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# Electronics I PHYS 252

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This 38 page Class Notes was uploaded by Hayley Jenkins on Thursday October 29, 2015. The Class Notes belongs to PHYS 252 at College of William and Mary taught by Staff in Fall. Since its upload, it has received 12 views. For similar materials see /class/231185/phys-252-college-of-william-and-mary in Physics 2 at College of William and Mary.

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Date Created: 10/29/15

Chapter 4 Passive Analog Signal Transmission Chapter 4 Passive Analog Signal Processing In this chapter we introduce lters and signal transmission theory Filters are essential components of most analog circuits and are used to remove unwanted signals ie noise from the actual signal Transmission lines are essential for sending signals from one device to another such as from a detector to a data acquisition module 1 Filters Filters are ubiquitous in analog electronic circuitry In fact if you see a capacitor or an inductor in a circuit there is a good chance it is part of a lter Filters are frequently used to clean up ie remove high frequency noise power supplies and remove spurious frequencies from a signal frequently 60 Hz switching power supply noise in computers display screen noise ground loop noise and RadioFrequency RF pickup A RC Filters RC lters are by far the most common lters around They are simple to make ie just a resistor and a capacitor reliable and involve relatively simple design calculations 1 The Low Pass RC Filter The lowpass RC lter or integrator is used to remove high frequencies from a signal Applications include the removal of RF pickup noise and reducing ripple voltages on power supplies A generic RC low pass lter circuit is shown on the right We have already calculated its performance in the previous chapter using Fourier analysis and complex impedances We recall the results equation 28 V VC V0 sin e 1 011 Where sin 1 1aRC2 and Vm V0e quantities we can compute the Gain and Phase performance of the lter The gain is de ned as Gain VM as go 7r2 this is just the part after the wt in the exponent From these and the phase VIN E C VOUT The RC lter is just a voltage divider with complex impedances so we can calculate the gain easily Chapter 4 Passive Analog Signal Transmission Vili C 1 sin 2 Vm Rl1wC l1a1RC l1aRCZ The phase of the output voltage with respect to the input is easily computed and is given b tan 7 tan 7 7r 2 cot wRC 3 At a l RC the output voltage drops to 1 J5 of the input voltage and consequently the power transmission drops to 50 or 3dB1 At this frequency the voltage across the resistor and the voltage across the capacitor are equal in amplitude but i Z out of phase with the drive voltage The average value of V across either the resistor or the capacitor is down by a factor of 2 from the drive voltage Consequently this frequency characterizes the RC circuit completely and is called the 3dB frequency We can rewrite the gain and phase equations in terms of the 3dB frequency l gain 4 tan ff345 5 V1ffgd5z with f 1272RC The Bode plots loglog or semilog plots for the gain and phase ofthe low pass filter are the following 0 0 Logoa 3 2 0 D g 3 E l2 1RC L 9m 7 1RC 1 A dB or decibel is a notation for quantifying a ratio of two numbers For power a dB is de ned as dB lOlogmPPu From this de nition We can see that a ratio of05 is roughly 3 Hence 3db is the same as halving a signal For Voltage or current a dB is de ned as dB ZOlogmVVU So at 3dB the Voltage or current has dropped to 1 15 of is input Value Chapter 4 Passive Analog Signal Transmission Past the 3 dB point the loglog slope for the gain is 20 dBDecade or 6 dBOctave The lowpass RC lter is also called an integrator because it integrates currents with frequencies above fm In other words V0 r I1tdt see chapter 3 equation 8 for currents with frequency components above fm 2 The HighPass RC Filter The highpass RC lter or differentiator is used to remove low frequencies from a signal Applications include the removal of DC bias voltages and 60 Hz pickup voltages A generic RC highpass lter circuit is shown on the right Mathematically highpass lter can be treated the same VW way a their low pass cousins they are almost identical except C that Vout measures the voltage drop across the resistor insteadof the capacitor Qualitatively the capacitor blocks DC and low frequency signals VOW If we treat the highpass lter as a complex impedance voltage divider then we obtain immediately R VIM 7 R 7 imRC 7 wRC 7 005w 6 Vm RliaC limRC 1wRCZ tanw lwRC 7 We can rewrite these using only fMB to produce more compact design equations 7 ffm gamiJ UrfzdgZ 8 tanw dBf 9 The Bode plots loglog or semilog plots for the gain and phase of the highpass lter are the following rr2 2 2 E E RC Logm 1RC Loam Chapter 4 Passive Analog Signal Transmission 3 RC Filter Design When designing an RC lter you need to think about two things 1 Choose an appropriate f3 2 Make sure the impedance is appropriate for your desired output load Step 1 is straightforward and depends on the frequencies you want to pass and block In Step 2 you pick R so as to satisfy the impedance requirements of your signal source and your signal destination Typically the signals we want to pass are around f3 so the impedance of the capacitor and the impedance of the resistor will be about the same When filtering an input choose the resistance to be about 10 times smaller than the input impedance of the next stage This will prevent the next stage from loading the filter You would also like to choose the resistance to be at least 10 times larger than the previous stage of your circuit s output impedance Choose the capacitor for the appropriate f3 lZnRC o If the signal you want to pass is low frequency choose f3 m 2fpm and hence C 147 fm 0 Ifthe signal you want to pass is high frequency choose f3 m fpm 2 and hence C Wm 3 Combination Filters If you put several RC filters in a row you can make a more sophisticated signal filter For example several similar lowpass filters placed on after the other will produce a steeper fall off of the gain One can also use a lowpass followed by a high pass to Pmduce a bandpass lter f3dBJuwpass gt f3dBhzghpass Design tip When you are designing a bandpass filter you should make sure that the resistor in the second stage is larger than the first stage by about a factor of 10 Otherwise the second stage will load the first stage and shift the effective f3 frequency of the first stage 4 Applications Here are some specific applications of RC filters Blocking Capacitor This is a high pass filter that is used to eliminate DC Suppose that you want to measure small timedependent signals that happen to float on a high voltage If you use a blocking capacitor then the high voltage DC will not get through to your detection Chapter 4 Passive Analog Signal Transmission electronics but the signal will get through Choose f3 low to insure that your entire signal gets through Ripple Eliminator This is a low pass lter used to build power supplies Since most of our power is 60 Hz AC our DC power supplies will convert AC to DC but there will always be some residual 60 Hz ripple A low pass lter with f3 set well below 60 Hz will work You do not use a resistance in this case but let the combination of the loading resistance RL and the Thevenin resistance of the previous components serve as your R This usually requires a large capacitor since RL might be quite small when you use the power supply If the capacitor is not big enough then f3 will then shift to a higher frequency and the 60 Hz ripple will reappear Chip Supply Clean Up Frequently the voltage which you supply to a chip component such as an opamp we will study these later in the semester may be clean when it comes out of the power supply but will pick up noise by the time it reaches the component In this case a 10 7 100 nF capacitor placed at the supply leads of the component will remove the high frequency pickup noise Noise Eliminator Any signal line is susceptible to picking up high frequency transients especially if there are motors or switching power supplies or FM radio stations nearby A noise eliminator is a low pass filter with a high value of f3 dB Integrator If you build an RC filter but set the value of f3 much higher than the highest frequency in your signal the filter integrates your signal From our earlier analysis when f ltlt f3 we can see that each low frequency voltage component will see a 7r2 same phase shift and its amplitude will be proportional to l f This is exactly the prescription for integration So a high pass filter sends high frequencies out on the resistor and the integral of very low frequencies on the capacitor Differentiator If you build an RC filter with f3 lower than the lowest frequency in your signal the filter differentiates your signal From our earlier analysis when f gtgt f3 each high frequency voltage component will see a 