CIRCUITS LABORATORY EXPERIMENT 2

Transcription

1 CIRCUITS LABORATORY EXPERIMENT 2 The Oscilloscope and Transient Analysis 2.1 Introduction In the first experiment, the utility of the DMM to measure many simple DC quantities was demonstrated. However, in the vast majority of electrical circuits having practical use, the quantities of interest vary with time. While the DMM can be used to make measurements on such signals (and you will be introduced to its use for this purpose in future experiments), it is often desired to visually observe such signals as a function of time. By converting time into distance on the face of a cathode ray tube (CRT), signals much faster than the human eye can follow can be displayed. The resulting instrument is an oscilloscope whose unique features are invaluable to the electrical engineer in activities that range across the whole spectrum of the profession. This experiment describes the oscilloscope and illustrates its use with several exercises involving series RC, RL, RLC circuits plus an electrical relay. 2 1

2 2.2 Objectives At the end of this experiment, the student will be able to: (1) use the oscilloscope to measure the voltage, period, and rise/fall times of periodic signals, (2) adjust the oscilloscope probe so that it gives optimum response at high frequencies (3) sketch the equivalent circuit of the oscilloscope plus probe connection for both the 1x and l0x probes, (4) choose the best probe to use for a given signal of interest, (5) explain how the equipment in our lab is grounded, (6) analyze the transient response of series RC, RL, and RLC circuits, (7) design a circuit to determine the coil inductance of an electrical relay, and (8) use the oscilloscope to measure the switching times of a Single Pole Single Throw (SPST) electrical relay. 2.3 Theory General Considerations The cathode ray tube (CRT) is one of the most useful electronic devices. It is familiar to all of us in its everyday use as a television display and also as a device for displaying alphanumeric information when used as a computer monitor. Usage of the term 'cathode ray' dates from Faraday (1865) who observed luminescence between electrodes in a rarefied atmosphere. A number of early day scientists including 2-2

3 Crookes and Thomson worked to establish that 'cathode rays' were actually electrons moving at high speed in a vacuum. When the moving electrons impinge on a 'phosphor' material, in particular zinc sulfide, the electron kinetic energy is partially converted to light energy and the phosphorescence produced can be utilized for display purposes. The term 'phosphorescence' refers to the production of light after excitation as contrasted to 'fluorescence' which implies light production during excitation. Most of the light output from CRTs is due to phosphorescence. Phosphors differ widely in their 'persistence'; a P16 phosphor decays to the 10% level within one microsecond while a P26 phosphor requires 16 seconds to reach this level. The efficiency with which a CRT converts electrical energy to light is rather good, perhaps 50 lumens/watt for some phosphors, and may be two or more times larger than a good incandescent lamp. A moving electron experiences a force from both electric and magnetic fields. This is the 'Lorentz force' of amount F = q(e + v x B) (2.1) This leads to two types of CRTs, which differ in their deflection method. Tubes used for TV display and computer monitors employ magnetic deflection of the electron beam while tubes for oscillographic purposes use electrostatic deflection. The basic structure in the two cases is shown in Figure 2.1. Several reasons exist for this preference in the two applications. For one, magnetic deflection allows a substantially wider linear deflection angle to be realized which is desirable for large screen displays, as in a television receiver. On the other 2-3

4 hand producing an appropriate, relatively large, deflection current and forcing it through deflection coils requires inordinately high voltages at higher frequencies and indeed may not be possible at frequencies above which the deflection coil is self resonant. Commonly then, tubes for oscillographic purposes are relatively long and narrow so that precision in the amount of deflection per unit voltage applied to the deflection plates is obtained. The electron beam itself is produced by a heated cathode which is held at a negative potential, some several hundred volts below the potential of an accelerating electrode. Beam focusing also takes place in the accelerating electrode after which the beam passes through the deflection plates. Further acceleration occurs due to a large potential difference (the 'high voltage supply') between the phosphor screen and the accelerating electrode. Depending on the secondary emission characteristic of the phosphor, this potential may range up to 20 kv. Generally speaking, 2-4

5 there is a range of electron energies where, if the electrons impinge on a phosphor, more electrons are ejected from the phosphor than impinge upon it. Consequently, a potential difference cannot be maintained between phosphor and cathode that exceeds the upper extreme of the range over which the secondary emission ratio exceeds unity. Although screen brightness can be varied by adjusting the phosphor potential, more commonly the intensity of the beam is varied by a grid operated negative relative to the cathode so as to reduce or cut off the number of electrons reaching the phosphor per unit time. This grid represents the 'z-axis' input, as it is commonly called in oscillography, and allows intensity modulation of the CRT with an external electrical signal. The main function of an oscilloscope is to display, as a function of time, the waveform of a voltage applied to its terminals. Since the deflection sensitivity of a CRT, defined as the deflection plate voltage required for the beam to trace out unit distance on the face of the CRT, is on the order of 30 Volts/inch, substantial amplification of the input signal is required to obtain a usable deflection and this is obtained via the vertical amplifier in the oscilloscope. Figure 2.2 (a) shows a partial block diagram of a conventional analog oscilloscope. The diagram omits such details as attenuators and some of the switching circuits. Within the last few years, the analog oscilloscope, as described above, has been largely displaced by the digital oscilloscope. In contrast to the analog scope, which displays its trace as it is generated, the digital scope collects data for the entire waveform and then displays it. Figure 2.2 (b) shows a block diagram of a digital ` 2-5

