# Electrical System Elements

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1 Electrical System Elements Application areas include: Electromechanical (motor) Electro-optical (phototransistor) Electrothermal Electro-mechano-acoustic (loudspeaker, microphone) Measurement Systems and Controls Systems Here we focus on strictly electrical systems 1

2 Electrical components are described in terms of their voltage / current relations. Classification: Network vs. field concept Passive vs. active device Linear (proportional) vs. digital (on-off) device Network vs. Field Classification Essentially that of lumped vs. distributed parameters Based on wavelength / physical size criterion: 2

3 If the physical size of a device is small compared to the wavelength associated with signal propagation, the device may be considered lumped and a network model employed. Wavelength = (velocity V of wave propagation) / (signal frequency f) The velocity of propagation for electrical waves in free space is 186,000 miles / second. Example: Audio Systems: 20 to 20,000 Hz frequency range λ = (186,000 miles/sec) / (20,000 cycles/sec) = 9.3 miles/cycle Typical resistor or capacitor < 1 inch long Audio electrical systems can be treated with the lumpedparameter (network) approach 3

4 The wavelength / physical size concept is applicable to any physical system which exhibits wave propagation, e.g., mechanical vibrating systems, acoustic systems. In treating electrical elements we will take strictly the lumped (network) approach and eliminate the consideration of high-frequency phenomena. This restriction is not a severe one. Passive vs. Active Devices Distinction is based on energy considerations 4

5 Resistors, capacitors, and inductors are not sources of energy in the sense of a battery or a generator. They are called passive elements since they contain no energy sources. It is true that capacitors and inductors can store energy, but some energy source was needed initially to charge the capacitor or establish the current in the inductor. Resistors dissipate into heat all the electrical energy supplied to them. Basic Active Elements are energy sources: Batteries (electrochemical source) Generators (electromechanical source) 5

6 Solar cells (electro-optical source) Thermocouples (thermoelectric source) When these basic sources are combined with a power modulator, the transistor, we obtain active devices called controlled sources, whose outstanding characteristic is the capability for power amplification. The transistor does not itself supply the power difference between the input and output; it simply modulates, in a precise and controlled fashion, the power taken from the basic source (battery, etc.) and delivered to the output. 6

7 The combinations of transistors with their power supplies are called active devices. Because of their amplification capability, they are the fundamental base of all electronic systems. The single most useful active linear device is the operational amplifier. It is now considered an inexpensive circuit element. Its ease of application makes it a basic building block for many different types of useful circuits. While an op-amp is not strictly an element (it contains resistors, transistors, etc.), it is treated like a component or element. 7

8 Linear (proportional) vs. Digital (on-off) Devices Digital electronic devices perform on-off switching-type functions needed to implement the logic operations required in digital computation. +2 V to +5 V represents ON state 0 V to +0.8 V represents OFF state Digital devices are very tolerant of noise voltages and need not be individually very accurate, even though the overall system can be extremely accurate. 8

9 Digital devices are small, cheap, and fast. In linear devices, the specific waveform of input and output signals is of vital importance. In digital devices, it is simply the presence (logical 1) or absence (logical 0) of a voltage within some wide range that matters; the precise value of the signal is of no consequence. Like op-amps, digital devices are not really elements, as they contain resistors, transistors, diodes, etc.; however, they are treated as basic building blocks. 9

10 Since a properly functioning digital system operates in the realm of arithmetic rather than differential equations, its modeling, analysis, and design do not fit the pattern of linear system dynamics and thus we do not treat digital elements per se. We can, however, model those aspects of computer behavior that influence the performance of the overall computer-aided system. These aspects have to do mainly with: Sampling Quantization Computational Delays 10

11 The Resistance Element Pure and ideal resistance element has a mathematical model: Strict linearity between e and i Instantaneous response of i to e or e to i All electrical energy supplied is dissipated into heat Real resistors Non-ideal (not exactly linear) i = e R 11

12 Impure they exhibit some capacitance and inductance effects which make themselves known only when current and voltage are changing with time. A steady-state experiment will reveal departures from ideal behavior, but will not reveal impurity of a resistor. Definition of Resistance R (ohms) and Conductance G (siemens) R e i G i e 12

13 Instantaneous electric power P ( ) 2 2 e e 2 P ei = i ir = ir = e = = eg R R Power is always positive; the resistor always takes power from the source supplying it. Since the resistor cannot return power to the source, all the power supplied is dissipated into heat. Electric power (watts) is the heating rate for the resistor. 13

