Experiment 2 Complex Impedance, Steady State Analysis, and Filters

Transcription

1 Experiment 2 Cmplex Impedance, Steady State Analysis, and Filters Purpse: The bjective f this experiment is t learn abut steady state analysis and basic filters. Backgrund: Befre ding this experiment, students shuld be able t Determine the transfer functin f a tw resistr vltage divider. Determine the real and imaginary parts f a cmplex number, write cmplex numbers in plar frm, and lcate a cmplex number in the cmplex plane (real and imaginary axes) Determine the cmplex impedance f capacitrs and inductrs frm the values f the cmpnents and the perating frequency f whatever pwer supply is being used. Determine the values f capacitrs and inductrs frm the infrmatin printed n them. (Review the Quiz frmula sheet.) Review the backgrund fr the previus experiment. Learning Outcmes: Students will be able t D a transient (time dependent) simulatin f RC, RL and RLC circuits using Capture/PSpice D an AC sweep (frequency dependent) simulatin f RC, RL and RLC circuits using Capture/Pspice, determining bth the magnitude and the phase f input and utput vltages. Determine the general cmplex transfer functin fr RC, RL and RLC circuits and simplify fr high and lw frequencies. Be able t define what is meant by high and lw frequencies in the cntext f RC, RL and RLC circuits. Identify whether an RC, RL r RLC circuit is a lw-pass filter, a high-pass filter, a band-pass filter r a bandreject filter Find the crner frequency fr RC and RL circuits and the resnant frequency fr RLC circuits. Find a practical mdel fr a real inductr and determine the range f frequencies in which the real inductr behaves nearly like an ideal inductr. Measure inductance and capacitance using a cmmercial impedance bridge. Equipment Required DMM (Digital Multimeter) Analg Discvery (with Wave Frms) Oscillscpe (Analg Discvery) Functin Generatr (Analg Discvery) DC Pwer Supply (Analg Discvery) Impedance Bridge (lcated n center table) Cmpnents: kω resistr, F capacitr, 00mH inductr, 0.068F (683) capacitr Prtbard Helpful links fr this experiment can be fund n the Links by Experiment page. Pre-Lab Required Reading: Befre beginning the lab, at least ne team member must read ver and be generally acquainted with this dcument and the ther required reading materials listed under Experiment 2 n the EILinks page. Hand-Drawn Circuit Diagrams: Befre beginning the lab, hand-drawn circuit diagrams must be prepared fr all circuits either t be analyzed using PSpice r physically built and characterized using yur Analg Discvery bard. K.A. Cnnr, S. Bnner, P. Schch - -

2 Part A RC circuits, RL circuits, and AC Sweeps Backgrund Cmplex plar crdinates: Cmplex numbers allw yu t express a single number in terms f its real and imaginary parts: z = x + jy. j (the symbl i is used in mathematics) is used t represent the square rt f -. Yu can als represent a number in the cmplex plane in terms f plar quantities, as shwn in Figure A-. The length f the line segment between a pint and the rigin and the angle between this segment and the psitive x axis frm a new cmplex number: z = Acs + jasin where 2 tan y A x y and. 2 x Figure A-. Impedance and basic circuit cmpnents: Each basic circuit cmpnent has an effect n a circuit. We call this effect impedance. Yu shuld remember that impedance causes a circuit t change in tw ways. It changes the magnitude (amplitude) and the phase (starting psitin alng the time axis) f the vltages and currents. When a resistr is placed in a circuit, it affects nly the amplitude f the vltages. When capacitrs and inductrs are placed in a circuit, they influence bth. We can use cmplex plar crdinates t represent this influence. A capacitr will change the amplitude by /C and shift the phase by -90. An inductr will change the amplitude by L and shift the phase by +90. (Recall that is called angular frequency and is equal t 2f.) We can represent these changes easily in the cmplex plar plane. We represent impedance by the letter Z. Therefre, j Z R R Z C Z L jl C jc Experiment The Influence f a Capacitr In this sectin, we will examine hw a capacitr influences a circuit using Capture/PSpice. Create the simple RC circuit shwn in Figure A-2 in Capture Lcate the VSIN surce in the SOURCE library. Set the amplitude t 200mV, frequency t khz and ffset t 0V. Lcate the resistr (R) and capacitr (C) in the ANALOG library. Leave the resistr at k, but change the value f the capacitr t uf. [In PSpice, u represents micr, (0-6 ).] Chse the 0 grund frm the grund SOURCE library and put the wires int the circuit. K.A. Cnnr, S. Bnner, P. Schch - 2 -

3 V V R VOFF = 0 VAMPL = 00mV 200mV FREQ = K V k C u 0 Figure A-2. Set up a simulatin fr this circuit. Chse a Transient Analysis. Run t time 4ms. Chse a step size f 4us. Place tw vltage markers n the circuit: ne between the surce and the resistr and ne between the resistr and the capacitr. The leftmst marker in the circuit will display the input t the circuit. The rightmst marker will display hw the circuit is influencing this input, i.e it displays the utput. Run the simulatin. Yu shuld get an utput with tw sinusids n it. Yu shuld see that the circuit has influenced bth the amplitude and the phase f the input. Hw clse is the phase shift t -90? Nte: Use ne f the later cycles t determine this. Cpy this plt and include it in yur reprt. Hint/Suggestin: The impedance expressins abve fr R, L and C wrk in steady-state. That is, they will describe the behavir f the vltages and currents nly after the circuit has gne thrugh its initial transients and settled int the state it will remain in frever (its steady-state). Thus, when we are using transient analysis t help understand steady-state, we ignre at least the first cycle r tw f the sinusids and fcus n the latter part f the signal. Yu will find at times that the signal des nt reach its steady-state in the number f cycles yu are displaying. T be sure yu are in steady-state, yu can set up yur simulatin prfile s that it begins at 0ms and runs t 4ms r even start at 00ms and run t 04ms. Because the circuits we analyze in this curse usually have nly a small number f cmpnents, the extra time necessary t run t 04ms is usually nt large. Fr nw, try starting at 0ms and see hw things lk. Fr reference, the full simulatin f the prcessr in yur cmputer can take several cmputer years t cmplete. This is usually dne with many cmputers and still can take weeks. PSpice has anther type f analysis that lets yu lk at the behavir f a circuit ver a whle range f frequencies. It is called an AC sweep. We knw that the influence f the capacitr depends n and this is related t the frequency f the input signal. Shuld the behavir f the circuit change at different frequencies? Let s set up an AC sweep and find ut. Edit yur simulatin by pressing n the edit simulatin buttn. Chse AC Sweep/Nise frm the drp dwn list bx. Chse a lgarithmic sweep type with a start frequency f and an end frequency f Meg. Set the pints per decade t 00. (This will give yu plenty f pints and the plt will be nice and smth.) Yu need t d ne mre imprtant thing befre yu run the simulatin. Yu cannt d an AC sweep withut setting a parameter fr the VSIN surce called AC. Set the AC parameter t yur amplitude, 200mV (400mVp-p). Nte that each cmpnent in PSpice has many mre parameters than thse that appear n the screen. If yu duble click n yur surce, yu will pen the spread sheet fr the VSIN surce. Parameter values can be changed in the spreadsheet in additin t clicking n the parameter value next t the cmpnent symbl in the circuit diagram. Run the simulatin. Yu shuld see tw traces. These traces are shwing yu the amplitude f yur input and utput at all frequencies between and MegHz. Nte that the hrizntal scale f the plt is nw in Hertz. The input trace is a straight line at 200mV. This makes sense because the amplitude f the input is 200mV at all frequencies. K.A. Cnnr, S. Bnner, P. Schch - 3 -