7r2 same phase shi and its amplitude will be proportional to f This is exactly the prescription for differentiation So a low pass filter Chapter 4 Passive Analog Signal Transmission sends low frequencies out on the capacitor and the derivative of high frequencies on the resistor 5 High frequency performance of capacitors In principle one could use an LR inductorresistor circuit instead of an RC circuit to make lowpass and highpass lters and obtain similar performance However in practice generally avoids inductors if possible Inductors tend to be physically larger more expensive and deviate further from ideal performance than capacitors While capacitors generally offer superior performance to inductors they also show signi cant deviations from the ideal Z liwCimpedance at high frequencies Capacitors will generally have a little bit of spurious resistance ie like a resistor and inductance ie like an inductor at high frequencies In fact circuit designers will o en model a real capacitor with the following simple circuit though more complex circuits are sometimes necessary C C R L The resistance is due to the nite the high frequency conductivity of the dielectric material separating the two conductor plates of the capacitor The inductance is due to two effects 1 the design of the capacitor especially the leads will contribute inductance and 2 Maxwell s equations for electrodynamics require that a capacitor behave have an inductance at high frequenciesZ The capacitor manufacturer will provide the speci cations for the spurious resistance and inductance of their capacitance The plot below shows the frequency dependence of the impedance of a Cornell Dubilier acrylic surface mount lm capacitor Impedance ohms 1 mm moan mo Frequency lkHi 2 The Feynman Lectures on Physics Vol 2 by R P Feynman R B Leighton and M Sands p 232 6 Chapter 4 Passive Analog Signal Transmission A common remedy for dealing with the inherent inductance of a capacitor at high frequencies is to place a small capacitor 10 7 100 pF in parallel with the main capacitor in RC lter The high frequency performance of the small capacitor will generally much better than that of the main capacitor the small will pickup the high frequency signal when the main larger capacitor begins to have a significant inductance The gain falloff of the RC filter will no longer be 20 dBdecade but at least a lowpass filter will not start to behave like a highpass filter or viceversa B LC Filters RC filters are by far the simplest and the most common type of filter found in analog circuits however they suffer from a relatively slow roll off of the gain while the gain or attenuation slope can be made steeper than 20 dBdecade the transition region or knee of the curve the region where the gain changes from at to a loglog slope will always have the same shape and frequency width LC filters are more compleX but can be engineered to produce much sharper features and steeper falloff regions The standard design for LC filters is an LC ladder with an uninterrupted ground line such as in the 5Lh order filters shown below RTH L1 L2 L3 L4 c1 02 cs 04 c5 RLOAD T T T T T v 539 order lowpass LC lter Rm 0 02 c3 0 lt5 g L g L2 g L3 g L4 E RLOAD 5 order highpass LC lter The algebra required for computing the gain and phase Bode plots for these filters is generally quite cumbersome and a computer program ie Maple or Excel is generally useful for helping with the design A number of webapplets can also be found on the intemet for determining all the inductor and capacitor values that will produce the required filter performance Chapter 4 Passive Analog Signal Transmission Example 1 53911 order ButterWoth LC lowpass lter with fm10 kHz for 50 Q input impedance and 50 Q output impedance 1205 1206 1207 i205 1209 124 12711 123 lt24 lt25 tems tame W This ButterWoIth lter uses C1 C5 06946 pF C2 C4 30642 pF C3 4 pF L1 L4 5 mH andL2 L3 9397 mHButterWo11h lters have aver at passband and a reasonably regular phase change across the knee of the curve Example 2 53911 order Chebyshev LC lowpass lter with fm10 kHz for 50 Q input impedance and 50 Q output impedance 123 194 125 ms ems w This Chebyshev lteruses C1 C5 1125 pF C2C4 1486 pF C3 1505 pF Chapter 4 Passive Analog Signal Transmission Ll L4 0617 mH and L2 L3 0646 mH Chebyshev lters have a very sharp knee and fast cutoff but suffer from irregular transmission in the passband They also have a highly irregular phase variation at the knee of the lter The above Chebyshev and Butterworth lters can be scaled to another frequency or load impedance with the following rules me Rmme famesz JLM CW Rzuamzd f3dBn2w JCM 10 Rimde Kfsmmew Rluadn2w kf3dBpld Some remarks on LC lters i Generally the higher the order of the LC lter the sharper the cutoff will be however this usually requires a tradeoff in the regularity of the output phase ii LC lters do not have resistive elements and consequently they do not consume power and do not load the source signal However the load of the input device into which the lter sends its output will load the LC lter and can shift the f3 point considerably 7 always remember to include the load resistor in your calculations iii LC lters are widely used in RF circuits where active lters do not have the bandwidth to respond to high frequencies iv One cannot construct a lter with arbitrary gain and phase pro les While we have treated lters in Fourier space in the time domain lters must obey causality In fact one can derive KramersKronig relations for lters 11 Transmission Lines Signals are sent from one device to another or from one part of a circuit to another with transmission lines The quality of your transmission determines the quality of your signal whether you are connecting one device to another or a resistor to a capacitor LC Ladder Model of a Transmission Line A transmission line consists of two parallel conductors separated by a xed distance d which gives rise to an effective capacitance C per unit length The conductors will also produce a magnetic eld which is gives the transmission line an effective inductance L per unit length We can model the transmission line as a repeated network of series inductors and parallel capacitors or LC ladder as depicted in the gure below L L L L L C Chapter 4 Passive Analog Signal Transmission If the LC ladder is in nite and has an impedance Zg then if we add an extra LC ladder rung the total impedance should not change and we obtain the following relation ZOLZOC ZOW 11 icoLZ0lz39aC From this relation we can extract an expression for Zn and we obtain icoL L 0sz 0 2 C 4 If we consider that L and C are the 39 J and quot I quot 39J of a short 1 Z 12 section A6 of transmission line then as we take the limit A gt0 we have L gt0 and C gt0 but LC gt constant In this limit the equation for Zn becomes 20 13 where L and C are the inductance and capacitance respectively of the transmission line and Z is called the characteristic impedance of the transmission line It may seem surprising that a network of inductors and capacitors can have a real impedance and consequently consume power The explanation for this apparent paradox is that since the network is in nite power is owing from one LC ladder rung to the next ad infinitum so that power is constantly moving down the transmission line though it is not dissipated in either the inductors or capacitors Of course the power is consumed at the end of the transmission line when we attach a load resistor The LC ladder model is a high frequency model of transmission line and does not include the wire resistance which can contribute to signal attenuation Transmission Line Impedance Matching A transmission line with a characteristic impedance of Zn should be terminated with an load impedance of Zn if the transmission line is longer than 110 of the wavelength of the signal recallwavelength 6 f where c is the speed of light in the transmission line and f is the signal frequency If the transmission line is not properly terminated then the signal will be partially re ected back towards the source upon arrival at the load There are three main types of transmission lines wires twisted pairs and coaxial cables In this section we go over their performance characteristics 1 Wires 10 Chapter 3 Capacitors Inductors and Complex Impedance Chapter 3 Capacitors Inductors and Complex Impedance In this chapter we introduce the concept of complex resistance or impedance by studying two reactive circuit elements the capacitor and the inductor We will study capacitors and inductors using differential equations and Fourier analysis and from these derive their impedance Capacitors and inductors are used primarily in circuits involving timedependent voltages and currents such as AC circuits I AC Voltages and circuits Most electronic circuits involve timedependent voltages and currents An important class of timedependent signal is the sinusoidal voltage or current also known as an AC signal Alternating Current Kirchhoff s laws and Ohm s law still apply