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.