6 oscilloscope. The function of the vertical system in this figure is to adjust the amplitude of the signal to place it in the range of the analog-digital converter in the Figure 2.2 (a): Partial block diagram of a conventional analog oscilloscope. acquisition system block. In the acquisition system the analog-digital converter samples the signal at discrete points in time and converts the signals voltage at these points to digital values called sample points. These are then stored in memory to be later displayed. The horizontal systems sample clock determines how often the converter takes a sample. The rate at which the clock operates is called the sample rate and is measured in samples/second. When stored in memory, the sample points become waveform points. More than one sample point may make up a waveform point using, for example, averaging methods. Together, the waveform points make 2-6

7 up one waveform record. The number of waveform points used to make a waveform record is called the record length. The trigger system determines the start and stop points of the record. The display receives these record points after they are stored in memory. Fig. 2.2 (b): Digital Oscilloscope Block Diagram As you will notice when you begin this experiment, the oscilloscope is a com- plicated instrument with a wide range of features. In this course, you will use the Agilent 54622D Digital Oscilloscope. 2 7

8 2.3.2 Making Measurements Using the Oscilloscope The input impedance of the vertical and horizontal amplifiers (see Figure 2.2) of a typical oscilloscope is high. This serves to minimize the loading of a circuit under test and the consequent distortion of the observed waveform. In our lab oscilloscopes, the input impedance can be represented by an equivalent circuit which is a 1 MΩ resistor in parallel with a 14 pf capacitor for both the vertical and horizontal amplifiers. This equivalent circuit is shown in Figure 2.3, where v i (t) is the waveform at the input terminals of the oscilloscope. Note that this input impedance can cause objectionable noise and interference on the display if measurements are made on a high impedance circuit via unshielded wires. Furthermore, ordinary coaxial cable has a capacitance on the order of 14 pf/ft so shielded cable effectively adds capacitance in parallel with the 14 pf inherent in 14 pf the instrument. Nevertheless, if the impedance level of the circuit being measured is low, say less than 10 kω, and if the rise and fall times being measured are not too 2-8

9 fast, say greater than roughly lμs, then shielded cable or a lx Probe may be used to connect the oscilloscope to the circuit. If the circuit conditions lie outside these ranges, then the l0x probe gives better performance at some sacrifice in sensitivity. The equivalent circuit of the l0x probe is shown in Figure 2.4. The l0x probe presents an effective impedance to the circuit of about 10 MΩ in parallel with less than 14 pf of capacitance (depending on the capacitance of the connectors and the probe wire) and so introduces less error in the measuring process than does the 1x probe. It should be noted though that the sensitivity of the l0x probe is only one tenth of that of a 1x probe. Most l0x probes provide an adjustment so that the capacitance, C, can be varied to allow optimum scope response at high frequencies where the capacitive portion of the input impedance dominates. All oscilloscopes provide a calibration signal for checking if the l0x probe has been properly adjusted. To ensure optimum measurement 14 pf 2-9

10 accuracy, always check that the l0x probes are critically compensated before making any measurements. In many cases, little error is incurred in a measurement if ordinary banana plug terminated wires are used to connect the scope to a circuit. Generally, if the signal level is large enough so that the noise introduced by the unshielded wires is negligible, the convenience of banana plug wires may justify their use RC Transient Circuit Analysis In this section, we will review the fundamentals of RC transient circuit analysis. Analysis of this sort is important for a number of reasons; for example, many nonlinear devices used in switching circuits can be modeled in terms of RC circuits. Also, the equivalent circuits of the wires, connectors, and equipment used to measure circuit behavior are RC circuits and it is this situation with which we will be primarily concerned in this experiment. Consider the circuit shown in Figure 2.5. This circuit consists of a resistor R in series with an ideal capacitor C. We shall assume that v(t) is a square wave 2-10

11 switching between 0 and +V volts with a period T. Using Kirchhoff's voltage law, we can write the following equation. v( t) + i ( t) R+ v ( t) = 0. C C (2.2) Making use of the voltage-current relationship for a capacitor, namely, that i C (t) = Cdv C (t)/dt, and then rearranging terms, this equation becomes dvc ( t) RC + vc ( t) = dt v( t) (2.3) From your introductory course in circuit analysis, recall that this equation can easily be solved for v C (t) by separating the variables (v C (t) and t) for any interval of time for which v(t) is constant. (That is, for 0 < t < T/2, v(t) is equal to +V, and for T/2 < t < T, v(t) is equal to 0.) This analysis is left as an exercise for the student. A general approach to analyzing circuits of this type can be obtained if one considers the initial and final value of the capacitor voltage over any interval of time for which v(t) is constant. Suppose that at time t 0 the voltage v(t) switches from V a to V b and remains equal to V b until time t 1. From your introductory circuit analysis course, recall that it was shown that the general solution for v C (t) for times between t 0 and t 1 is given by v ( t) = V C F + ( V I V ) e F ( t t0 ) τ, (2.4) where τ = RC, the RC circuit time constant. In this equation, V I is the initial value of the capacitor voltage (i.e., the value at t 0 ), and V F is the final value of the capacitor voltage assuming the input voltage remains constant (i.e., the value at t = ). Note 2-11