they always apply but one must be careful to differentiate between time averaged and instantaneous quantities AnAC voltage or signal is of the form Vt Vp coser 1 where a is the angular frequency VP is the amplitude of the waveform or the peak voltage and t is the time The angular frequency is related to the freguency f by f27rco and the period T is related to the frequency by T1f Other useful voltages are also commonly defined They include the peak to peak voltage VPP which is twice the amplitude and the RMS voltage VRMS which is VRMS Vp Average power in a resistive AC device is computed using RMS quantities PIRMSVRMS IPVPZ 2 This is important enough that voltmeters and ammeters in AC mode actually return the RMS values for current and voltage While most real world signals are not sinusoidal AC signals are still used extensively to characterize circuits through the technique of Fourier analysis Fourier Analysis One convenient way to characterize the rate of change of a function is to write the true function as a linear combination of a set of functions that have particularly easy characteristics to deal with analytically In this case we can consider the trigonometric functions It turns out that we can write any function as an integral of the form Vt Ifcoslta2t da 3 where I7 and are functions of the frequency a This process is called Fourier analysis and it means that any function can be written as an integral of simple sinusoidal 1 Chapter 3 Capacitors Inductors and Complex Impedance functions In the case of a periodic waveform this integral becomes a sum over all the harmonics of the period ie all the integer multiplicative frequencies of the period Vt Z Aquot cosncot gun 4 An implication of this mathematical fact is that if we can gure out what happens when we put pure sinusoidal voltages into a circuit then we will know everything about its operation Complex Notation In compleX notation we replace our sinusoidal functions by eXponentials to make the calculus and bookkeeping easier still Then we can include both phase and magnitude information We ll define eW Ecosgzzisingzz 5 where 1392 E l The general procedure for using this notation is 1 Change your problem into compleX algebra ie replace cos at with em 2 Solve the problem 3 Take the real part of the solution as your answer at the end II Capacitors One of the most basic rules of electronics is that circuits must be complete for currents to ow This week we will introduce an exception to that rule The capacitor is actually a small break in a circuit Try measuring the resistance of a capacitor you will find that it is an open circuit However at the inside ends of the capacitor s lead it has little plates that act as charge reservoirs where it can store charge For short times you do not notice that the break is there Negative charge initially ows in to one side and out from out the other side just as if the two leads were connected For fast signals the capacitor looks like a shortcircuit But after a while the capacitor s reservoirs fill the current stops and we notice that there really is a break in the circuit For slow signals a capacitor looks like an open circuit What is fast and what is slow It depends on the capacitor and the rest of the circuit This week you will learn how to determine fast and slow for yourselves Capacitors serve three major roles in electrical circuits although all three are just variations of one basic idea 0 Charge integrators 0 High or low frequency lters 0 DC isolators Chapter 3 Capacitors Inductors and Complex Impedance To be able to do this analytically we will need to introduce a number of new concepts and some signi cant mathematical formalism In this process we will also develop a number of new concepts in analyzing electronic circuits Capacitance A capacitor is a device for storing charge and electrical energy It consists of two parallel conducting plates and some nonconducting material between the plates When voltage is applied positive charge collects on one plate and negative charge collects on the other plane Since they are attracted to each other this is a WW stable state until the voltage is changed again A capacitor s charge capacity or capacitance C is de ned as QC V 6 which relates the charge stored in the capacitor Q to the voltage across its leads V Capacitance is measured in Farads F A Farad is a very large unit and most applications use uF nF or pF sized devices Many electronics components have small parasitic capacitances due to their leads and design The capacitor also stores energy in the electric eld generated by the charges on its two plates The potential energy stored in a capacitor with voltage V on it is E lCV2 7 2 We usually speak in terms of current when we analyze a circuit By noting that the current is the rate of change of charge we can rewrite the de nition of capacitance in terms of the current as l l V dt 8 CQ CI 01 1Cd VCV 9 air This shows that we can integrate a function Ht just by monitoring the voltage as the current charges up a capacitor or we can differentiate a function Vt by putting it across a capacitor and monitoring the current ow when the voltage changes A Simple RC Circuit We will start by using looking in detail at the simplest capacitive circuit An RC circuit is made by simply putting a resistor and a capacitor together as a voltage divider To be concrete we ll put the resistor in rst can connect the capacitor to ground 3 Chapter 3 Capacitors Inductors and Complex Impedance By applying Kirchhoff s Laws to this circuit we can see VIN VOUT that l The same current ows through both the resistor and the R capacitor and 2 The sum of the voltage drops across the two elements equal C the input voltage 1 This can be put into a formula in the following equation VIN 1Rij1dt 10 C which can also be written as V l IEjldt 11 We can also put this into the form of a differential equation in the following way dstW R g 12a or CV W RCI39 1 12b These equations show that times are measured in units of RC and that what you see depends on how quickly things change during one RC time interval If the current changes quickly then most of the voltage will show up across the resistor while the voltage across the capacitor slowly charges up as it integrates the current If the voltage changes slowly then most of the voltage shows up across the capacitor as it charges Since this usually requires a small current the voltage across the resistor stays small But what happens at intermediate times To determine this quantitatively we will have to develop some more sophisticated mathematical techniques Solutions to RC Circuit Rather than produce the general solution we will concentrate on two special cases that are particularly useful The first will be for a constant voltage and the second will be a sinusoidal input To study a constant supply voltage on an RC circuit we set the left side of Equation 5 equal to a constant voltage Then we have a simple homogeneous equation with the simple solution for the current of a decaying exponential I loam 13 which will account for any initial conditions After a time of a few RC time periods this solution will have decayed away to the supply voltage Well isn t that kind of boring Chapter 3 Capacitors Inductors and Complex Impedance And now let s consider the other solution In the prior section we argued that if we can understand the RC circuit s behavior for sinusoidal input we can deal with any arbitrary input Therefore this is the important one Let s look at our simple RC circuit and suppose that we apply or drive a simple sine wave into the input VIN V0 cosat 14 In compleX notation this means that we will set the drive voltage to VIN V0 eXpiat 15 and we just have to remember to take the Real part at the end of our calculation If we put this drive voltage into the differential equation Equation 5 then it becomes a relatively simple inhomogeneous differential equation C dVIN dt d CVOimeXpz39cotRC7 16 t This is relatively simple because it shows up so often in physics that you might as well memorize the solution or at least the way to get the solution Note that mathematically it looks just like a driven harmonic oscillator We can obtain the solution by using the standard recipe for first order linear differential equations We start by rewriting equation as d 1 icoVO dt RC eXpz39 at 17 which we then multiply by eXpt RC to obtain eXptRC 6Xpl RC sz0 dt RC R The left handside of this equality can be rewritten under the form of a total derivative multiplication rule so that we now have eXpz39 a t 18 d t imV 1 1 t 0 39 t 19 all eXPRC R eXPW RCH This equation is easily integrable and can be rewritten as t icoV 1 teX 0 eX ia tdt 20 NRC R j p RCH The integral is straightforward and yields the following expression in 1 t It eX iwt CsteX 21 R c w P P RC The first term represent the steady state oscillatory behavior of the driven circuit while the second term describes the transient behavior of the current after switching on the driving voltage