4 The amplitude f the utput, hwever, changes with frequency. Fr what range f frequencies is the amplitude f the utput equal t the amplitude f the input? Fr what range f frequencies is it near zer? What is the amplitude at khz? Des this match the amplitude f the transient yu pltted at khz? Cpy the AC sweep plt f yur RC circuit and include it in yur reprt. We nw knw that a capacitr will influence the amplitude f a circuit and this amplitude influence changes at different frequencies. What abut the phase? Capture/PSpice prvides special markers fr displaying the phase. First, remve bth vltage markers frm yur circuit. Frm the Capture main menu chse PSpiceMarkersAdvancedPhase f Vltage. Place tw phase markers (marked with VP) n yur circuit in the same lcatins as the vltage markers were befre. Rerun the AC sweep simulatin. Nte: Yu can als change yur display frm yur plt using the Add Trace ptin. Nw yu shuld be lking at tw traces that represent the phase f the input and the utput f the circuit at all frequencies between and MegHz. Nte that the y axis is nw in degrees. The input phase des nt change. It is always zer because the sine wave in PSpice is drawn starting at zer by default. Hwever, the phase fr the utput ver the capacitr des change with frequency. At what frequencies is the phase f the utput the same as the phase f the input? At what frequencies is the phase shift -90? What is the phase shift at khz? Des this crrespnd with the phase shift f the later cycles that yu gt in yur transient at khz? Cpy the AC sweep plt f the phase f yur RC circuit and include it in yur reprt. The Influence f an Inductr In this sectin, we will repeat the prcedure abve fr the simple circuit with an inductr shwn in Figure A-3 belw. R VOFF = 0 VAMPL = 00mV 200mV FREQ = K V k V V L 00mH 0 Figure A-3. Create the RL circuit in Capture Delete the capacitr in yur circuit. Lcate the inductr (L) in the ANALOG library. Change the value f the inductr t 00mH. Nte: It is better t create a new prject fr each circuit yu analyze rather than just mdifying the same ne ver and ver. The circuit prject mdels are very small s they will nt take up much space n yur cmputer and it will be easier t cmpare ntes with yur partners and re-run things, if necessary, when yu are writing yur reprt. Return t yur transient analysis by chsing Transient in the drp dwn bx. Yur simulatin values (run t 4ms and step size 4us) shuld still be there. Remve the phase markers and replace the vltage markers n the circuit Run the transient simulatin. Again, the circuit has influenced bth the amplitude and the phase f the input. Hw clse is the phase shift t +90? Nte: Use ne f the later cycles t determine this. Cpy this plt and include it in yur reprt. K.A. Cnnr, S. Bnner, P. Schch - 4 -

5 Nw return t yur AC sweep analysis by changing the value in the drp dwn list bx. Again the parameters yu set befre (frm t Meg with 00 pints per decade) shuld nt have changed. Run the simulatin. Yu shuld see tw traces. At what frequencies is the amplitude f the utput equal t the amplitude f the input? At what frequencies is it near zer? What is the amplitude at khz? Des this match the amplitude f the transient yu pltted at khz? Hw is this sweep different than the sweep yu created using the circuit with the capacitr? Cpy the AC sweep plt f the RL circuit and include it in yur reprt. The final thing we need t d is plt the phase. Remve bth vltage markers frm yur circuit. Frm the Capture main menu chse PSpiceMarkersAdvancedPhase f Vltage. Place tw phase markers (marked with VP) n yur circuit in the same lcatins as the vltage markers were befre. Rerun the AC sweep simulatin. Again, yu will see tw traces. At what frequencies is the phase f the utput the same as the phase f the input? At what frequencies is the phase shift +90? What is the phase shift at khz? Des this crrespnd with the phase shift yu gt in yur transient at khz? Cpy the AC sweep plt f the phase f yur RL circuit and include it in yur reprt. Summary In this part f the experiment, yu have learned that capacitrs and inductrs influence the behavir f a circuit. They change bth the phase and the amplitude. The degree f influence depends n the frequency f the input surce. Yu als learned hw t examine the behavir f a circuit ver a range f frequencies using the AC sweep feature in PSpice. Part B Transfer Functins and Filters In this sectin, we will cntinue ur analysis f the tw simple circuits we created in part A and intrduce the cncepts f transfer functins and filters. Backgrund Transfer functins: We knw that a circuit with nly resistrs will behave the same at any frequency. A vltage divider with tw k resistrs divides a vltage in half at 0Hz as well as it des at 00kHz. We als knw that circuits cntaining capacitrs and/r inductrs behave very differently at different frequencies. What if we culd find a functin that, when applied t any input signal, wuld give yu the utput signal? This cannt be dne easily in the time dmain, hwever, it is quite simple in the cmplex plar dmain we intrduced in part A. We can define a functin H(j) fr any circuit such that ut V H ( j) Finding transfer functins: Fr a simple series circuit, we can find the transfer functin using a cncept similar t the vltage divider rule. We simply need t expand the idea f resistance t include cmplex impedance. Hw and why we can d that is discussed in detail in the curse ntes. Nw, we can cmbine impedances in the cmplex plar dmain in just the same way that we cmbine resistances in the time dmain. Nw we can als easily define ur transfer functin using the vltage divider rule, as illustrated in Figure B-. V in K.A. Cnnr, S. Bnner, P. Schch - 5 -