12 that if the half-period (T/2) of the square wave is sufficiently long, the value of v C (t) will come "close to" the final value V F before the value of v(t) changes again. The definition of "close to" is of course arbitrary, but we will take this term to mean within 1 % of the final value. It can be shown that it takes roughly 5 time constants (5τ) for v C (t) to come within 1 % of its final value, so we assume that v C (t 1 ) = V F when (t 1 - t 0 ) > 5τ. To find the values of V F and V I, first recall that the voltage across a capacitor cannot change instantaneously with time. Also, clearly if v(t) is constant for a long time, then i C (t) = 0, and hence v C (t) = v(t). Therefore, one can see that at time t 0 the value of v C (t) is V a, and, since the voltage across a capacitor cannot change instan- taneously with time, the value of v C (t) at t 0 + is also V a. Therefore, V I = V a. Similarly, one can argue that V F = V b at t 1, provided (t 1 - t 0 ) > 5τ. Thus, Equation (2.4) becomes v ( t) = V C b + ( V a V ) e b ( t t0 ) τ. (2.5) Using this general solution, we can directly write down the expression for v C (t) when v(t) is a square wave switching between 0 and +V volts with a period T. If the condition 5τ < (T/2) is satisfied, then the capacitor voltage is v C ( t) = V(1 e v ( t) = Ve C t /τ ( t T /2)/τ ) for 0 < t < T/2, and (2.6) for T/2 < t < T. (2.7) If one wished to go further and obtain an expression for the current i C (t), it could easily be obtained using the voltage-current relationship for the capacitor. 2 12

13 2.3.4 Probe Effects In the first experiment, we learned that when using the DMM to measure voltage, it is possible to affect the voltage being measured. This occurs because the (DC) input resistance of the DMM is finite, which allows current to flow through the DMM, thereby changing the voltage being measured. In a similar way, the oscilloscope does not present an infinite impedance to the circuit under test, so its use in measuring a voltage waveform often alters the waveform being measured. Furthermore, the impedance presented to the circuit when using the lx probe and the l0x probe is not the same, so these two probes will alter the waveform being measured in different ways. In this section, we will quantify the way in which these two probes can alter the waveform under test lx Probe Consider the circuit shown in Figure 2.6 having a scope, probe and external resistor R ex connected in series to a 33120A function generator. Now, suppose that the 33120A Function Probe Fcn Gen Grounding Wire to Scope Figure 2.6 Circuit Connection Using 1x Probe function generator has been set to produce a square wave and that the 1x scope probe has been used to connect the resistor to Channel 1 input of the oscilloscope. 2-13

14 An equivalent circuit for this connection is shown in Figure 2.7. The output of 1 14 pf Probe Figure 2.7: Equivalent circuit for the 1x probe connection. the signal generator is represented by an ideal voltage source v sig (t) and an internal generator source resistance R sig. The input impedance of the scope is represented by a 1MΩ resistor in parallel with a 14 pf capacitor. The unknown capacitance introduced by the wires and connectors is represented by the capacitor labeled C p. Note that C P for a 1x Probe is approximately 60 pf. Finally, v i (t) is the input voltage into the scope and v 0 (t) is the resultant voltage which is displayed on the CRT. The equivalent circuit shown in Figure 2.7 can be simplified as shown in Figure 2.8. Note that R t = (R sig + R ex ) and C t = (14 pf + Cp). Notice also that this is an RC circuit driven by the input signal v sig (t.) 1 Figure 2.8: Simplification of the equivalent circuit for the 1x probe connection. 2-14

15 The equivalent circuit for the 1x probe can be further simplified by redrawing it as shown in Figure 2.9 (a) and then taking the Thevenin's equivalent circuit across the capacitor C t, i.e., across Terminals a and b, as shown in Figure 2.9 (b). Figure 2.9: Further simplification of the equivalent circuit of the 1x probe connection: (a) After interchanging the capacitor and the 1 MΩ resistor, and (b) after taking the Thevenin equivalent circuit across the capacitor. The Thevenin's equivalent circuit shown in Figure 2.9 (b) is obtained as a result, where: and 1MΩ vt ( t) vsig ( t), 1M R = Ω + t ( Rt )(1M Ω) RT = Rt 1MΩ = R + 1MΩ t (2.8). (2.9) In the above equations, v T (t) and R T represent the Thevenin voltage and resistance, respectively. The circuit in Figure 2.9 (b) is a series RC circuit, and it follows that the time constant of this circuit is given by τ = R T C t. Assuming v sig (t) is a square wave switching between 0 and +V volts with a period T, it can be shown that if T/2 > 5τ, then the voltage across the capacitor during the positive half cycle when v sig (t) = V is given by 1MΩ v0( t) = V 1MΩ + Rt (1 e t τ ) for 0 t T/2. (2.10) 2-15

16 Similarly, the voltage across the capacitor during the negative half cycle when V sig (t) = 0 is given by 1MΩ v0( t) = V e 1M R Ω + t T ( t ) / τ 2 for T/2 t T. (2.11) Assuming that R sig = 50Ω, C t = 74 pf (i.e., C p = 60 pf), and v sig (t) is the square wave defined above, one can then determine the expression for v 0 (t) for 0 t T. It follows that this expression represents the "expected" waveform for v 0 (t). Variations in the actual waveform will result from any additional stray capacitance and resistanceintroduced by the wires and connectors x Probe Again consider the series circuit shown in Figure 2.6, but suppose now that the l0x probe has been used to connect the resistor R ex to the Channel 1 input of the scope. Note that the digital scope recognizes that a 10x probe is connected and applies a gain of 10 to compensate for the voltage divider effect of the 9 MΩ resistance of the 10x probe. As before, the function generator has been set to produce a square wave. + v i (t) pf Figure 2.10: Equivalent Circuit for the l0x Probe Connection. 2 16