Since we are only interested in the longterm behavior of the circuit we 5 Chapter 3 Capacitors Inductors and Complex Impedance neglect the second term and concentrate on the first After a little bit of algebra we can rewrite the steadystate current as COVOC oRC z39 icoV l t 0 v z cot R a w H J1mRC2J1mRC2 xpacor 22 The second fraction can be interpreted as a phase term with tan C so that the a expression for the current becomes 1t 10 expiat 23 with V C V 10 0 0 cos 24 11coRC2 R The real solution of this simple RC circuit can be obtained by taking the Real part of equation 23 and is left as an exercise to the reader The solution of the simple RC circuit appears to be rather complicated and involved however it simpli es considerably when we plug equation 23 back in to the original integral equation from Kirchhoff s loop law equation 10 After integrating the exponential and a little bit of algebra we obtain Vntw1tR1t 25 1a This remarkably simple expression looks a lot like the standard Kirchhoff s loop law for resistors except that the capacitor term behaves with a frequency dependent imaginary resistance RC Impedance We will obtain the same solution as the one we obtained for the original voltage divider as long as we assign an imaginary frequency dependent resistance to the capacitor The imaginary part just means that it will produce a 7r 2 phase shift between the voltage and the current for a sinusoidal input We will call this impedance Z 26 C iwC I Now the solution for an RC divider becomes somewhat simplified We can compute the total current owing through the circuit as I Vm Vm Voe iaCl0e quot 2 V0 z39mRCoe t ECOWWM 20 RZC R1z39aC 1z39a2RC R 1z39a2RC R Chapter 3 Capacitors Inductors and Complex Impedance The voltage across an element is just this current times the element s impedance For the voltage drop across the resistor it is largely the same as before V R V0 cos e 27 For the capacitor we get the following voltage drop 1 V0 cos e VC IZC I z39V0 sin e 10C 12le 28 COS dummy2 1at 77r2 e V Sin 6 0 wRC 0 If everything is correctly calculated then the sum of the voltage drops across the two elements should be equal to the input voltage Let s try it V VC V0 cos isin e me e Voe Remember you get the actual waveforms by taking the real parts of these compleX solutions Therefore VR Vg cos cosat and Vc Vgsin coswt L7r2 Vgsin sinat This looks complicated but the limits of high frequency and low frequency are easy to remember At high frequencies gt 0 the capacitor is like a short and all the voltage shows up across the resistor At low frequencies gt 7r 2 the capacitor is like an open circuit and all the voltage shows up across the capacitor If you consider the leading terms for the elements with the small voltages you find that chmmqnj MM 1a2RC a RC wRCltimRCgt VR V 2 gtzaRClIas a gt0 1a2RC Thus at high frequency the voltage across the capacitor is the integral of the input voltage while at low frequency the voltage across the resistor is the derivative of the input voltage This says that as long as all the important frequencies are high the capacitor will integrate the input voltage If all the important frequencies are small the resistor will differentiate the voltage If there are intermediate frequencies or a mixture of some high and some low frequencies the result will not be so simple but it can be determined from the voltage divider algebra using compleX notation We finish be noting that the voltage on the capacitor is always 1I2 out of phase with the voltage on the resistor Chapter 3 Capacitors Inductors and Complex Impedance III Inductors An inductors is a coil of wire or solenoid which can be used to store energy in the magnetic field that it generates 1 is mathematically similar to a capacitor but has exactly the opposite behavior it es as a hort circuit for low frequencies and as an open circuit for high fre uencies ie it passes low frequency signals and blocks high frequency signals e energy stored in the field of an inductor with inductance L is given by the following formula E in 2 29 2 The unit of SI unit of inductance is the Henry H Commercially available inductors have inductances that ran e from nH to mH Small millimeter size and centimeter size solenoids typically have inductances in e ce in the m a and can sometimes have inductances o u to several H Most electronics components have small parasitic inductances due to their leads and design for example wire woun power resistors n an electric circuit a voltage or electromotive potential is generated across the terminals of the inductor when the current changes due to Faraday s law The voltage drop is given by the following simple expression d V L 30 dt From this equation we see that the inductor operates exactly opposite to a capacitor an inductor differentiates the current and integrates the voltage The LR circuit e can analyze the LR circuit in much the same way that we derived the operation of e RC circuit We VIN start by applying Kirchhost loop law to the LR circuit on the right and we find that R d VIN 1R Li 3 l Vour dt If we apply a constant voltage the solution can be calculated using the techniques developed for the RC circuit and we calculate that 1t10 eexpetj 32 The circuit approaches the steady state current IoV11vR with a time constant of DR 8 Chapter 3 Capacitors Inductors and Complex Impedance LR impedance Instead of solving the differential equation for the LR circuit with a sinusoidal applied input voltage such as that given by equations 14 and 15 as we did with the RC circuit we will just assume that the current has the form It 10 expiat 33 We plug this ansatz solution back into the differential equation of equation 31 and find that VIN ItR 139le 34 from which we deduce that the inductor behaves as a resistor with frequency dependent imaginary resistance The impedance of an inductor is therefore Z L icoL 35 Just as with the RC circuit we can apply Ohm s law to the circuit to calculate the total current Since R and L are in series we obtain V V m V m V V m oe oe 2E0 R exmchos exate 36 It 0 2m RZL Ria2L where the phase is given by tan a We calculate the voltage drop across the resistor using the expression for the current and find that VR ItR V0 cos e 37 The voltage drop across the inductor is calculated the same way and we find VL mum imLcos el We sin equot V0 cos equot 38 If everything is correctly calculated then the sum of the voltage drops across the two elements should be equal to the input voltage Let s try it V VL V0 cos isin e Voe e Voe You get the actual waveforms by taking the real parts of these complex solutions Therefore VR Vg cos cosat and VL V0 sin cosat 7V2 V0 sin sinat This looks complicated but the limits of high frequency and low frequency are easy to remember At high frequencies gt 7r2 the inductor is like an open circuit and all Chapter 3 Capacitors Inductors and Complex Impedance the voltage shows up across the inductor At low frequencies gt 0 the inductor is like a short circuit or just a plain wire and all the voltage shows up across the resistor It should also be pointed out that the voltage on the capacitor is always 12 out of phase with the voltage on the resistor IV Transformers Transformers are an ingenious combination of two inductors They are used to transfer power between two circuits by magnetic coupling The transformer changes an input voltage without affecting the signal shape similar to the voltage divider of last week However it has several important differences 0 It can increase as well as decrease a signal s amplitude ie AC voltage 0 It requires a timevarying AC input to work 0 It is much harder to fabricate It usually does not work well for very fast signals since inductors block high frequencies VIN VOUT Transformers are commonly used as a major component in a DC power supplies since they can convert a 120V AC wall voltage into a smaller voltage that is closer to the desired DC voltage eg 5V or ilS V Transformers are passive devices that simultaneously change the voltage and current of a circuit They have at least four terminals two inputs called the primary and two outputs called the secondary There is no real difference between the input and output for a transformer you could simply ip it around and use the secondary as the input and the primary as the output However for the sake of clarity we will always assume that you use the primary for input and the secondary for output The coupling between the input and output is done magnetically This allows transformers to have a number of interesting benefits including 0 There is no DC connection between input and output so transformers are often used to isolate one circuit from another 0 Transformers only work for time varying signals when the inductive coupling between the coils is greater than the resistive losses Since they have no external power the output power can not be greater than the input power P VP 2 V515 39 Usually we will assume equality but there are small resistances and hence resistive