14 Internal heat generation causes the resistor temperature to rise. When the resistor temperature is higher than that of its surroundings, heat transfer by conduction, convection, and radiation causes heat to flow away from the resistor. When the resistor gets hot enough, this heat transfer rate just balances the e 2 /R heat generation rate and the resistor achieves an equilibrium temperature somewhere above room temperature. In a real resistor this temperature cannot be allowed to get too high, or else the R value changes excessively or the resistor may actually burn out. 14

15 Instantaneous dynamic response is characteristic of a pure resistance element (zero-order dynamic system model). Sinusoidal transfer function i 1 ( iω ) = 0 e R Real resistors are always impure and this prevents the instantaneous step response, the perfectly flat amplitude ratio, and the zero phase angle. 15

16 Since practical systems always deal with a limited range of frequencies, if a real resistor behaves nearly like a pure/ideal model over its necessary range, the fact that it deviates elsewhere is of little consequence. Resistance elements can be pure without being ideal. Useful nonlinear resistors are semiconductor diodes and the Varistor ( a semiconductor element with a symmetrical e/i relation of approximate 4 th -power shape (i Ke 4 ). 16

17 Resistance Element e = Ri 17

18 Resistors in series and parallel If the same current passes through two or more resistors, those resistors are said to be in series, and they are equivalent to a single resistor whose resistance is the sum of the individual resistances. If the same voltage difference exists across two or more resistors, those resistors are said to be in parallel and they are equivalent to a single resistance whose reciprocal is equal to the sum of the reciprocals of the individual resistances. 18

19 The Capacitance Element Two conductors separated by a nonconducting medium (insulator or dielectric) form a capacitor. Charging a Capacitor Process of removing charge from one conductor and placing an equal amount on the other. The net charge of a capacitor is always zero and the charge on a capacitor refers to the magnitude of the charge on either conductor. ( ) C farads ( ) e( volts) q coulombs 19

20 In a pure and ideal capacitance element, the numerical value of C is absolutely constant for all values of q or e. Real capacitors exhibit some nonlinearity and are contaminated by the presence of resistance and/or inductance. Mathematical Model 1 de 1dq 1 de e = q = = i i = C C dt Cdt C dt e t t 0 e de = i dt de = ( idt ) e e = ( i) dt C C C 0 20

21 i e ( D) ( D ) CD e 1 i = = CD Operational Transfer Functions Energy Stored The pure and ideal capacitance stores in its electric field all the electrical energy supplied to it during the charging process and will give up all of this energy if completely discharged, say by connecting it to a resistor. 21

22 The work done to transfer a charge dq through a potential difference e is (e)dq. The total energy stored by a charged capacitor is: q q 2 2 ( ) edq 0 0 This is true irrespective of how the final voltage or charge was built up. q q Ce dq C 2C 2 = = = There is no current through a capacitor; an equal amount of charge is taken from one plate and supplied to the other by way of the circuit external to the capacitor. 22

23 Capacitance Element i = de C dt 23

24 Approximate and Exact Impulse Functions If e s =1.0 (unit step function), its derivative is the unit impulse function with a strength (or area) of one unit. This non-rigorous approach does produce the correct result. 24

25 A step input voltage produces a capacitor current of infinite magnitude and infinitesimal time duration. Real physical quantities are limited to finite values. A true (instant rising) step voltage cannot be achieved. A real capacitor has parasitic resistance and inductance which limit current and its rate of change. Thus, a real capacitor will exhibit a short-lived (but not infinitesimal) and large (but not infinite) current spike. Impulse functions appear whenever we try to differentiate discontinuous functions. 25

26 The Inductance Element An electric current always creates an associated magnetic field. If a coil or other circuit lies within this field, and if the field changes with time, an electromotive force (voltage) is induced in the circuit. The magnitude of the induced voltage is proportional to the rate of change of flux dφ/dt linking the circuit, and its polarity is such as to oppose the cause producing it. 26

27 If no ferromagnetic materials ( e.g., iron) are present, the rate of change of flux is proportional to the rate of change of current which is producing the magnetic field. The proportionality factor relating the induced emf (voltage) to the rate of change of current is called the inductance. The presence of ferromagnetic materials greatly increases the strength of the effects, but also makes them significantly nonlinear, since now the flux produced by the current is not proportional to the current. 27

28 Thus, iron can be used to get a large value of inductance, but the value will be different for different current levels. The pure inductance element has induced voltage e instantaneously related to di/dt, but the relation can be nonlinear. The pure and ideal element has e directly proportional to di/dt (e = L di/dt), i.e., it is linear and free from resistance and capacitance. A real inductor always has considerable resistance. At DC and low frequencies, all real inductors behave like resistors, not inductors. 28