17 An equivalent circuit of this connection is shown in Figure In this case, the l0x probe introduces an additional capacitor and resistor as shown previously in Figure 2.4. The capacitor C p again represents the stray capacitance introduced by the connectors and the probe wire. If C p and the 14 pf scope input capacitance are combined in parallel, then the equivalent circuit of Figure 2.11 results. (C p + 14 pf) Figure 2.11: Simplification of the equivalent circuit of the l0x probe connection. As mentioned previously, before using the l0x probe, it is important that it be critically compensated. It can be shown that when the l0x probe is critically compensated, the following condition must be satisfied: (9 MΩ) C = 1 MΩ (C p + 14 pf). (2.12) This means that when we critically compensate the l0x probe, we are adjusting C so that C C = p + 14 pf 9 (2.13) With C equal to this value, it can he shown using phasor analysis that the circuit 2-17

18 in Figure 2.11 can be replaced with the equivalent circuit in Figure 2.12, where C t = (C p + 14 pf) / 10. Note that C P for the 10x probe is approximately 136 pf. + v i 10 Figure 2.12: Simplification of the equivalent circuit of the l0x probe connection when the probe is compensated. We now have an equivalent circuit for the l0x probe connection, which can be further simplified and analyzed in the same manner as the lx probe connection discussed previously. It follows that the circuit in Figure 2.12 can be simplified to the circuit given in Figure 2.13 (a) where R t = (R sig + R ex ). This circuit can be (a) (2) (b) Figure 2.13: Further simplification of the equivalent circuit of the l0x probe connection: (a) After combining the signal generator resistance with the external resistor, and (b) after taking the Thevenin equivalent circuit across the capacitor. further simplified as shown in Figure 2.13 (b) by taking the Thevenin equivalent circuit across the capacitor, where 2-18

19 and v R T T 10MΩ ( t) vsig ( t) 10M R = Ω + t ( Rt )(10MΩ) = Rt 10MΩ = R + 10MΩ t (2.14). (2.15) One can again solve for the expected waveform v 0 (t) using this equivalent circuit and assuming that C t = 15 pf corresponding to C P = 136 pf Grounding of Electrical Equipment This section provides a brief introduction on how electrical equipment is grounded. Grounding is an important consideration in safety, and it is also one of the important ways to minimize noise. Furthermore, grounding is a critical consideration when configuring test equipment to make even the simplest of measurements. Figure 2.14 shows the standard three-wire AC distribution system that is used for electrical equipment. Figure 2.14: Three-lead distribution system. 2-19

20 In general, current flows through the black wire and the fuse and returns through the white wire. The green wire, which is connected to earth ground in the AC distribution system, is connected to the equipment enclosure or chassis. In addition, the neutral and ground wires are connected at only one point in the AC distribution system so that none of the neutral wire's current flows through the ground wire. The fact that the chassis is connected to the ground wire ensures that no voltage potential difference can exist between the chassis and earth ground and this prevents any potential shock hazard to the user when he/she touches the equipment. The AC input voltage is connected to the internal circuitry within the equipment chassis and some arbitrary output signal generated by the internal circuitry can result as shown in Figure Normally, the output signal is electrically isolated from the AC input voltage since the internal circuitry uses a transformer or some other isolation technique. This means that the impedance between either of the output signal's leads and the AC ground is large. Thus, if you use an ohmmeter to measure the resistance between the negative (-) return lead of the output signal and the chassis, it will appear as an open circuit. An alternative is to connect the signal return lead (-) to the chassis within the equipment enclosure. This is shown by the dotted line in Figure In this case, the signal output return lead is NOT ISOLATED from earth ground since there exists a short circuit between the signal return lead and earth ground through the respective connections of each lead to the chassis. It follows that with this alternative, if you use an ohmmeter to measure the resistance between the negative (-) return lead of the output signal and the chassis, it will appear as a short circuit. 2-20

21 If the signal return leads of two instruments are NOT ISOLATED from chassis ground, this results in both leads being connected together through the green ground wire of the AC distribution system as illustrated by examining Figure 2.15 (a). On the other hand, if the signal return lead of one of the instruments is ISOLATED, i.e., it is not connected internally to the chassis, then the return leads are not connected through the ground wire as illustrated in Figure 2.15 (b). In the Bryan Hall, Room 316 laboratory, the 33120A Function Generator output is not internally grounded whereas the Digital Scope input is internally grounded as shown in figure 2.15 (b), so a reference wire must be connected between the Function Generator's (-) signal terminal and earth ground in order for the Digital Scope to be able to measure signals. This results in a ground path equivalent to that shown in Figure 2.15 (a). V out out V out VV in in V in in (a) (b) Figure 2.15: Illustration of a instrument grounds and signal return leads. 2-21

22 The DMM's signal return lead is isolated from chassis ground and thus a large impedance exists between this lead and chassis ground or the signal return leads of both the function generator and the scope. Whenever the signal return lead is NOT ISOLATED from chassis ground, which is the case for the Digital Scope, care must be taken when connecting the instrument at any point in the circuit to make a measurement since you could be shorting the connected point in the circuit to chassis ground without realizing it. This is best understood by considering the circuit shown in Figure 2.16, which is assembled for test purposes using the circuit of Figure 2.17 a b Figure 2.16: RC circuit used to demonstrate the oscilloscope measurement problems that can occur when grounding is not properly understood. 2-22