losses in the coils and a poorly or cheaply designed transformer many not have the input 10 Chapter 3 Capacitors Inductors and Complex Impedance and output sufficiently strongly coupled to each other Depending on the device and the signal the output power may well be less than the input power Transformers are most commonly used to change line voltage 120 V RMS at 60 Hz into a more convenient voltage High power transmission lines use transformers to increase the voltage and decrease the current This reduces 12R power losses in the transmission wires For our circuits we will use a transformer that reduces the voltage and increases the current Transformers are characterized by the ratio of the number of turns on the input and output windings The magnetic coupling in an ideal transformer will insure that the number of turns times the current owing is the same for the input and output I N N I N I 3 S P 40 P P S S 11 NS Since the voltage must change in the opposite manner to keep the input and output power the ratio of the voltages is the same as the ratio of the turns V N S S 41 VP Np Transformers are usually called stepup or stepdown according to whether the output voltage increases or decreases A transformer also transforms the impedance of a circuit since it changes the ratio of VI Using our rules above the ratio of output impedance to input impedance is the square of the ratio of turns 2 ZS IVS 1 NS ZP IS 1 NP So if you use a transformer as a stepup transformer it increases the voltage and the impedance at its output relative to its input If you use a transformer as a stepdown transformer it decreases the voltage and the impedance at its output Design Exercises Design Exercise 31 Using Kirchhoff s laws derive a formula for the total capacitance of two capacitors in parallel and a formula for the total capacitance of two capacitors in series Design Exercise 32 Using Kirchhoff s laws derive a formula for the total inductance of two capacitors in parallel and a formula for the total inductance of two inductors in series 11 Chapter 3 Capacitors Inductors and Complex Impedance Chapter 3 Capacitors Inductors and Complex Impedance In this chapter we introduce the concept of complex resistance or impedance by studying two reactive circuit elements the capacitor and the inductor We will study capacitors and inductors using differential equations and Fourier analysis and from these derive their impedance Capacitors and inductors are used primarily in circuits involving timedependent voltages and currents such as AC circuits I AC Voltages and circuits Most electronic circuits involve timedependent voltages and currents An important class of timedependent signal is the sinusoidal voltage or current also known as an AC signal Alternating Current Kirchhofi s laws and Ohm s law still apply they always apply but one must be careful to differentiate between time averaged and instantaneous quantities AnAC voltage or signal is of the form Vt VP coscot 31 where a is the angular frequency VP is the amplitude of the waveform or the peak voltage and t is the time The angular frequency is related to the freguency f by 0an and the period T is related to the frequency by T1f Other useful voltages are also commonly defined They include the peak to peak voltage VPP which is twice the amplitude and the RMS voltage VRMS which is VRMS Vp Average power in a resistive AC device is computed using RMS quantities PIRMSVRMS IPVPZ 32 This is important enough that voltmeters and ammeters in AC mode actually return the RMS values for current and voltage While most real world signals are not sinusoidal AC signals are still used extensively to characterize circuits through the technique of Fourier analysis Fourier Analysis One convenient way to characterize the rate of change of a function is to write the true function as a linear combination of a set of functions that have particularly easy characteristics to deal with analytically In this case we can consider the trigonometric functions It turns out that we can write any function as an integral of the form Vt jfcosmt dm 33 where I7 and are functions of the frequency a This process is called Fourier analysis and it means that any function can be written as an integral of simple sinusoidal 17 Chapter 3 Capacitors Inductors and Complex Impedance functions In the case of a periodic waveform this integral becomes a sum over all the harmonics of the period ie all the integer multiplicative frequencies of the period Vt Z A cosncot gun 34 An implication of this mathematical fact is that if we can gure out what happens when we put pure sinusoidal voltages into a linear circuit then we will know everything about its operation even for arbitrary input voltages Complex Notation In complex notation we replace our sinusoidal functions by exponentials to make the calculus and bookkeeping easier still Then we can include both phase and magnitude information We ll define eW Ecosgzzisingzz 35 where 1392 E l The general procedure for using this notation is 1 Change your problem into complex algebra ie replace cos at with em 2 Solve the problem 3 Take the real part of the solution as your answer at the end II Capacitors One of the most basic rules of electronics is that circuits must be complete for currents to ow This week we will introduce an exception to that rule The capacitor is actually a small break in a circuit Try measuring the resistance of a capacitor you will find that it is an open circuit However at the inside ends of the capacitor s lead it has little plates that act as charge reservoirs where it can store charge For short times you do not notice that the break is there Negative charge initially ows in to one side and out from out the other side just as if the two leads were connected For fast signals the capacitor looks like a shortcircuit But after a while the capacitor s reservoirs fill the current stops and we notice that there really is a break in the circuit For slow signals a capacitor looks like an open circuit What is fast and what is slow It depends on the capacitor and the rest of the circuit This week you will learn how to determine fast and slow for yourselves Capacitors serve three major roles in electrical circuits although all three are just variations of one basic idea 0 Charge integrators 0 High or low frequency lters 0 DC isolators 18 Chapter 3 Capacitors Inductors and Complex Impedance In order to perform these functions analytically we will need to introduce a number of new concepts and some signi cant mathematical formalism In this process we will also develop a number of new concepts in analyzing electronic circuits Capacitance A capacitor is a device for storing charge and electrical energy It consists of two parallel conducting plates and some nonconducting material between the plates as shown in gure 31 on the right When voltage is applied positive charge collects on one plate and negative charge collects on the other plane Since they are attracted to each other this is a stable state until the voltage is changed again A capacitor s charge capacity or capacitance C is de ned as QC V 36 which relates the charge stored in the capacitor Q to the voltage across its leads V Capacitance is measured in Farads F A Farad is a very large unit and most applications use uF nF or pF sized devices Many electronics components have small parasitic capacitances due to their leads and design Figure 31 A capacitor consist of two parallel plates which store equal and opposite amounts of charge The capacitor also stores energy in the electric eld generated by the charges on its two plates The potential energy stored in a capacitor with voltage V on it is l E CV2 37 2 We usually speak in terms of current when we analyze a circuit By noting that the current is the rate of change of charge we can rewrite the de nition of capacitance in terms of the current as 1 1 V 1dr 38 CQ CI 01 ICd VCV 39 dt This shows that we can integrate a function Ht just by monitoring the voltage as the current charges up a capacitor or we can differentiate a function Vt by putting it across a capacitor and monitoring the current ow when the voltage changes 19 Chapter 3 Capacitors Inductors and Complex Impedance A Simple RC Circuit We will start by looking in detail at the simplest VIN capacitive circuit which is shown in gure 32 on the right An RC circuit is made by simply putting a resistor R and a capacitor together as a voltage divider We will put the resistor in first so we can connect the capacitor to Vour ground C By applying Kirchhoff s Laws to this circuit we can see that l The same current ows through both the resistor and the capacitor and Figure 32 Asimple RC 2 The sum of the voltage drops across the two Clrcult thh Integrates current elements equal the input voltage This can be put into a formula in the following equation 1 VIN IREj1dt 310 which can also be written as VIN l I 1dr 311 R RC J We can also put this into the form of a differential equation in the