6 Z2 Vut Vin Z Z2 Figure B-. H Z2 Z Z2 Filters: Mst electrical signals are made up f many frequencies. In the first experiment, yu learned that sund waves within human hearing range cver a range f frequencies frm very lw t very high. Smetimes we may nt want t include all these frequencies in ur signal. We may want t filter ut the very high frequencies that sund like nise, fr instance, s that we nly hear the part f the sund that we want t. In electrnics a circuit that filters ut certain frequencies while allwing thers t remain unchanged is called a filter. Figure B-2. The tw basic types f filters we will cnsider in this part are lw pass filters (LPF) and high pass filters (HPF). An idealized representatin f these tw types f filters is shwn in Figure B-2. Lw pass filters filter ut high frequencies while allwing lw frequencies t pass thrugh unchanged. High pass filters blck ut lw frequencies while allwing high frequencies t pass thrugh unchanged. In an ideal wrld, the transfer functin wuld be fr the frequencies that yu want t remain unchanged (a signal multiplied by is the same) and 0 fr the frequencies yu want t filter ut (a signal multiplied by 0 is 0). In an ideal filter, the transitin between and 0 is instantaneus. In a real filter, this transitin is less exact. Using transfer functins t determine filter behavir: We can determine what kind f filter the simple RC circuit pictured in Figure B-3 is, by finding the transfer functin and examining its behavir at lw and high frequencies. First d the cmplex algebra t determine the capacitr vltage. Yu will ntice that nce again we have a vltage divider circuit, except that ne f the impedances is imaginary and ne is real. Fr an perating frequency 2f, the impedance f the capacitr C is equal t ZC, while the impedance f the resistr R is jc Z R. R K.A. Cnnr, S. Bnner, P. Schch - 6 -

7 Figure B-3. Applying the usual vltage divider relatin fr this series cmbinatin f tw impedances gives us ZC V Z Z V jc j C R V B A A C R Nte that the relatinship between the input vltage V A and the utput vltage V B is nw cmplex. It is useful t be able t set up these expressins and then simplify them fr very lw and very high frequencies. We will be able t use the resulting expressins t see if ur PSpice plts make any sense. We will first lk at very lw frequencies. We cannt set frequency equal t zer, since parts f ur frmula will blw up. Rather, we will assume that the frequency is very small, but nt zer. Then the capacitive impedance Z C will be very much larger than the resistive impedance Z R and we can neglect the latter term. We have then, at lw frequencies, that V ut ZC V Z V V V B C A A in V ut r, mre simply, that Vin. Nte that the input and utput vltages are essentially identical. At high frequencies, the capacitive impedance will be the small term, s we can neglect it in the denminatr. We cannt neglect it in the numeratr, since it is the nly term there. Thus, at high frequencies, V V j R C V j R C V V ut ut B A in r that. This rati clearly ges t zer as the Vin jrc frequency ges t infinity. It is als negative imaginary and, thus, has a phase f 90. Nte abut Lw and High : It may seem that terms like lw and high, when applied t smething like frequency, are fuzzy, ill-defined terms. Hwever, that is definitely nt the case. Here, we mean smething quite specific by the expressin lw frequency. A frequency is nly lw when the capacitive impedance is s much larger than the resistive impedance that we can neglect the resistive term. We usually need t define the required accuracy t make a statement like this. Fr example, we can say that a frequency is lw as lng as the apprximate relatinship we fund between the input and utput vltages is within 5% f the full expressin. Mst f the parts we use in circuits are n mre accurate than this, s there is n particular need t d ur calculatins with better accuracy. In ther applicatins we might need better accuracy. Crner frequency: When we design a high r lw pass filter, we nly need t knw three things: hw it behaves at lw frequencies (0 r ), hw it behaves at high frequencies (0 r ), and at what frequency it switches (0 t r t 0). Fr a simple RC r RL filter, the frequency at which it switches is called the crner frequency. It is defined as the pint at which the magnitude f the transfer functin is equal t /2. Experiment There are fur standard markers available when we want t display the results f ur simulatins. We have used the first ne fr vltage at a pint. Next we will use the differential marker (the secnd chice with the tw prbes) t indicate the vltage difference between tw pints. The Transfer Functin Equatin K.A. Cnnr, S. Bnner, P. Schch - 7 -

8 In this sectin we will explre the impedance f the capacitr circuit. If we want t apply the vltage divider equatin t frm the transfer functin, then the vltage drp ver the tw cmpnents must add up t the input vltage at all frequencies. In Capture/PSpice, lad the capacitr circuit (Figure A-2) r recreate it by replacing the inductr (Figure A-3) with a uf capacitr. Remve the markers frm the circuit. T test the validity f applying a vltage divider t cmplex impedances, we want t shw that, just like with the vltage divider made with resistrs, the surce vltage (V) is equal t the sum f the vltages acrss the resistr and the capacitr (V R + V C). T determine V R and V C, use we will use vltage differential markers. Chse a differential marker. Place the psitive vltage marker (marked with a plus) at the end f the capacitr clsest t the surce. This fllws the cnventin that vltage is psitive at the end where the current enters. Place the negative marker n the end f the capacitr clsest t grund. Place a secnd differential marker n either side f the resistr. Place a regular vltage marker at the input (next t the surce). Rerun the transient analysis frm part A (4ms with a step size f 4us). When yu run yur simulatin, yu will btain three vltages traces. T see the sum f V R and V C, we must add a trace in PSpice. Find the symblic name fr the V R trace by lking at the bttm f yur plt. Each trace n the plt has a name that crrespnds t its clr. It shuld lk smething like V(R:,R:2). If yur resistr has a different name r a different plarity relative t grund, the expressin will nt be exactly the same. G t the Trace menu in the PSpice prgram and chse Add Trace. Type in the name f the V R trace frm the bttm f the plt. Type in + (plus) r click n the + symbl in the functins menu. Find the signal crrespnding t V C n the plt. Type in the name f the V C trace. The expressin in the bttm bx shuld lk smething like: V(R:,R:2)+V(C:2,0). (Since the names and plarities f yur cmpnents may vary, yur expressin may nt be identical.) Click n OK. When have added the trace crrectly, yu shuld see that the sum will equal the surce vltage. Cpy a plt f yur results, and include it in yur reprt. Run a transient simulatin fr frequencies f 0Hz (run time = 400ms and step size=400us) and 0kHz (run time = 0.4ms and step size=0.4us). Place the trace f V R and V C n each plt. Verify that the vltages add as expected. Cpy these plts and include them in yur reprt. Since we have a series circuit, we knw that the current, I, is the same thrugh all the cmpnents. Therefre, V=IZ fr each cmpnent and fr the entire circuit. This means that the transfer functin, H=Vut/Vin, becmes independent f current, and simplifies t an expressin in terms f the cmpnent values nly. H Vut Vin IZ R IZ C IZ C Z C Z Z Transfer Functin f the Capacitr Circuit In this part, we will find the transfer functin f ur capacitr circuit frm part A and determine what type f filter it is. R C Remve the differential markers frm the capacitr circuit (Figure A-2) and replace the tw riginal vltage markers. Run the AC sweep (frm t MegHz at 00 pints per decade). We knw that H is the rati f the utput vltage t the input vltage. We can bserve this directly by adding a trace t take the rati between the tw traces we have n ur plt. G t the Trace menu in the PSpice prgram and chse Add Trace. K.A. Cnnr, S. Bnner, P. Schch - 8 -