23 a Figure 2.17: Faulty Connection of the oscilloscope. As shown, the scope leads are connected across R 1 in order to observe the voltage v R1 (t). In this case, the result is that the components R 2 and C are effectively removed from the circuit once the negative return lead of the scope is connected to node b of the circuit. This is due to the fact that node b is now connected to the return lead of the function generator through the scope chassis, the AC ground wire, and the function generator ground wire as illustrated by the dotted line. The equivalent circuit resulting from connecting the scope in this manner is shown in Figure a b Figure 2.18: Equivalent circuit resulting from the connection in Figure

24 One way to correct this problem is to reconfigure the circuit such that one side of R 1 is connected to the negative return lead of the function generator (or chassis ground). This is illustrated in Figure It follows that the scope always v T Figure 2.19: Reconfiguration of the RC circuit so that the voltage across R 1 can be measured with the oscilloscope. measures the voltage differential between its positive lead (+) and chassis ground. This explains why you don't even have to connect the negative lead (-) of the scope when making measurements on circuits that use the 33120A Function Generator provided that its signal return line is grounded. On the other hand, special care must be taken when measuring the voltage across a component when using the scope so that you don't short out components and modify the resultant operation of the circuit you are trying to measure. This is achieved by always having one side of the component connected to chassis ground when connecting the scope across it. An alternative approach to measuring the voltage across R 1 without reconfiguring the circuit is to employ the Digital Oscilloscope's "Math Function" feature. This feature can be used to obtain the difference between two waveforms. To do this, consider the circuit connection shown in Figure From the above discussion, 2-24

25 v T Figure 2.20: Circuit connection used so that the voltage across R 1 can be measured without reconfiguration of the circuit. we can see that the voltage v R2 (t) will appear on Channel 2, and the voltage v R1 (t) + v R2 (t) will appear on Channel 1. Clearly, the desired voltage v R1 (t) is simply the waveform on Channel 1 minus the waveform on Channel 2. Thus, all that must be done to observe the desired voltage v R1 (t) is to use the "Math" function to subtract the waveform on Channel 2 from the waveform on Channel 1. In order for the Math function to work properly, be sure that the voltage scales being used on Channels 1 and 2 are the same, that both signals can be displayed without any clipping, and then adjust the math voltage scale to its highest value. Refer to the "Front Panel" in the Operating Manual. After pressing the "Math" button on the digital scope, choose "Channel 1 - Channel 2" from the menu. This will be the desired waveform V R1 (t). Therefore, this constitutes a second method for measuring this voltage using the oscilloscope. 2-25

26 2.3.6 Electrical Relay Analysis An electrical relay is essentially an electromagnet that, when energized, moves a ferrous armature causing one or more attached switches to close and/or open. Since the electromagnet coil has induction as well as resistance, and since the armature has inertia, there is a time lag involved between when the coil is energized or deenergized and when a switch opens or closes. These lags are termed the relay pull-in and dropout times, respectively. Additionally, the switch may exhibit "contact bounce" when it opens or closes. We want to observe all of these features using the oscilloscope. An equivalent circuit of the small relay to be used in this experiment is shown in Figure It consists of a relay coil that can be energized by a voltage source. Figure 2.21: Equivalent circuit of the electrical relay to be used in this experiment. 2-26

27 The relay coil is represented by an equivalent circuit consisting of an ideal inductor L C, which represents the inductance of the coil, in series with a resistor R C, which represents the resistance of the coil windings. When no excitation current is flowing in the coil, the relay is de-energized and the switch is in the Normally Closed (NC) position which results in a short circuit between the Common (C) switch contact and the NC contact. When excitation current exists in the coil, the relay is energized and the switch moves to the Normally Open (NO) position, which results in a short circuit between the C contact and the NO contact RL Transient Circuit Analysis First, the coil resistance and inductance will be determined. The coil resistance can be measured directly using the DMM as an ohmmeter. In order to find the value of the inductance, an R-L circuit must be designed and used to measure the necessary parameters. Figure 2.22: Series R-L circuit, Consider the circuit shown in Figure 2.22 above, which consists of a resistor R in series with an ideal inductor L. Assuming v(t) is a square wave switching between

28 and V volts with a period T, it follows that if 5τ < (T / 2), the inductor current is i ( t) = i L L V R V ( t) = R (1 e e ( t T t / τ / 2) /τ ) for 0 t T / 2, and (2.16) for T / 2 t T (2.17) where τ is the time constant of the R-L circuit and is given by τ = L/R. (2.18) The above result can be obtained using the same method that was used to derive the solution for the RC circuit. That is, suppose that at time t 0 the voltage v(t) switches from V a to V b and stays at the value V b until time t 1. Then, assuming that 5τ < (t 1 t 0 ), the current in the inductor for t 0 < t < t 1 is given by i ( t) = I L F + ( I I I F ) e ( t t0 ) τ (2.19) In this equation, I I is the initial value of the inductor current (i.e., i L (t) at time t 0 ) and I F is the final value of the inductor current (i.e., i L (t) at time t = ). Therefore, in order to write down the results given in Equations (2.16) and (2.17), all that we need to do is compute the initial and final values of the inductor current and plug these values into Equation (2.19). It can also be shown that if t r is the 10% to 90% rise time of the signal, then t r = 2.2(τ). (2.20) Similarly, if t f is the 90% to 10% fall time of the signal, then t f = 2.2(τ). (2.21) 2-28