following way dV IN R i 312a air air C or CV W RCi1 312b These equations show that times are measured in units of RC and that what you see depends on how quickly things change during one RC time interval If the current changes quickly then most of the voltage will show up across the resistor while the voltage across the capacitor slowly charges up as it integrates the current If the voltage changes slowly then most of the voltage shows up across the capacitor as it charges Since this usually requires a small current the voltage across the resistor stays small But what happens at intermediate times To determine this quantitatively we will have to develop some more sophisticated mathematical techniques Solutions to RC Circuit Rather than produce the general solution we will concentrate on two special cases that are particularly useful The first will be for a constant voltage and the second will be a sinusoidal input 20 Chapter 3 Capacitors Inductors and Complex Impedance To study a constant supply voltage on an RC circuit we set the left side of equation 312 equal to a constant voltage Then we have a simple homogeneous differential equation with the simple solution for the current of a decaying exponential I gal 313 which will account for any initial conditions After a time of a few RC time periods this solution will have decayed away to the supply voltage And now let us consider the other solution In the prior section we argued that if we can understand the RC circuit s behavior for sinusoidal input we can deal with any arbitrary input Therefore this is the important one Let s look at our simple RC circuit and suppose that we apply or drive a simple sine wave into the input VIN V0 cosat 314 In compleX notation this means that we will set the drive voltage to VIN V0 eXpz39at 315 and we just have to remember to take the Real part at the end of our calculation If we put this drive voltage into the differential equation equation 312 then it becomes a relatively simple inhomogeneous differential equation C dVIN d CV0ia2eXpiatRC7I 316 t This is relatively simple because it shows up so often in physics that you might as well memorize the solution or at least the way to get the solution Note that mathematically it looks just like a driven harmonic oscillator We can obtain the solution by using the standard recipe for first order linear differential equations We start by rewriting equation as d 1 39 V I ta eX iwt 317 dt RC R P which we then multiply by expt RC to obtain 39 V exptRC expo RC 1 0 dt RC R The left handside of this equality can be rewritten under the form of a total derivative multiplication rule so that we now have 1 eXpza Et 318 i Itex L ex iat 319 cit p RC R p RC 39 39 This equation is easily integrable and can be rewritten as t in 1 It eX 0 eX ia tdt 320 NRC R j p RCH 21 Chapter 3 Capacitors Inductors and Complex Impedance The integral is straightforward and yields the following expression 39 V 1Q m 0 4expiatCstexp L 321 a RC R XE 1 The first term represent the steady state oscillatory behavior of the driven circuit while the second term describes the transient behavior of the current after switching on the driving voltage Since we are only interested in the longterm behavior of the circuit we neglect the second term and concentrate on the first After a little bit of algebra we can rewrite the steadystate current as wVOC wRC z39 39 V t W 0 1 c Wag c Wag 322 R c w J1mRCZ J1wRC2 The second fraction can be interpreted as a phase term with tan Ilec so that the a expression for the current becomes t 10 expz39at 323 with 10 A V cos 324 quot1 QRC2 R The real solution of this simple RC circuit can be obtained by taking the real part of equation 23 and is left as an exercise to the reader The solution of the simple RC circuit appears to be rather complicated and involved however it simplifies considerably when we plug equation 23 back in to the original integral equation from Kirchhoff s loop law equation 10 After integrating the exponential and a little bit of algebra we obtain K r w IrR 1t 325 la This remarkably simple expression looks a lot like the standard Kirchhoff s loop law for resistors except that the capacitor term behaves with a frequency dependent imaginary resistance RC Impedance We will obtain the same solution as the one we obtained for the original voltage divider as long as we assign an imaginary frequency dependent resistance to the capacitor The imaginary part just means that it will produce a n39 2 phase shift between the voltage and the current for a sinusoidal input We will call this impedance 1 326 C iwC 22 Chapter 3 Capacitors Inductors and Complex Impedance Now the solution for an RC divider becomes somewhat simplified We can compute the total current owing through the circuit as 141 mt 141 I Vm Vm Voe szVOe 2 V0 szCoe Z coswklw p 327 2 RZC RliaC 1z39a2RC R 1inC R The voltage across an element is just this current times the element s impedance For the voltage drop across the resistor it is largely the same as before VR R V0 cos e 328 For the capacitor we get the following voltage drop L V0 cos e VCIZC1 z39V sin e lmquot sz imRC 0 at m 329 cos 2 V ezat 77rZ V Sin exat 77r2 0 TRC 0 If everything is correctly calculated then the sum of the voltage drops across the two elements should be equal to the input voltage Let s try it VR VC V0cos isin e Voe e Voem 330 Remember you get the actual waveforms by taking the real parts of these compleX solutions Therefore VR Vgcos cosat and 331 Vc Vgsin cosat L7r2 Vgsin sinat 332 This looks complicated but the limits of high frequency and low frequency are easy to remember At high frequencies gt 0 the capacitor is like a short and all the voltage shows up across the resistor At low frequencies gt n39 2 the capacitor is like an open circuit and all the voltage shows up across the capacitor If you consider the leading terms for the elements with the small voltages you find that chnmq V0 J M3 1a2RC a2 RC mRCltimRCgt R 0 2 gtzaRClLas w gt0 1a2RC 333 Thus at high frequency the voltage across the capacitor is the integral of the input voltage while at low frequency the voltage across the resistor is the derivative of the input voltage This says that as long as all the important frequencies are high the capacitor will integrate the input voltage If all the important frequencies are small the resistor will differentiate the voltage If there are intermediate frequencies or a mixture of some high 23 Chapter 3 Capacitors Inductors and Complex Impedance and some low frequencies the result will not be so simple but it can be determined from the voltage divider algebra using complex notation We nish by noting that the voltage on the capacitor is always 1t2 out of phase with the voltage on the resistor III Inductors An inductors is a coil of wire or solenoid which can be used to store energy in the magnetic eld that it generates see gure 33 on the right It is mathematically similar to a capacitor but has exactly the opposite behavior it behaves as a short circuit for low frequencies and as an open circuit for high frequencies ie it passes low frequency signals and blocks high frequency signals The energy stored in the eld of an inductor with inductance L is given by the following formula E 2L12 334 The SI unit of inductance is the Henry H Commercially available inductors have inductances that range from nH to mH Small millimetersize and centimeter consists Of a coiled Wire size solenoids typically have inductances in the range of also called a solenoid uH while magnetic eld coils can have a inductances in The dashed arrow B the mH range and can sometimes have inductances of up represent the magnetic to several H Most electronics components have small eld generated by the parasitic inductances due to their leads and design for current in the inductor example wirewound power resistors Figure 33 An inductor In an electric circuit a voltage or electromotive potential is generated across the terminals of the inductor when the cu1rent changes due to Faraday s law The voltage drop is given by the following simple expression d V L 335 dt From this equation we see that the inductor operates exactly opposite to a capacitor an inductor differentiates the cu1rent and integrates the voltage The LR circuit We can analyze the LR circuit in much the same way that we derived the operation of the RC circuit We start by applying Kirchhoff s loop law to the LR circuit in gure 34 below and we nd that 24 Chapter 3 Capacitors Inductors and Complex Impedance VIN R L 336 VIN If we apply a constant voltage the solution can be R calculated using the techniques developed for the RC circuit and we calculate that VOUT I0 1016Xp t 337 L The circuit approaches the steady state current IoV1NR with a time constant of HR Figure 34 A simple LR circuit LR impedance Instead of solving the differential equation for the LR circuit with a sinusoidal applied input voltage such as that given by equations 14 and 15 as we did with the RC circuit we will just assume that