9 Find the signal crrespnding t yur utput in the list n the left hand side. The name f this signal is written at the bttm f yur plt next t a symbl indicating the clr f the trace. The trace crrespnding t the utput is the ne that is nt cnstant. Click n the name f the utput trace. Type in / (divided by) r click n the / symbl in the functins menu. Find the signal crrespnding t yur input and click n it. The expressin in the bttm bx shuld lk smething like: V(R:2) / V(V:+) Click n OK. Anther trace shuld appear n yur plt. This trace is the transfer functin f the circuit. Is the transfer functin 0 r at lw frequencies? Is it 0 r at high frequencies? What kind f filter is this? The crner frequency f the circuit is the frequency at which the transfer functin is Use the cursrs t mark this lcatin n yur plt. Click n the cursr buttn n the PSpice tlbar. r Click n the clred symbl fr the trace yu added (H) at the bttm f yur plt. This places the cursr n this trace. Drag the cursr alng the trace until the y crdinate is as clse as pssible t Mark this pint with the buttn n the tlbar. r What is the crner frequency? Cpy this plt and include it in yur reprt. In the vide lectures r lecture slides yu were given an expressin fr the crner frequency f an RC circuit: C = /RC. Calculate the crner frequency. Dn t frget that f (in Hertz) is /2. Hw des this cmpare t the frequency yu fund with PSpice? The equatin fr the transfer functin f a series circuit is Z2 H Z Z2. Use this expressin t find the transfer functin f the capacitr circuit. Take the limit f this functin at high and lw frequencies. Des the plt yu made with PSpice behave the same at lw and high frequencies? Transfer Functin f the Inductr Circuit In this part, we will find the transfer functin f ur inductr circuit frm part A and determine what type f filter it is. In Capture/PSpice, recreate the inductr circuit (Figure A-3) by replacing the capacitr with a 00mH inductr. Run the AC sweep. Create the transfer functin by adding a trace that divides the utput vltage by the input vltage. Is the transfer functin 0 r at lw frequencies? Is it 0 r at high frequencies? What kind f filter is this? Use the cursrs t mark the lcatin f the crner frequency n yur plt. What is the crner frequency? Cpy this plt and include it in yur reprt. Can yu derive the equatin fr the crner frequency f this circuit? Calculate the crner frequency using the equatin yu derived. Dn t frget that f (in Hertz) is /2. Hw des this cmpare t the frequency yu fund with PSpice? Find the transfer functin f the inductr circuit using the equatin fr H. Take the limit f this functin at high and lw frequencies. Des the plt yu made with PSpice behave the same at lw and high frequencies? Build the circuit K.A. Cnnr, S. Bnner, P. Schch - 9 -

10 Nw yu can build ne f these circuits n yur prtbard, hk it t the functin generatr and cmpare its behavir t the simulatin. Yu may find the cursrs and the Measurements view will be useful. Build the capacitr filter circuit (Figure A-2) n yur prtbard using a uf capacitr and a k hm resistr. Set the functin generatr t a khz signal with an amplitude f 200mV, (400mVp-p). Display the input signal n channel and the utput signal n channel 2. Cpy the image f this signal with the edit functin in the sftware and include it in yur reprt. (We will call this the Camera Functin f the sftware.) Hw des the utput f this circuit cmpare (amplitude and phase) with the transient analysis yu made f this circuit at this frequency? Yu shuld ntice that the amplitude and phase f this circuit changes as yu change the frequency. At what frequencies des the utput lk rughly the same as the input? At what frequencies des the utput essentially disappear? At the crner frequency, the transfer functin is equal t The transfer functin is defined as Vut/Vin. If Vin is 200mV, what shuld the amplitude f the utput be at the crner frequency? Adjust the frequency f the input signal until the utput amplitude is 70.7% f the input amplitude. Recrd this frequency and take a picture f the signal with the camera functin in the sftware and include it in yur reprt. The crner frequency fr an RC circuit is given by f C=/(2RC). Hw clse is the crner frequency f the circuit yu built t the calculated ne? Summary Transfer functins relate the utput vltage t the input vltage fr all frequencies f a circuit. Yu can use transfer functins t determine the type f filter a circuit represents. Yu can derive the transfer functin fr a series circuit by using circuit analysis rules t cmbine the cmplex impedances int a rati H = Zut/Zin. Part C Transfer Functins, Filters and RLC Circuits Backgrund Phasrs: Circuits, like mst engineering systems, can be divided up int basic building blcks, each with its wn functin. If we assume that the functin f such a building blck is t change a vltage in sme way (i.e. filter it, amplify it, etc.), then we label the input vltage as V in and the utput vltage as V ut. Since it is easiest t reference the phase f the input t zer, we can write a sinusidal input vltage as V in = V cs(t) and the utput vltage as V ut = V cs(t + ) where = 2f. When wrking exclusively with AC steady state cnditins, it is generally easier t analyze circuits using what is called phasr ntatin. Euler's identity tells us that expnentials with imaginary arguments can be related t sines and csines by e j(t + ) = cs(t + ) + jsin(t + ) that permits us t write the utput vltage, fr example, in the equivalent frm V ut = Re{V e j(t + ) } = Re{V ut e jt } where V ut = V e j is a cmplex number and is called the phasr frm f V ut. Much f the time we find it cnvenient t cnsider the real and imaginary parts f this cmplex representatin separately V ut = V Re + jv Im where V Re = V cs() and V Im = V sin() Thus, we will als be keeping track f the real and imaginary parts f vltages and currents. When we run PSpice simulatins, we will be able t add vltage markers that will d this fr us. One can access these markers by starting at the PSpice menu, then ging t the Markers menu and, finally, the Advanced menu r use Add Trace n plts. K.A. Cnnr, S. Bnner, P. Schch - 0 -

