff’s Current Law Week 3:...
Phasors and Kirchhoff’s Current Law
Week 3: Experiment 23
Changes to Experiment
• Frequency of operation: 40 kHz – You will have to use the arbitrary function generator on the Velleman
scope
• Amplitude of voltage supply: 5 V • Inductor: 10 mH • Capacitor: 2.2 nF • R1: 4 kW
• R3: 2 kW
• Pick R2 and R4 (shunt resistors) appropriately. – Be aware that the stored energy in the inductor could send amount of
current out of phase with the voltage back into the scope if incorrect components are used (XC appraches -j ∞ W).
• Reference all phase measurements to the phase of the voltage supply.
Additions to Experiment
• Analysis
– Calculate the currents i1, i2, and i3 in rectangular and phasor notation.
• This can be by hand or using a mathematical program such as Excel or MATLAB
– Plot the currents as a function of time to show the effect of the different phase angles.
• Two complete cycles should be displayed for at least one of the currents.
• Measurements
– Import data from the oscilloscope into MATLAB.
– Plot i2 and i3 as a function of time and overlay plots of the expected i2 and i3 from your analysis.
Measurement Issues
• Thevenin Equivalent Circuit
• Amplitude vs. Voltage Peak-to-Peak
• True RMS
Thevenin Equivalent Circuit
• The arbitrary function generator can be modeled as a 50mV-5V source with an internal resistance of 40 W.
– Consider this when trying to deliver power to your circuit (Rload)
Function Generator
Thevenin Equivalent Circuit
• The oscilloscope can be modeled as a 1MW
load resistor in parallel with your circuit.
Your Circuit
Vpp
Amplitude and Peak-to-Peak Voltage
VM
RMS – Root Mean Square
T
RMS
T
RMS
dttiT
I
dttvT
V
0
2
0
2
)(1
)(1
RMS vs. True RMS
• RMS of a sinusoid is 0.707 VM
– Some instruments assume that the voltage measured is always a sinusoid
• Output RMS values are wrong for all other waveshapes – This is what your digital multimeter does.
– True RMS, which is what the Velleman outputs, is calculated using the equation on the previous slide.
New Circuit
Transient Analysis
• Source: Vsin
– Instructions state that you need to wait a few cycles before making any measurements from the plot.
• This is because the capacitor has an initial condition of 0V (no charge stored on the electrodes).
– This can be changed as IC (initial condition) is an attribute in the capacitor model.
» In certain circuits, the capacitor and inductor in a circuit can store energy extremely efficiently (i.e., the time constant of the circuit is much shorter than 1/f).
Transient Plot
Automatically generated when current markers are placed in the schematic.
To Obtain Smooth Curve
Set the Step Ceiling to a small
fraction of T, the period of one cycle.
Bode Plots
• Phase angles can be determined from PSpice by:
– Measuring the difference in the zero crossing of the voltage from the arbitrary function generator and the DUT using the transient analysis
– Displaying the phase angle on a plot generated during an AC Sweep.
• Note that the voltage source must be changed from Vsin to Vac.
• P() marco will display the phase angle of the parameter inserted [e.g., V(R2:2)]
Schematics
• PSpice Schematics uses superposition when performing the AC Sweep. So, both voltage sources may be put into the same circuit.
– You select which plot is generated during the simulation run.
To Plot Phasor Information
Add a Trace to the New Plot
Select P() in the List of Functions or Macros
Select Voltage or Current from the List of Simulation Output Variables
• It will appear within the paraphrases as the argument of the phase function. You can add multiple traces at once by putting a comma between each on the list at the bottom of the pop-up window. Then, click OK.
To Change the x axis to Log(f)
Phase Angle in Degrees Vs. Frequency
The angle in phasor notation should be between -180o to +180o.
Phase Angle Measurement
• Two techniques using the Velleman scope
– Waveform Parameters
• Measurement of relative phase to internal reference at the operating frequency of the arbitrary function generator.
– Bode Plot
• Measurement of the phase of the signal on Channel 1 with respect to the signal on Channel 2 over a frequency range specified by the user.
Waveform Parameters
Bode Plot
Select Phase Plot
from View menu
after having set the
Frequency Range,
Frequency Start,
and other
measurement
parameters and
then click Start to
obtain the phase
measurement.
Natural Frequency of Circuit
• The specified frequency of operation of the voltage source is close to the natural frequency of the RLC network.
– If a sharp square wave was obtained from the arbitrary voltage source, you would be able to see the ringing associated with the energy transfer between the inductor and capacitor before the system reached steady-state.
Transient Response
If you wanted to, you could
look at the transient
response to a square wave
input (i.e., see the ringing
associated with the natural
and forced response of this
RLC circuit), by adding
Vpulse to the circuit.
Set the amplitude of Vsin
to 0V. Then set the
amplitude of Vpulse to 5V
and the PW to 100us and
PER to 200us. The plot of
the response for a similar
circuit is shown on the
following slide.
Transient Response: Square Wave Input
Voltage at the node
after the inductor.
Currents (in mA)
through the inductor,
capacitor, and R1.