the current has the form I0 10 eXpiwt 338 We plug this ansatz solution back into the differential equation of equation 31 and nd that VIN ItR 139le 339 from which we deduce that the inductor behaves as a resistor with frequency dependent imaginary resistance The impedance of an inductor is therefore 2 imL 340 Just as with the RC circuit we can apply Ohm s law to the circuit to calculate the total current Since R and L are in series we obtain L Vm V0614 V0614 V0 l zm R 2 l m J R 2 RZL Ria2L L where the phase is given by tan m We calculate the voltage drop across the 1t e 1 c0s e 341 resistor using the expression for the current and find that V ItR V0 cos e 342 The voltage drop across the inductor is calculated the same way and we find V VL mug imLEOcos e 39 We sin e V0 cos e Z 343 25 Chapter 3 Capacitors Inductors and Complex Impedance If everything is correctly calculated then the sum of the voltage drops across the two elements should be equal to the input voltage Let s try it V VL V0 cos isin e 39 Voe e Voe 344 You get the actual waveforms by taking the real parts of these complex solutions Therefore VR Vg cos cosat and 345 VL V0 sin cosat 7V2 V0 sin sinat 346 This looks complicated but the limits of high frequency and low frequency are easy to remember At high frequencies gt IT 2 the inductor is like an open circuit and all the voltage shows up across the inductor At low frequencies gt 0 the inductor is like a short circuit or just a plain wire and all the voltage shows up across the resistor It should also be pointed out that the voltage on the inductor is always 1I2 out of phase with the voltage on the resistor IV Transformers Transformers are an ingenious combination of two inductors They are used to transfer power between two circuits by magnetic coupling The transformer changes an input voltage without affecting the signal shape similar to the voltage divider of last week However it has several important differences 0 It can increase as well as decrease a signal s amplitude ie AC voltage VIN VOUT It requires a timevarying AC input to work It is much harder to fabricate It usually does not work well for very fast signals since inductors block high frequencies Transformers are commonly used as a major Figure 3 5 The SChematlc component in a DC power supplies since they can SymbOI for atranSformer39 convert a 120 V AC wall voltage into a smaller voltage that is closer to the desired DC voltage eg 5 V or ilS V The schematic symbol for a transformer is shown in figure 35 above Transformers are passive devices that simultaneously change the voltage and current of a circuit They have at least four terminals two inputs called the primary and two outputs called the secondary There is no real difference between the input and output for a transformer you could simply ip it around and use the secondary as the input and the primary as the output However for the sake of clarity we will always assume that you use the primary for input and the secondary for output 26 Chapter 3 Capacitors Inductors and Complex Impedance The coupling between the input and output is done magnetically This allows transformers to have a number of interesting bene ts including 0 There is no DC connection between input and output so transformers are often used to isolate one circuit from another Transformers only work for time varying signals when the inductive coupling between the coils is greater than the resistive losses Since they have no external power the output power can not be greater than the input power PVPIPZVSIS 347 Usually we will assume equality but there are small resistances and hence resistive losses in the coils and a poorly or cheaply designed transformer many not have the input and output sufficiently strongly coupled to each other Depending on the device and the signal the output power may well be less than the input power Transformers are most commonly used to change line voltage 120 V RMS at 60 Hz into a more convenient voltage High power transmission lines use transformers to increase the voltage and decrease the current This reduces I R power losses in the transmission wires For our circuits we will use a transformer that reduces the voltage and increases the current Transformers are characterized by the ratio of the number of turns on the input and output windings The magnetic coupling in an ideal transformer will insure that the number of turns times the current owing is the same for the input and output I N N N13 S P 348 P P S S 11 N S Since the voltage must change in the opposite manner to keep the input and output power the ratio of the voltages is the same as the ratio of the turns V N S S 349 VP NP Transformers are usually called stepup or stepdown according to whether the output voltage increases or decreases A transformer also transforms the impedance of a circuit since it changes the ratio of VI Using our rules above the ratio of output impedance to input impedance is the square of the ratio of turns 2 Z V I N SSP 5 350 ZP IS VP NP So if you use a transformer as a stepup transformer it increases the voltage and the impedance at its output relative to its input If you use a transformer as a stepdown transformer it decreases the voltage and the impedance at its output 27 Chapter 3 Capacitors Inductors and Complex Impedance Design Exercises Design Exercise 31 Using Kirchhoff s laws derive a formula for the total capacitance of two capacitors in parallel and a formula for the total capacitance of two capacitors in series Design Exercise 32 Using Kirchhoff s laws derive a formula for the total inductance of two inductors in parallel and a formula for the total inductance of two inductors in series Design Exercise 33 Calculate Vom as a function of Vin in the RLC circuit of gure 36 on the right using the formulas for ZR Zc and ZL do not use Maple Mathematica MATLAB MathCad for these calculations and show all steps Vin is a perfect AC voltage signal with a frequency of a Plot the magnitude and phase of Vom as a function of a for R 1 k9 C 1 HF and L 10 pH What happens to the magnitude and the phase of Vom at a lVLC Maple Mathematica MATLAB MathCad are permitted for the plots Design Exercise 34 Calculate Vom as a function of Vin in the RLC circuit depicted on the right using the formulas for ZR Zc and ZL do not use Maple Mathematica MATLAB MathCad for these calculations and show all steps Vin is a perfect AC voltage signal with a frequency of a Plot the magnitude and phase of Vom as a function of a for R 1 k9 C 1 HF and L 10 pH What happens to the magnitude and the phase of Vom at a 1 MC Maple Mathematica MATLAB MathCad are permitted for the plots 28 VIN R VOUT C Egg Figure 36 An RLC filter circuit VIN VOUT Figure 37 Another RLC filter circuit Chapter 3 Capacitors Inductors and Complex Impedance Lab 3 AC signals Complex Impedance and Phase Section 1 Introduction to transformers In this section we use a transformer to change the impedance of an AC signal 1a Measure the output impedance of a signal generator with a 05V amplitude sinusoid output of 1 kHz and the input impedance of a speaker Remember you are using AC signals What does an AC current reading from a DVM mean in terms of the waveform How do you measure current with an oscilloscope Check this with the oscilloscope 1b Connect the signal generator to a speaker and measure the signal amplitude with and without connecting to the speaker The voltage drops so much because of the impedance mismatch Measure the power into the speaker lc Use a transformer to decrease the output voltage while increasing the output current into the speaker Measure Vom Vin Ii and 10m How well does the transformer transmit power Does VomVin Iin 10m Estimate the ratio of primary turns to secondary turns 1d Measure the output impedance of the signal generator plus transformer circuit Does the measured value agree with what you expect theoretically Section 2 The RC circuit In this section we take a first look at the classic RC circuit and the concept of phase 2a Get two capacitors and measure their individual capacitances Measure the total capacitance with a capacitance meter when they are in series and when they are in parallel Do you get good agreement with what you expect 2b Construct the RC circuit to the right with component VIN ranges Rl10 kg and C0001001 uF Set the function generator at approximately 030lRC with a square wave R and describe what you see Measure the time constant of the exponential and use it to determine the capacitance of C R VOUT should be determined with a multimeter C 2c Same setup Set the function generator to sinusoidal 1 output at 031RC and measure the magnitude of Vin and Vom Do you get what you expect Measure the phase of Vom Figure 3 8 An RC lter with respect to Vin and make a Lissajou plot of Vom and Vi circuit 29 Chapter 2 Kirchhoff s Laws and Th venin s Theorem Chapter 2 Kirchhoff s Laws and Th venin s Theorem In this course we will be using a variety of mathematical and conceptual models to describe the