11 If we apply the phasr frm t the equatins that characterize vltage and current fr capacitrs and inductrs, we btain the fllwing V L = Re{V e jt } = L di L/dt = L (d/dt) Re{I e jt } = jl Re{I e jt } = jl I L r mre simply, V L = jl I L and, fllwing the same kind f analysis, I C = jc V C Thus, instead f a differential equatin, we btain the same kind f algebraic relatinship we had fr resistrs, Ohms Law, except that the impedances (a generalizatin f resistance) f inductrs and capacitrs are given by Z L = jl and Z C = /(jc). Fr resistrs, Z R = R, while in general Z = R + jx. RLC circuits: Circuits with resistrs, inductrs and capacitrs are called resnant circuits. They find wide applicatin in electrnics because there are many circumstances in which we wish t prduce r blck a single frequency. There are tw cmmn types f resnant circuits, parallel and series cmbinatins f a resistr, an inductr and a capacitr. This figure shws series RLC circuits with 4 pssible input-utput chices. Figure C-. Oscillating systems usually have such a natural resnance. This frequency, where the inductive and capacitive impedances cancel, is called the resnant frequency. The expressin fr the resnant frequency is. Mre cmplex filters: Simple RLC circuits can be used t create mre types f filters than simple RC r RL circuits. Yu can create any f the fur generic circuit types shwn in Figure C-2. 0 LC Figure C-2. Each f the abve circuits has a lw frequency and a high frequency apprximatin, fund in the same manner as we have seen fr RC and RL circuits. Lking at the limiting cases f H(j) fr frequencies near zer and fr very large frequencies, we can determine which type f filters they are. Fr example, nte that at bth high and lw frequencies, the transfer functin fr case C-(d) is equal t ne, and thus the input vltage appears unchanged at the utput. [If yu want t verify this, redraw the circuit at lw and high frequencies replacing the capacitr and the inductr by the apprpriate apprximatin (shrt r pen). Yu shuld be able t see that in each case, the utput pint, C, is at the same vltage as the surce. This means that the input signal is appearing at the utput. The circuit is passing very lw and very high frequencies.] Als, since the impedance f a capacitr is negative imaginary K.A. Cnnr, S. Bnner, P. Schch - -

12 and that f an inductr is psitive imaginary, there will be a frequency where the net impedance acrss the utput terminals C and D will be zer. Near this frequency, the utput vltage will be zer r at least very small. These frequencies are rejected by this filter. It must be a band-reject filter. Mdeling electrical cmpnents: When we build circuits, we use a functin generatr t prduce the input vltage, a resistr, an inductr and a capacitr. Hwever, we cannt use just a vltage surce and three ther cmpnents when we mdel them in PSpice, since real devices usually cannt be mdeled by a single parameter. Let us g thrugh each f the fur cmpnents f circuits and see what is necessary fr realistic analysis r simulatin. Functin Generatr Mst cmmercial functin generatrs have an internal resistance f 50Ω. Thus, in a realistic circuit, we must use an ideal sinusidal vltage surce and a resistr t represent the functin generatr. In the case f the FG n the Analg Discvery this resistance can be neglected. Resistr Except at very high frequencies, resistrs behave in an essentially ideal manner. Thus it is almst always sufficient t represent the resistr in a circuit as a single resistr. Inductr Since inductrs are made with a lng piece f wire, they usually have a significant resistance, in additin t their inductance. Thus, we must include an additinal resistr t mdel the resistance f a real inductr. Capacitr A capacitr typically cnsists f tw large metal plates separated by a thin insulatr. If the insulatr is very gd, almst n current will flw between the plates. Then, like the resistr, the capacitr will behave in an essentially ideal manner and we dn t need t add any extra cmpnents in a circuit t represent a capacitr. N cmpnents we can make are really ideal. Resistrs als have inductance and, smetimes, capacitance. Inductrs have capacitance. Capacitrs have resistance and inductance. Frtunately, we have figured ut hw t make these devices s that they behave in a nearly ideal manner fr quite a brad range f frequencies. Hwever, when we push the limits f a circuit, we have t remember that it may behave as if it cnsists f mre cmpnents that we can see when we build it. When the RLC circuit is built, we will nly see the three cmpnents R, L and C. Experiment Simulatin f an RLC Circuit In this sectin, we will use PSpice t simulate an RLC circuit and plt the magnitude and phase f its transfer functin. The PSpice mdel f the inductr must include bth L and R Set up the circuit as shwn in Figure C-3 belw. V V V Vin L 2 00mH R 90 Inductr L with intrinsic R 2 2 Vut C uf Figure C-3. Assume that the amplitude f V is 200mV, (400mVp-p). Nte that there is nthing t represent the impedance f the functin generatr (assumed zer) and R represents the resistance f the 00mH inductr. Add a vltage marker at the input f L (tp f V). This is the input vltage. Add a secnd marker at the upper right hand crner between R and C. This is the utput vltage. Perfrm an AC sweep frm 0Hz t 00kHz. Since we have nw dne several AC sweeps, can yu explain why we d Decade sweeps rather than Linear sweeps? Create a duble plt (tw plts in the same windw) f the magnitude and phase f the transfer functin. K.A. Cnnr, S. Bnner, P. Schch - 2 -