29 It follows that if we could measure t r or t f of i L (t), the value of the inductance can be determined from Equation (2.20) or Equation (2.21). Unfortunately, the oscilloscope will only enable us to measure and observe voltage signals. Now, the voltage across the resistor R is given by v R (t) = R i L (t), (2.22) and it follows that the voltage across R is directly proportional to the current through the inductor and thus the rise and fall times of this voltage will be the same as the rise and fall times of the current. In developing the design of the required circuit, an external resistor must be added in series with the relay coil so that the rise t r and fall time t f of the voltage can be measured across this resistor using the oscilloscope. The external resistor must be added because there is no way to measure the voltage across R C, the equivalent coil resistance. The circuit connection to be used to measure the value of the coil inductance L C is shown in Figure In this circuit, the output of the signal generator, V T, can be observed on Channel 1 and the voltage across the external resistor R E can be observed on Channel 2 of the scope, which allow measurement of t r and t f for the circuit. Note that R E must be connected as shown, i.e., between the relay and the negative return of the function generator, in order to obtain proper ground reference for the scope, as explained previously in the section concerning the grounding of electrical equipment. The equivalent circuit of the connection in Figure 2.23 is shown in Figure

30 33120A Function Gen. Figure 2.23: Circuit connection that can be used to measure the relay coil inductance Figure 2.24: Equivalent circuit of the circuit connection used to measure the relay coil inductance. 2-30

31 2.3.8 Relay Switching Times In order to further examine the operation of the relay, we will measure the pull-in and drop-out times plus observe any contact bounce when the switch moves between the NO and NC contacts. This can be done by constructing the circuit shown in Figure 2.25 and observing the indicated waveforms on the scope. In this circuit, suppose that the signal generator is a square wave voltage that switches between 0 and V pp at a frequency of approximately 20 Hz. The value of V pp is chosen to be large enough to cause the relay to switch on and off, and a frequency of 20 Hz is slow enough so that the relay can operate properly (i.e., as the frequency is increased, the relay will eventually stop operating since the relay is a mechanical device and requires a minimum amount of time to energize and de-energize). Analysis of this circuit indicates that when the signal generator is at V pp, the relay coil will activate and the switch will move from the NC contact to the NO contact A Function Gen. Figure 2.25: Circuit used to measure the relay pull-in and drop-out times, as well as to observe any contact bounce when the switch moves between the NO and NC contacts. 2-31

32 Likewise, when the signal generator is at 0 volts, the relay coil will deactivate and the switch will move back to the NC contact from the NO contact. When the switch is in the NC position, the voltage across R 4 is V 7 and the voltage across R 3 is 0 volts. Likewise, when the switch is in the NO position, the voltage across R 4 is 0 volts and the voltage across R 3 is V 7. Finally, when the switch is open (i.e., between the two contacts), the voltage across R 3 is R 3 V 7 /(R 3 + R 4 ) and the voltage across R 4 is R 4 V 7 /(R 3 + R 4 ). When R 3 = R 4 the voltage across both R 3 and R 4 is equal to V 7 /2 when the switch is open. Now consider the signal connected to Channel 2 of the oscilloscope. When the square wave output of the signal generator goes positive, the relay coil is energized, but there will be a delay before the switch completes the transition from the NC position and makes a permanent closure on the NO contact. This results from the rise time of the energizing current due to the inductance and resistance of the relay coil plus the inertia of the armature. In addition, the switch will "bounce" on the NO contact before it comes to rest and makes permanent closure. The relay pull- in time is defined as the total time required for the switch to move from the NC contact to the NO contact and make permanent closure on the NC contact after the relay coil is initially energized. In terms of the voltage seen on Channel 2 of the oscilloscope, the relay pull-in time is defined as the time from when the voltage first leaves V 7 until the voltage settles at 0 volts. Similarly, when the square wave output of the signal generator goes back to 0 volts, the relay coil is de-energized. Again, there will be some delay before the 2-32

33 switch completes the transition from the NO position to the NC contact and makes a permanent closure. This time delay is defined as the relay drop-out time. In terms of the voltage seen on Channel 2 of the oscilloscope, the relay drop-out time is defined as the time from when the voltage first leaves 0 volts until the voltage settles at V RLC Transient Circuit Analysis The final subject of this experiment is the transient behavior of a series RLC circuit. This discussion is important because it illustrates the ringing phenomena (often unwanted) that can occur in many electronic circuits, particularly in integrated circuits operating at high speeds. Consider the circuit shown in Figure 2.26, which consists of a resistor R, a capacitor C, and inductor L, all connected in series. We shall assume that v(t) is a square wave switching between 0 and V volts with a period of T. Applying Kirchhoff's voltage law to this circuit, one can show that the differential equation for the capacitor voltage is given by Figure 2.26: Simple Series RLC Circuit. 2-33

34 2 d v 2 dt R dv + L dt vc + LC C C = Vs LC. (2.23) where V s = V for 0 t T / 2, and V s = 0 for T / 2 t T. The differential equation shown in Equation (2.23) is an ordinary, second-order differential equation with constant coefficients. To solve a differential equation of this type, we must first form the characteristic equation, which is s 2 + R L 1 s + LC = 0. (2.24) The roots of the characteristic equation are 2 2 1,2 = α ± α ω0 s. (2.25) In this equation, α is the neper frequency in radians/second. For the series RLC circuit, neper frequency is defined as α = R 2L. (2.26) Note that ω 0 is the resonant radian frequency in radians/ second for the series (or parallel) RLC circuit and is defined as ω = 0 1 LC. (2.27) Furthermore, one can also define ω d, the damped radian frequency, which is given by ω = d 2 2 ω 0 α. (2.28) 2-34