electrical components and circuits that we will encounter Ohm s Law was the first such model In this chapter we will consolidate some of the concepts from last week into a generic model of a linear device known as a Th venin equivalent device inth impedance output impedance and internal voltage The Th venin equivalents will also be generalized and used to describe properties of signals A signal will be described not only by its voltage but also by the output impedance of the device that produced or transported it The concept of the impedance of a signal will be used extensively through this course as we discover various methods to transform the properties of a signal This week we will also examine network analysis in a complete framework known as Kirchhoff s Laws These rules will allow us to analyze the properties of any combination of resistors and power supplies EU u N I Kirchhoff s Laws If you connect lots of power supplies and resistors together in a complicated network then V1 currents will ow through all the various elements so as to insure that charge is conserved energy is conserved and Ohm s Law is satis ed for each resistor Simultaneously satisfying all these conditions will give you exactly one solution The method for writing down equations to represent these conservation laws is called Kirchhoff s Laws To explore these Laws let s consider the sample circuit shown to the right 5 AAAUM AAAHM VV 9 VVVVVVV E K VVVVV 4 J S 9 I Kirchhoff s Point Law I Kirchhoft s Point Law says that all of the current that ows into a junction must come out of the junction This means that charge is conserved 7 none falls out of the circuit or pools in any location This concept is illustrated for a junction in example circuit In this case 12713 0 Note the sign conventions We say that a current is positive if we define it to ow into the junction and negative if we define it to ow out the junction Chapter 2 Kirchhoff s Laws and Thevenin s Theorem Kirchhoff s Loop Law Kirchhoft s Loop Law says that if you sum the voltage drops around any closed loop in a circuit the total must be zero Since the voltage drops represent the energy per unit charge this means that all the energy which comes out of the circuit in heating resistors for example must come from some power sources such as power supplies This is just conservation of energy EU S AAAAAA g The sign conventions are that the voltage increases across a power supply and decreases across a resistor in the direction of current ow The direction of current ow should be the same as the one de ned for the Point Law In our example circuit there are three loops that could be considered From these three loops we could write down three relationships 0 V1 11R1 IZR2 V2 0 V1 11R1 13R3 0 V2 12R2 13R3 Any two of these relationships since all three are not independent and the point relationship can be used to solve for the current in each resistor and hence the power in each resistor It can be a real nightmare to solve more complicated networks and we will develop models and methods to avoid solving complicated networks wherever we can We will usually do this by making one part of a circuit relatively independent of another part So our most common use of Kirchhoff s Laws will usually be to say that VIR and apply the simple relationships for resistors connected in series or parallel In the next section we will begin to explore how this is possible 11 Th venin Equivalents The Thevenin equivalents are models to describe the input and output properties of a device We will assume the input of a device looks like a resistor with the other end connected to ground We will assume that an output looks like an ideal voltage source followed by a resistor Both of these are shown in the gure to the right Note that for this week s discussion the term resistance and impedance are treated as synonymous One can determine the input impedance Chapter 2 Kirchhoff s Laws and Thevenin s Theorem ZIN by simply applying a voltage to the input measuring the current owing into the device and applying Ohm s Law Given any electrical device with two output terminals we will assume one is called ground there are a number of you might do to try to determine its properties 1 Measure the voltage across the terminals 2 Attach a resistor across the terminals and then measure the voltage 3 Change the resistor and measure the voltage again repeat Clearly these are all variations on one simple measurement What is the minimum number of measurements you must do to characterize this black box completely Thevenin s answer is just two Th venin s Theorem Any linear ie Ohmic system with two terminals can be completely characterized as an ideal voltage source VTH or Thevenin voltage in series with one resistor ZTH or Thevenin resistance and also called internal resistance sometimes No matter what resistance RLOAD you connect across the two terminals it just forms a voltage divider and so the voltage across that load resistor is given by R VLOAD VTH LOAD A special case is when there isn t any load applied ie option 1 above where RLOAD 00 then the output is simply VTH From the results of these two measurements we can solve for RTH Voltage Divider as a Thevenin Device Let s consider what happens to our voltage divider from last week when we connect a load resistance to its output From last week we computed the output of a loaded voltage divider was given by R R VLOAD VIN 2 R1 R2 R If we equate factors from in this relationship with the ones in the previous section we can see that for a loaded voltage divider produces Thevenin equivalents of R RIR2 2 and Z R1R2 TH R12 R1 R2 12 RLOAD VTH VIN The relationship for VTH makes sense It is simply the unloaded output voltage as describe in the previous section The output impedance is simply the parallel equivalent of the two resistors on the voltage divider Chapter 2 Kirchhoff s Laws and Thevenin s Theorem Designing a Stiff Voltage Divider A stz39ffoutput implies that the output voltage stays relatively close to its Thevenin or unloaded voltage when a load is implied A rule of thumb for designing a stiff voltage source is that the output should not deviate by more than about 10 when loaded From the relationships in the previous section we can see that this implies that the load resistance must be more than a factor of 10 larger than the output impedance in the voltage divider case RTH R1 m We can think of this at a Rule of 10 We will usually design it so that R1 itself is smaller than the expected load by a factor of 10 Generalization and Application of the Rule of 10 and Notable Exceptions Since we are going to design and construct some very complicated circuits this year we need to be able to focus on parts of circuits which we will call sub circuits For example we will often build amplifiers in three stages The basic idea is to always insure that the downstream elements do not load the upstream elements 1 The first stage will be a buffer amplifier that takes a signal and amplifies the current but not the voltage This will allow us to connect it to the next part of the ampli er without changing 139 6 loading the characteristics of whatever generated the signal The second stage will be a voltage amplifier that amplifies a signal s voltage to whatever level we need The final stage will be another buffer amplifier that again amplifies the current so that we will not load our voltage amplifier when we use its output This gains us two major advantages We will never have a large number of interdependent simultaneous equations to solve and our circuits will work under a wide range of differing operational conditions Here is the general procedure 0 Any subcircuit can be modeled using Thevenin s Theorem as an ideal voltage source VTH in series with a resistance Rm 0 When we connect this to the next subcircuit we can represent that by a load resistance RLOA D thus making a voltage divider We can keep our equations simple so that we need not employ the full Kirchhoff s Laws formalism simply by following a factor of 10 rule D 1 n summary anything we design to produce a reliable voltage should have RTH smaller RLOAD by least a factor of 10 That is we do not want the voltage to change significantly when the subcircuit is connected to subsequent parts of the circuit Constant Voltage Sources The most common subcircuit we will use will supply a voltage to a later part of a circuit It may be a power supply to power all the other subcircuits or it may be a biasing network to keep a transistor in its normal operating region For these type sub circuits we want VOUT to be relatively independent of RL so this means R m A D gt lORTH Constant Current Sources We will later see that transistors are current amplifiers This means that we will often want to drive an amplifier with a current source A current source can also be modeled as ideal voltage source in series with an RTH Again the complete circuit will look like an 4

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