13 Use the Add Plt t Windw menu under the Plt menu in Prbe. Add a trace f the abslute value f the rati f V ut t V in, t the tp plt. Yu can use the abs() functin fr this purpse. [The trace expressin shuld lk smething like abs(v(c:)/v(r:)). T find the actual names f yur input and utput vltage traces, lk at the bttm left hand crner f yur plt.] This is the magnitude f the transfer functin. On the bttm plt display the trace f the phase f the transfer functin Vut/Vin. Add a trace f the phase f the rati f Vut t Vin. Yu can use the phase functin (p) fr this purpse. [The trace expressin shuld lk smething like p(v(c:)/v(r:)). T find the actual names f yur input and utput vltage traces, lk at the bttm left hand crner f yur plt.] What is the phase shift between the utput and input at lw and high frequencies? Cpy this plt. Label the resnant frequency (extreme pint) with the cursrs. Yu shuld als calculate the value fr the theretical resnant frequency and mark it n the plt. f 0 2 LC Yu will als mark the actual resnant frequency f the circuit yu build in the next sectin n this plt. Nte that the calculated value is clse t, but nt exactly at, the resnant frequency. The equatin gives yu a simple mathematical way t get clse t the resnant frequency f a circuit. Smetimes the expressin will give the exact value but it depends upn the cnfiguratin f the circuit. The exact value f the resnant frequency can be fund by examining when denminatr f the transfer functin ges t zer. We d nt g int these details in this class. Fr us the apprximatin is sufficient. What type f filter is this? Build the Circuit Nw yu can build the circuit in Figure C-3 n yur prtbard, hk it t the functin generatr and cmpare its behavir t the simulatin. Befre yu build the circuit, yu shuld measure the exact values f the cmpnents. Measure the DC resistance f the inductr s resistr using the DMM. D nt use the impedance bridge fr this measurement. Measure the inductance f the inductr and the capacitance f the capacitr using the impedance bridge n the center table. The cnnectrs may be a bit lse. Be sure they are making gd cntact. Change yur PSpice simulatin t crrespnd t the actual parameters f yur circuit. Run the AC sweep again t see what it lks like nw. Cpy the AC sweep plt and include it in yur reprt. Are there any significant differences between the tw simulatins? Build the circuit n yur prtbard. Remember that yu need nly tw cmpnents: the 00mH inductr (with an internal resistance f abut 90Ω) and the F capacitr. The F capacitr will have 05 written n it if it desn t have F written directly. Remember that R represents the resistance f the 00mH inductr. DO NOT add a separate resistr t yur circuit fr this. Find the resnant frequency f the circuit. This is the pint where the vltage f the utput is at a maximum. It will exceed the input vltage at this pint. Yu shuld find it is clse t the resnant frequency yu fund using PSpice and the ne yu calculated using the theretical equatin. It will nt fall in exactly the same place because f errrs intrduced by the tlerances f the cmpnents and the influence f ther parts f the circuit. Mark the actual resnant frequency n the AC plt yu generated using the theretical cmpnent values. Nte that depending upn hw yu fund the resnance, the same circuit gave yu three similar, but nt exact, values. Determine 5 representative frequencies that cver the range f yur simulatin: a lw frequency, a high frequency, the resnant frequency, sme pint between the resnant frequency and the high frequency, and sme pint between the resnant frequency and the lw frequency. D nt chse a pint where the utput disappears cmpletely. K.A. Cnnr, S. Bnner, P. Schch - 3 -

14 At each frequency, determine the rati f the input and utput amplitudes (magnitude f the transfer functin) and the phase shift. There is n magic buttn fr determining either the transfer functin r the phase shift between tw signals. Yu will have t d this by displaying the input and utput vltages n the scpe and manually determining the rati f the amplitudes and the phase shift. Cursrs will be useful. Put the values that yu measured n the PSpice AC sweep plt yu made using the actual cmpnent values. Summary In this sectin, yu extended yur knwledge f transfer functins and filters t include circuits with all three cmpnent types: R, L and C. These circuits are called resnant circuits. Fr these circuits the center f the band fr band pass and band reject filters ccurs at the resnant frequency. In high and lw pass filters, the resnant frequency can be used t determine the lcatin f the transitin between passed and rejected frequencies. Part D Equivalent Impedance Backgrund Cmbined Impedance f Parallel Cmpnents: We have cvered several series circuits with a variety f different cmpnent cmbinatins. What abut parallel circuits? We knw that we cannt use a simple vltage divider analysis n a resistive circuit with parallel cmpnents. We have t cmbine the parallel cmpnents until we nave a series circuit and then we can apply the vltage divider. The same is true f cmplex impedance. If we cmbine the impedances until we have a series circuit, then we can find the transfer functin fr the simplified circuit using the same methd already described in this experiment. The rules fr cmbining cmplex impedances are the same as thse fr cmbining resistrs: Experiment series parallel Z T Z n Z Z Z 2 Z 2 Z n Z n Cmbining Impedances In this part f the experiment, we will cnsider what happens when we cmbine tw impedances in parallel. Figure D-. Set up a vltage divider in Capture with tw 50Ω resistrs and a VSIN surce with 200mVamplitude, khz frequency and n DC ffset. Add a F capacitr in parallel with the secnd resistr, as shwn in Figure D-. Since we are ging t d an AC sweep, dn t frget t set the AC parameter t the amplitude f yur signal, 200mV. Again, yu can d this directly frm the parameter displayed r using the spreadsheet. Set up an AC sweep f this circuit. D a lgarithmic sweep with a start frequency f Hz and an end frequency f 5MEGHz. Yu shuld use abut 00 pints per decade. The AC Sweep Type has been chsen as Lgarithmic - Decade since frequency effects usually nly becme bvius when we change rders f magnitude. This generates a lg K.A. Cnnr, S. Bnner, P. Schch - 4 -