35 There are three possible solutions for v C (t), depending on the relationship between the values of ω 0 and α. If α 2 > ω 0 2 then the solution is overdamped, and it follows that if 5/ [smaller of ( s 1, s 2 )] < (T / 2), the capacitor voltage is for 0 t T / 2 (2.29) and for T / 2 t T. (2.30) Furthermore, if α 2 < ω 0 2, then the solution is underdamped, and it follows that if 5/α < (T / 2), the capacitor voltage is for 0 t T / 2 (2.31) and, for T / 2 t T,. (2.32) Finally, if α 2 = ω 0 2, then the solution is critically damped, and it follows that if 5/α < (T / 2), the capacitor voltage is for 0 t T / 2 (2.33) and = t s t s C e s s s e s s s V t v ) ( + = ) 2 / ( ) / ( ) ( T t s T t s C e s s s e s s s V t v + = ] sin[ ] cos[ ) ( t t Ve V t v d d d t C ω ω α ω α + = 2)] / ( sin[ 2)] / ( cos[ ) ( 2) / ( T t T t Ve t v d d d T t C ω ω α ω α 1) ( ) ( + = t Ve V t v t C α α

36 v ( t) = Ve C α ( t T / 2) [ s( t T / 2) + 1] for T / 2 t T. (2.34) Thus, from these equations, if one knows the relationship between α 2 and ω 2 0, then one can readily predict the capacitor voltage. Furthermore, the current i(t) as a function of time can be easily obtained by using the voltage-current relationship for the capacitor given by dv i = C dt C. (2.35) If we examine the capacitor voltage for the case when the solution is underdamped, we see that the response will include a damped sinusoid. Now suppose that we wish to measure the parameters α and ω d from this response. By measuring the time between any two positive or negative peaks of this sinusoid, we can easily obtain its period, which gives us the value of ω d since ω d = 2π/period. To measure α, consider the sinusoidal component of the capacitor voltage that is superimposed on the applied square wave. Suppose that this amplitude is V 1 at time = t 1 and V 2 at time = t 1 + 2πn/ω d, where n is an integer. It can be shown that for the interval when the signal is decaying to zero, ω d V α = ln πn V T/2 < t < T (2.36) Thus, all that must be done to determine α is to measure v C (t) at two peak values, which are inherently an integral number of periods apart, when the signal is decaying and then use Equation (2.36) to calculate α. 2-36

37 2.4 Advanced Preparation The following advanced preparation is required before coming to the laboratory: (1) Thoroughly read and understand the theory and procedures. (2) Analyze the circuit shown in Figure 2.27 and derive the expressions for both v C (t) and v R1 (t) over one period using the parameters given by your instructor. (3) For the circuit of Figure 2.30, determine the expression for v 0 (t), the signal seen on the oscilloscope, for 0 t T when the 1x probe is used. See Figure 2.7. Sketch a plot of this signal for one period carefully labeling the period and voltage levels. (4) Repeat the above calculation when the 10x probe is used. (5) Perform a PSpice Transient simulation for the circuit shown in Figure See Section for signal and electrical element values. Make copies of the transient voltages across C R and R EC and the current through L R. 2.5 Experimental Procedures x Probe Adjustment Using the calibration signal generated by the oscilloscope, check that the 10x probe is critically compensated. To do this, connect the probe connector to Channel 1 input connector and connect the probe tip to the calibration signal on the face of the oscilloscope. The calibration signal is a 5V peak-to-peak square wave with a frequency of 1.2 khz. The observed signal should be a sharp square wave when the probe is critically compensated. Your instructor will adjust the variable capacitor on the probe using the screw adjustment provided to illustrate when the probe is overcompensated, undercompensated, and critically compensated and provide hardcopies indicating the period and voltage levels for each waveform. 2-37

38 2.5.2 RC Transient Analysis Construct the circuit shown in Figure 2.27 using the values given to you by your v T (t) Figure 2.27: Simple RC circuit whose transient response is to be measured. instructor for R 1, R 2, and C. The time-varying voltage v T (t) should be a periodic square wave with a peak-to-peak voltage varying from V 1 to V 2 at a frequency of f 1 (again as specified by your instructor); you should use the HP33120A function generator to produce this voltage. Now observe the waveforms v C (t) and v R1 (t) using the scope. Connect the scope as shown in Figure Use Channel 1 to trigger the scope and to observe the output of the function generator v T (t) and use Channel 2 to observe v C (t). Note that you can not directly measure v R1 (t) with this set-up because of the common grounding of the function generator and the scope as shown in Figure In order to observe v 1 (t) with this set-up, you must use the math function to subtract v C (t) from v S (t). Make a copy of the scope display showing v T (t), v R1 (t), and v C (t). Now reconnect the scope as shown in Figure 2.29 in order to dierctly observe v R1 (t). Note you have to change to circuit in order to directly measure v R1 (t). This 2-38

39 is true because the signal return leads (-) of both the function generator and the scope are connected to chassis ground as shown in Figure Now use the math function to observe v C (t). Make a copy of the scope display showing v T (t), v R1 (t), and v C (t). Function Gen Grounding Wire Figure 2.28: Connection used to measure the voltage across the capacitor. Function Gen Grounding Wire Figure 2.29: Connection used to measure the voltage across R

