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EQUIVALENT EQUIPMENT CIRCUITS

INTRODUCTION

The student will analyze the internal properties of the equipment
used in lab.  The input resistance of the oscilloscope and digital
multimeter when used as a voltmeter will be measured.  The output
resistance of the function generator will similarly be determined. 
The student will also determine the Thevenin and Norton equivalent a
complex circuit using SPICE.

BACKGROUND

When an electrical instrument is connected to a circuit to provide
power or take measurements it becomes part of the circuit.  Often
the resistance of the connected instruments is neglected as they have
been designed to not interfere with most circuits. Even though electricity
flows through multiple elements inside of the instrument, these components
may be modeled as a simple resistor or resistor and source.

To determine the internal resistance of an instrument it is usually
only necessary to vary a single component of an exterior connected circuit. 
Enough measurements are available throughout the exterior circuit to
provide information for basic circuit analysis techniques to calculate
the internal properties of an instrument.

Root Mean Square (RMS) 

The current and voltage in alternating current (AC) systems is not
constant.  Thus, one cannot easily apply ohms law to a circuit
with an AC source.  If one thinks of a resistive element, current
traveling forward will heat the element up just as much as current traveling
backwards.  Taking the average of the absolute value of an AC waveform
will result in the DC equivalent.  If one desires to use ohms
law to analyze an circuit with an AC source, RMS values for voltage
and current must be calculated or measured.

Thevenin and Norton Equivalents

R<sub>TH

V<sub>TH

Thevenins theorem states that a two terminal circuit can be replaced
by an equivalent circuit consisting of a voltage source V<sub>TH
in series with a resistor R<sub>TH where V<sub>TH is the
open-circuit voltage V<sub>OC at the terminals and R<sub>TH
is equivalent to the resistance at the terminals when all independent
sources are turned off. 

R<sub>N

I_N</span><span class="Normal--Char" style=" font-family: 'Arial', 'Arial';
">Nortons theorem states that a two terminal circuit can be replaced
by an equivalent circuit consisting of a current source I<sub>N
in parallel with a resistor R<sub>N, where I<sub>N is the
short -circuit current I<sub>SC through the terminals and R<sub>N
is the input or equivalent resistance at the terminals when the independent
sources are turned off.

Mathematically these relationships can be described as follows:

V<sub>TH
= V<sub>OC  I<sub>N = I<sub>SC  R<sub>IN
= R<sub>TH = R<sub>N = ( V<sub>TH / I<sub>N
)

PROCEDURE

Determining the internal resistance of the oscilloscope

Figure 1:  Input Resistance Measurement

Connect the oscilloscope directly to the DC power supply.  In
this manner the circuit in Figure 1 is constructed with the variable
resistance R<sub>v set to 0 volts and R<sub>i denoting the
internal resistance of the oscilloscope.

Adjust the DC power supply until the oscilloscope measures 8 volts.

V<sub>PS
= _______ Volts

Select a nominal 10 M resistor and measure its resistance using
the digital multimeter.

R<sub>V
= ________ Ohms

Using the nominal 10 M resistor as R<sub>V, construct
the circuit displayed in Figure 1 and measure the voltage across the
oscilloscope terminals V<sub>i.

V<sub>i
= _________ Volts

Knowing three of the four variables of the circuit displayed in Figure
1, calculate the internal resistance of the oscilloscope, denoted as
R<sub>i in the figure.  Include calculation in lab report.

Oscilloscope
Internal Resistance = R<sub>i
= _________ Ohms

Determining the internal resistance of the multimeter
used as a voltmeter

Set the multimeter to measure voltage and connect directly to the
DC power supply.  In this manner the circuit in Figure 1 is constructed
with the variable resistance R<sub>v set to 0 volts.

Adjust the DC power supply until the multimeter reads 16 volts. 
Make sure the multimeter is set to measure DC, not AC voltages.

V<sub>PS
= _______ Volts

Record the measured value of the nominal 10 M resistor used
in the previous section.

R<sub>V = ________ Ohms

Using the nominal 10 M resistor as R<sub>V, construct
the circuit displayed in Figure 1 and measure the voltage across the
multimeter terminals V<sub>i.

V<sub>i
= _________ Volts

Knowing three of the four variables of the circuit displayed in Figure
1, calculate the internal resistance of the multimeter, denoted as R<sub>i
in the figure.  Include calculation in lab report.

Multimeter
Internal Resistance = R<sub>i
= _________ Ohms

Measuring the output resistance of the function
generator

Figure 2:  Output Resistance Measurement

Connect the digital multimeter directly to the function generator. 
In this manner the circuit in Figure 2 is constructed with the load
resistance R<sub>L set to 0 volts.  We will neglect the internal
resistance of the voltmeter for the purposes of this experiment and
assume that the circuit is open, thus V<sub>O is equal to V<sub>fg
with R<sub>L removed.

Adjust the function generator to a sinusoidal frequency of 50Hz. 
Adjust the amplitude until the digital multimeter displays 1.6 volts. 
Make sure the digital multimeter is set to measure AC RMS, not DC.

V<sub>fg = _________ Volts RMS

Select a nominal resistor of about 50 and measure its resistance
using the digital multimeter.

R<sub>L
= __________ Ohms

Using the 50 resistor as R<sub>L, construct the circuit
displayed in Figure 2 and measure the voltage displayed on the digital
multimeter V<sub>O.

V<sub>O
= __________ Volts RMS

Knowing three of the four variables of the circuit displayed in Figure
2, calculate the output resistance of the function generator, denoted
as R<sub>O in the figure.  Include calculation in lab report.

Function
Generator Output Resistance = R<sub>O
= _________ Ohms

Norton and Thevenin Analysis by means of SPICE

Figure 3:  DC Circuit for SPICE Analysis

Assume that th