15 scale fr frequency. The start frequencies and end frequencies are chsen t cver an interesting range. Usually this range is selected frm sme knwledge f the expected perfrmance f the circuit. Hwever, since we are assuming that we knw very little abut this circuit, we can set the range t be rughly that cvered by a typical functin generatr. Fr what range f frequencies des the capacitr change the vltage acrss R2 by less than 5%? Use the cursr t find a reasnably precise answer t this questin. Mark the lcatin n yur plt. Cpy this plt and include it in yur reprt. Let us see if we can figure ut at least the magnitude f the equivalent impedance f the cmbinatin f R2 and C at the frequency f MegHz. G back t yur schematic and change the value f the surce (V) frequency t MEG. Set up a transient analysis. Click n the Edit Simulatin Settings buttn and set up fr transient analysis using smaller times. Since MHz is 000 times larger than khz, yu will have t make the run time and step size 000 times smaller t prduce three cycles f the scillating vltage signal. Run the transient analysis. In the circuit yu are analyzing, R2 and C tgether have a different impedance at different input frequencies. This means that at any ne given frequency, we culd replace the cmbinatin by a single resistr (that we will call Z). Nte that at the higher frequencies the vltage is very small and the cmbinatin f the capacitr and R2 shuld lk like a very small resistr indeed. Yu shuld see that the surce scillates as it did befre, althugh at a much higher rate and that the vltage acrss R2 and the capacitr seems t nt change with time at all. Actually, the latter vltage is still scillating, but at such a small amplitude that yu cannt see it. Delete the trace that shws the surce vltage. Click n the nde label at the bttm f the PROBE windw that crrespnds t the side f resistr R cnnected t the surce. Hit the Delete buttn n yur cmputer keybard. Using the cursrs, determine the amplitude f the sine wave scillatin acrss R2 and the capacitr. The vltage will als have a DC level, but we nly want t determine the sine wave amplitude. The peak-t-peak amplitude can be determined by subtracting the vltage at a minimum frm the vltage at a maximum. The actual amplitude f the sine wave will be half the peak-t-peak value. Write dwn the amplitude yu determined here. Frm this amplitude and yur knwledge f hw vltage dividers wrk, determine the magnitude f the equivalent impedance f the R2/C cmbinatin, that we are calling Z. [Hint: Figure D-2 shws the vltage divider. Yu can use the vltage divider equatin: Vz =V*(Z)/(R+Z)] R 50 V Z Equivalent Impedance 0 Figure D-2. Check yur answer by replacing the capacitr/resistr cmbinatin with a single resistr with the value f Z yu calculated. Rerun the simulatin. Des the utput have the same amplitude as the cmbinatin? Nw we can check if the transfer functin gives us the same amplitude. Use the parallel rules fr impedance t cmbine /jc and R. Then set up the transfer functin fr the circuit. Determine its magnitude at MegHz. (Dn t frget that =2f). Multiply the value f H at this frequency by the input vltage amplitude (200mV). Is the utput amplitude f yur plt cmparable t the ne yu calculated? K.A. Cnnr, S. Bnner, P. Schch - 5 -

16 Summary In this sectin yu explred what happens when yu cmbine tw impedances in parallel. Yu have learned that it is pssible t find transfer functins fr and analyze circuits with cmpnents in parallel. Yu simply need t cmbine the impedance, Z, f the parallel cmpnents int an equivalent impedance using rules similar t thse used t cmbine resistrs. K.A. Cnnr, S. Bnner, P. Schch - 6 -

17 Checklist and Cnclusins Prvide the fllwing packet. Include the cver/signature page attached t the end f this handut. The signatures are required fr all statements with a signature line next t it. They must be signed by TAs r Prfessr(s) after seeing the results n the cmputer screen. Give all required results and answer the questins cncisely. It is intended that these be quick t write. The fllwing shuld be included in yur experimental checklist. Everything shuld be labeled and easy t find. Partial credit will be deducted fr pr labeling r unclear presentatin. ALL PLOTS SHOULD INDICATE WHICH TRACE CORRESPONDS TO THE SIGNAL AT WHICH POINT. Hand-Drawn Circuit Diagrams fr all circuits that are t be analyzed using PSpice r physically built and characterized using yur Analg Discvery bard. Part A RC circuit, RL circuits, and AC Sweeps (22 pints) Include the fllwing plts:. PSpice transient plt f RC circuit (in Figure A-2). (2 pt) 2. PSpice AC sweep plt f the RC circuit vltage. (2 pt) 3. PSpice AC sweep plt f the RC circuit phase. (2 pt) 4. PSpice transient plt f RL circuit (in Figure A-3). (2 pt) 5. PSpice AC sweep plt f the RL circuit. (2 pt) 6. PSpice AC sweep plt f RL circuit phase. (2 pt) Answer the fllwing questins:. What is the amplitude and phase f the utput f the RC circuit at khz? (2 pt) 2. In what frequency range is the amplitude f the utput f the RC circuit abut equal t the input amplitude? In what frequency range is the amplitude f the utput abut zer? (2 pt) 3. In what frequency range is the phase f the utput f the RC circuit abut equal t the input phase? In what frequency range is the phase f the utput abut -90? (2 pt) 4. What is the amplitude and phase f the utput f the RL circuit at khz? (2 pt) 5. In what frequency range is the amplitude f the utput f the RL circuit abut equal t the input amplitude? In what frequency range is the amplitude f the utput abut zer? ( pt) 6. In what frequency range is the phase f the utput f the RL circuit abut equal t the input phase? In what frequency range is the phase f the utput abut +90? ( pt) Part B - Transfer Functins and Filters (26 pints) Include the fllwing plts:. PSpice plt f capacitr and resistr sum adding t the input vltage at khz. (2 pt) 2. PSpice plt f capacitr and resistr sum adding t the input vltage at 0Hz. ( pt) 3. PSpice plt f capacitr and resistr sum adding t the input vltage at 0kHz. (2 pt) 4. PSpice plt f transfer functin f RC circuit (Figure A-2) with crner frequency marked. (2 pt) 5. PSpice plt f transfer functin f RL circuit (Figure A-3) with crner frequency marked. (2 pt) 6. Wave Frms picture f RC circuit at khz. ( pt) 7. Wave Frms picture f RC circuit at crner frequency. ( pt) Answer the fllwing questins:. Write ut the mathematical expressins fr the utput vltage f the capacitr fr the first RC circuit case yu cnsidered (plt ). Write it in the frm V(t)=A Sin(t+). ( pt) 2. What kind f filter is the RC circuit? ( pt) 3. At what frequency did yu find the crner n the PSpice plt f the transfer functin f the RC circuit? What frequency did yu calculate using f=/(2rc)? Hw d the tw cmpare? (3 pt) K.A. Cnnr, S. Bnner, P. Schch - 7 -

