ECE 2006 University of Minnesota Duluth Lab 4 ECE Department Page 1 February 12/14, 2007 Equivalent Equipment Circuits 1. Introduction The student will analyze the internal properties of the equipment used in lab. The input resistance of the oscilloscope and Digital MultiMeter (DMM) 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 equivalents of a complex circuit using PSpice®. 2. 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. 2.1. Root Mean Square (RMS) The current and voltage in alternating current (AC) systems is not constant. Thus, one cannot easily apply ohm’s 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. If one desires to use Ohm’s law to analyze a circuit with an AC source, RMS values for voltage and current must be calculated or measured. An equation for this value can be derived by examining the average power. For resistive circuits, the instantaneous power can be expressed as: RtvtitvtPinst)()()()(2== When the voltage is periodic (e.g. a sinusoidal wave), the average power can then be expressed as: ∫+==TttavgdtRtvTperiodoverRtvavgP00)(11)(22 If the input waveform is periodic, the average power will be constant. As a result, there is a DC voltage which would produce the same average power as a given periodic waveform.ECE 2006 University of Minnesota Duluth Lab 4 ECE Department Page 2 February 12/14, 2007 This voltage is termed the Root-Mean-Square voltage—VRMS. Since VRMS is a constant voltage, the calculation of the average power is simply, =RVPRMSavg2 Setting the two average power equations equal to each other and solving for VRMS produces: ∫+=TttRMSdttvTV00)(12 which is the square root of the mean of the squared voltage (hence the name). For single sinusoidal voltages, this can be shown to be the amplitude (A) of the sinusoid divided by the square root of two, i.e. 2AVRMS= This can be shown graphically by noting the average of the square of a sinusoid is simply 0.5A2 (since the maximum value of the waveform is A2, the minimum is zero, and the waveform spends equal amounts of time above and below the average value). Taking the square root of this then gives us the previous equation. 2.2. Thevenin and Norton Equivalents Thevenin’s theorem states that a two terminal circuit can be replaced by an equivalent circuit consisting of a voltage source, VTH, in series with a resistor, RTH, where VTH is the open-circuit voltage, VOC, at the terminals and RTH is equivalent to the resistance at the terminals when all independent sources are turned off (see the equations below for the calculation of the resistance when the circuit contains both independent and dependent sources. Norton’s theorem states that a two terminal circuit can be replaced by an equivalent circuit consisting of a current source, IN, in parallel with a resistor, RN, where IN is the short-circuit current, ISC, through the terminals and RN is the input or equivalent resistance at the terminals when the independent sources are turned off. Mathematically these relationships can be described as in following equations: VTH = VOC IN = ISC RIN = RTH = RN = ( VTH / IN ) RTH VTH IN RNECE 2006 University of Minnesota Duluth Lab 4 ECE Department Page 3 February 12/14, 2007 Figure 1: Input Resistance Measurement 3. Procedure 3.1. Equipment • Tektronix TDS 3012B Digital Phosphor Oscilloscope • Agilent E3631A DC Power Supply • Agilent 33120A Waveform Generator • Fluke 8050A Digital MultiMeter (DMM) • Resistors as Needed 3.2. Internal Resistance of the Oscilloscope Connect the oscilloscope directly to the DC power supply. In this manner the circuit in Figure 1 is constructed with the variable resistance Rv set to 0 volts and Ri denoting the internal resistance of the oscilloscope. Adjust the DC power supply until the oscilloscope measures 8 volts. Record the corresponding value of the DC power supply in Table 1. Select a nominal 10 MΩ resistor and measure its resistance using the Digital MultiMeter (DMM). Record this value in Tables 1 and 2. Using the 10 MΩ resistor as RV, construct the circuit displayed in Figure 1 and measure the voltage across the oscilloscope terminals Vi (with the same value for Vps). Record this value in Table 1. Knowing three of the four variables of the circuit displayed in Figure 1, calculate the internal resistance of the oscilloscope (denoted as Ri in Figure 1) and record this value in Table 1. Include this calculation in lab write up. 3.3. Internal Resistance of the MultiMeter Used as a Voltmeter Set the DMM to measure voltage and connect it directly to the DC power supply. In this manner the circuit in Figure 1 is constructed with the variable resistance Rv set to 0 volts. Adjust the DC power supply until the DMM reads 16 volts. Make sure the DMM is set to measure DC, not AC voltages. Record the corresponding value of the DC power supply in Table 2. Using the (nominally) 10 MΩ resistor as RV, construct the circuit displayed in Figure 1 and measure the voltage across the DMM terminals Vi. Record this value in Table 2. Knowing three of the four variables of the circuit displayed in Figure 1, calculate