40 2.5.3 Probe Effects In this experimental section, you will investigate the effects of using two different types of scope probes. First, set the output of the HP33120A function generator to a square wave with a frequency of f 2 and an amplitude of V 3 peak-to-peak with the output voltage switching between V 4 and V 5. Use the HP5384A frequency counter to measure the frequency and the oscilloscope to measure both the peak-to-peak voltage and frequency. Now construct the circuit shown in Figure Connect the external resistor R ex 33120A Function Generator + Probe Fcn Gen Grounding to Scope Figure 2.30: Circuit to be used for examining probe effects. to the function generator and use the 1x probe to connect it to Channel 1 input of the scope. Note that the Function Generator negative terminal must be grounded. Now observe v i (t) on the scope using the circuit you constructed as shown in Figure Make a copy of the waveform and record its period and voltage levels plus note the ground reference. In addition, measure and record the 10-90% rise time (t r ) and the 90-10% fall time (t f ). These values can be directly measured by the digital scope. Next, change the frequency of v sig (t) to f 3 and measure and record the voltage levels, rise time, and fall time of v i (t). Now, change the frequency of v sig (t) back to f 2, replace the 1x probe with a 10x probe, and observe v i (t) on the scope. Make a copy of this waveform, and record 2-40

41 its period, voltage levels, ground reference, plus measure and record the rise and fall times. Finally, change the frequency of v sig (t) to f 3 and again measure the voltage levels plus the rise and fall times of v i (t) at this frequency Electrical Relay Analysis Measurement of Relay Parameters First, use an ohmmeter to measure the coil resistance R C of your electrical relay. Record the serial number of the electrical relay and its measured coil resistance. Now, before constructing the circuit of Figure 2.31, set the output of the signal 33120A Function Gen. Figure 2.31: Circuit connection that can be used to measure the relay coil inductance. generator to a 8 volt peak-to-peak square wave switching between 0 and +8 volts with a frequency of approximately 20 Hz. Next, leaving the setting of the function 2-41

42 generator fixed, construct the circuit of Figure 2.31 with R E as given by your instructor. Recall that R sig = 50 Ω and use the value you measured for R C in your computations. If you have connected the circuit correctly and you have set the voltage levels correctly, you should be able to hear the relay switching. If you do not hear the relay switching, increase the peak-to-peak amplitude of the function generator output until you do hear the relay switching. Note also how the relay is affected by the signal frequency. If you decrease the signal frequency, the switching sound should "slow down" and if you increase the frequency, the sound should "speed up". Eventually, the relay will stop operating as the frequency is increased since the switch is a mechanical device and requires a minimum amount of time to energize and deenergize With the frequency set to approximately 20 Hz, observe the waveforms indicated in Figure 2.31, i.e., V T and V RE, on Channels 1 and 2, respectively, of the oscilloscope. Be sure to include measurements of the period, amplitude, rise time, and fall time of V RE on the scope display. Make a copy of this scope display. Now look at Channel 1, which is the square wave output denoted by V T, and label its period and the voltage levels. Now look at Channel 2, which is the voltage across the external resistor, and label its period and voltage levels. Compute the coil inductance L C using the larger of the rise time or fall time measurements. Now disconnect the function generator output from the circuit, which can be accomplished by unplugging the wire connecting the relay to the (+) terminal of the function generator in Figure Now connect the function generator output directly 2-42

43 to Channel 1 of the oscilloscope. Copy the waveform observed, making sure to note the period and voltage levels. Measurement of Relay Switching Times With no circuit assembled, set the output of the signal generator to a square wave with a frequency of approximately 20 Hz. The amplitude of this square wave should be the same as that used previously. Now construct the circuit shown in Figure 2.32 using the values given by your instructor. Observe the signal on Channel 1 of the 33120A Function Gen. Figure 2.32: Circuit used to measure the relay pull-in and drop-out times, as well as to observe any contact bounce when the switch moves between the NO and NC contacts. oscilloscope and set the scope triggering to occur at the positive going edge of this signal. This will trigger the scope display (as well as the signal on Channel 2) when the square wave input reaches approximately 8 volts and the relay coil is just beginning to he energized. Adjust the time scale of the scope to show only pull-in. 2-43

44 Make a copy of the scope display. Now observe the signal on Channel 2 noting the voltage levels as well as the time scale. Also, by observing this waveform, determine and measure the relay pull-in time and label this time on your sketch. Also, clearly identify any contact bounce present in the signal. Now change the scope triggering to the negative going edge of Channel 1. This will trigger the display when the square wave input returns to 0 volts and the relay coil is just beginning to be de-energized. Adjust the time scale of the scope to show only dropout. Make a copy of this scope display. Determine and measure the relay dropout time by observing the waveform on Channel 2. Again, make a copy of this waveform and label the dropout time plus identify any contact bounce in the signal RLC Transient Circuit Analysis Now, before constructing the circuit of Figure 2.33, set the output of the signal generator to a square wave with a frequency of approximately 20 Hz. The amplitude of the square wave should be the same as the amplitude used in the previous section. Next, leaving the setting of the function generator fixed, construct the circuit of Figure 2.33 with values for R EC, C R, and L R as given by your instructor. Use decade boxes for the resistance, capacitance, and inductance. Now observe the voltage across the capacitor as indicated in Figure If you constructed this circuit correctly, you should see an underdamped voltage response, i.e., the transient response should be a damped sinusoid. Make a hardcopy of this display and label the signal period and voltage levels. 2-44