18 4. Find the transfer functin f the RC circuit in terms f R, C and j. Take the limit f this functin at very lw and very high frequencies. Shw that these results are cnsistent with the PSpice plt f the transfer functin. (3 pt) 5. What kind f filter is the RL circuit (Figure A-3)? ( pt) 6. Derive the equatin fr the crner frequency f the RL circuit. (2 pt) 7. At what frequency did yu find the crner n the PSpice plt f the transfer functin f the RL circuit? What frequency did yu calculate using the equatin yu derived in the previus questin? Hw d the tw cmpare? (3 pt) 8. At what frequencies did the utput f the RC circuit yu built lk rughly the same as the input? At what frequencies did the utput disappear int the nise? ( pt) Part C (6 pints) Include the fllwing plts:. PSpice plt f the transfer functin f the RLC circuit in Figure C-3 (magnitude and phase), with three resnant frequency values marked [PSpice value, calculated value, experimental value] (riginal cmpnent values). (3 pt) 2. PSpice plt f the transfer functin f the RLC circuit in Figure C-3 (magnitude and phase), with the resnant frequency marked (real cmpnent values). Als put the 5 experimental pints n this plt. (3 pt) Answer the fllwing questins:. Why is it necessary t plt the phase and the magnitude f the transfer functin separately, rather than n the same plt? ( pt) 2. Fr the three RLC circuits labeled (a), (b) and (c) in Figure C- indicate what type f filter the circuit is (high pass, lw pass, band reject, r band pass filter) and explain why each is the filter it is. [Filter (d) is discussed in the Backgrund sectin fr part C.] Recall that a capacitr can be mdeled as an pen circuit al lw frequencies and a shrt at high frequencies. Als recall that an inductr can be mdeled as a shrt at lw frequencies and an pen circuit at high frequencies. Redraw the circuits at lw and high frequencies and cnsider the value f the utput between C and D fr each case. Yu can check yur answers in PSpice if yu want. (3 pt) 3. What is the phase shift between the utput and input f plt abve at lw and high frequencies? Des the phase shift change when yu adjust the values f the cmpnents t create plt 2? Why r why nt? (2 pt) 4. Determine the resnant frequency f the RLC circuit yu analyzed with PSpice. (This ccurs at the extreme pint.) Calculate the resnant frequency with the equatin f = /[2(LC)]. Hw similar are they? What factrs d yu think accunt fr the discrepancy? (3 pt) 5. Why d yu suppse it is that, in practice, we generally use filters designed with capacitrs and nt inductrs? ( pt) Part D (2 pints) Include the fllwing plts:. PSpice AC Sweep plt f circuit in Figure D- with 5% pint marked with cursr. (2 pt) Answer the fllwing questins:. When the capacitr is added, fr what range f frequencies des the capacitr change the vltage acrss R2 by less than 5%? ( pt) 2. What is the amplitude f the utput f the parallel RC circuit? What is the value yu calculated fr the equivalent impedance (Z) f the parallel cmbinatin at MegHz? (3 pt) 3. Find the transfer functin fr the parallel RC circuit. Determine the magnitude f the transfer functin at MegHz. Calculate the amplitude f the utput vltage fr this circuit at MegHz? Hw well des it agree with the utput amplitude yu fund using PSpice? (6 pt) List member respnsibilities (4 pt) Nte that this is a list f respnsibilities, nt a list f what each partner did. It is very imprtant that yu divide the respnsibility fr each aspect f the experiment s that it is clear wh will make sure that it is cmpleted. Respnsibilities include, but are nt limited t, reading the full write up befre the first K.A. Cnnr, S. Bnner, P. Schch - 8 -

19 class; cllecting all infrmatin and writing the reprt; building circuits and cllecting data (i.e. ding the experiment); setting up and running the simulatins; cmparing the thery, experiment and simulatin t develp the practical mdel f whatever system is being addressed, etc. New Summary/Overview Sectin There are tw parts t this sectin, bth f which require revisiting everything dne n this experiment and addressing brad issues. In this experiment, yu must cmplete this sectin and yu will receive feedback n yur wrk, but n frmal grade. That will begin with Exp3.. Applicatin: Identify at least ne applicatin f the cntent addressed in this experiment. That is, find an engineered system, device, prcess that is based, at least in part, n what yu have learned. Yu must identify the fundamental system and then describe at least ne practical applicatin. 2. Engineering Design Prcess: Describe the fundamental math and science (ideal) picture f the system, device, and prcess yu address in part and the key infrmatin yu btained frm experiment and simulatin. Cmpare and cntrast the results frm each f the task areas (math and science, experiment, simulatin) and then generate ne r tw cnclusins fr the practical applicatin. That is, hw des the practical system mdel differ frm the riginal ideal? Example frm Experiment : Engineering Design Prcess. One f the key circuit cnfiguratins we will see is the vltage divider. Yu shuld be n the lkut fr it in every experiment and prject because almst all circuits invlve sme cmbinatin f cmpnents in series. Mst applicatins f a vltage divider use this device t prvide a reference vltage. Fr example, in a thermstat, a temperature sensr will utput a vltage prprtinal t temperature. This vltage must be cmpared t a reference t see if a furnace r air cnditiner shuld be turned n. If the sensr vltage is abve he reference then the A/C is turned n, etc. This can als be used fr a light sensr system r fr the cntrl f anything that can be measured by a sensr that utputs a vltage. 2. The ideal peratin f a vltage divider can be easily derived frm Ohm s Law and simple circuit analysis. Fr the cnfiguratin shwn at the right, the relatinship between the input and utput vltages is given by this equatin. If yu have a vltage surce V in, then a reference vltage at any level frm V in t 0 can be prduced by this circuit. In practical applicatin, it is necessary t measure the vltage V ut in rder t use it t cntrl smething like a lamp r furnace. In bth simulatin and experiment, cnnecting an scillscpe r meter acrss Z 2 will add a resistr t the circuit in parallel with Z 2. Bth the experiment and simulatin shwed that this additinal resistr must be much larger than Z 2 r the reference vltage will be changed. Thus, frm experiment and simulatin, we cnclude that a mre cmplete mdel f the practical vltage divider (at least at this pint in this curse) shuld include the lad and the restrictin that the lad resistance must be much larger than the divider resistrs. The ideal divider inf cmes frm Wikipedia and the laded divider frm a website at Tufts. The laded divider is the practical system mdel that gives us cnfidence t reliably use the divider in a real wrld applicatin. K.A. Cnnr, S. Bnner, P. Schch - 9 -