29 At high frequencies, all real devices (R, C, L) exhibit complex behavior involving some combination of all three pure elements. Thus, real inductors deviate from the pure/ideal model at both low and high frequencies, whereas R and C deviate mainly at high frequencies. One can expect real inductors to nearly follow the pure model only for some intermediate range of frequencies and, if the inductance value is small enough to be achieved without the use of magnetic material, the behavior may also approximate the ideal (linear). 29

30 Self-Inductance and Mutual-Inductance Self-inductance is a property of a single coil, due to the fact that the magnetic field set up by the coil current links the coil itself. Mutual inductance causes a changing current in one circuit to induce a voltage in another circuit. Mutual inductance is symmetrical, i.e., a current changing with a certain di/dt in coil 1 induces the same voltage in coil 2 as would be induced in coil 1 by the same di/dt current change in coil 2. This hold for coils in the same circuit or in separate circuits. The induced voltage in circuit A due to current change in B can either add or subtract from the self-induced voltage in A. This depends on actual geometry. 30

31 e = e + e A A1 A2 di di di = L ± M ± M dt dt dt A B A 1 B/A1 A2/A1 di di di + L ± M ± M dt dt dt A B A 2 B/A2 A1/A2 ( L L M M ) = + ± ± 1 2 A2/A1 A1/A2 ( M M ) + ± ± B/A1 B/A2 di B dt M = M = mutual inductance of coils 1 and A2/A1 A1/A2 L = self-inductance of coil 1 L = self-inductance of coil 2 M = mutual inductance of coils B and A M B/A1 1 = B/A2 2 di dt mutual inductance of coils B and A A 31

32 Energy Stored The pure and ideal inductance stores energy in its magnetic field. The energy stored, irrespective of how the current i is achieved, is: If we connect a current-carrying inductor to an energyusing device (e.g., resistor) the inductor will supply energy in an amount i 2 L/2 as its current decays from i to 0. During this decay process, i if originally positive stays positive, but di/dt (and thus e) becomes negative, making power negative. di Power = ei = L i dt t i 2 di il Energy = il dt = ( Li) di = dt

33 Inductance Element di e = L = LDi dt i 1 e ( D ) = LD i 1 1 ( iω ) = = ( i) e iωl ωl 1 = ω L 90 33

34 At very low frequencies, a small voltage amplitude can produce a very large current and thus an inductance is said to approach a short circuit in this case. At high frequencies, the current produced by any finite voltage approaches zero, and thus an inductance is said to approach an open circuit at high frequencies. For a capacitance, the reverse frequency behavior is observed: the capacitance approaches a short circuit at high frequencies and an open circuit at low frequencies. 34

35 One can often use these simple rules to quickly estimate the behavior of complex circuits at low and high frequency. Just replace L s and C s by open and short circuits, depending on which frequency you are interested in. Remember for real circuits that real L s always become R s for low frequency. 35

36 Electrical Impedance Electrical impedance is a generalization of the simple voltage/current relation called resistance for resistors. It can be applied to capacitors, inductors, and to entire circuits. It assumes ideal (linear) behavior of the device. Electrical impedance is defined as the transfer function relating voltage and current: Z D e D Z i e i Z s e s i i i ( ) ( ) ( ω) ( ω) ( ) ( ) 36

37 The impedances for the pure/ideal electrical elements are: Z D = R Z iω = R R L ( ) ( ) 1 1 ZC( D ) = ZC( iω ) = CD i ω C ( ) ( ) Z D = LD Z iω = iωl The impedances for the pure/ideal mechanical element are: f ZB( D) ( D) = B ZB( iω ) = B v f 1 1 ZS( D) ( D ) Z ( i ) v CD i C 37 R L = S ω = S ω f ZM( D) = ( D) = MD ZM( iω ) = iωm v S

38 Impedances Of Mechanical & Electrical Elements force velocity damper spring mass voltage current resistor capacitor inductor 38

39 Impedance is most useful in characterizing the dynamic behavior of components and systems. It is also useful in the solution of routine circuit problems. A e Z = φ= M φ= Mcosφ+ imsinφ= ZR + iz A i R Z = resistive impedance R X Z = reactive impedance X If Z X is a positive number, the reactive impedance is behaving like an inductor and is called inductive reactance; if negative, it is called capacitive reactance. X 39

40 Given R and X, one can always compute the magnitude and phase angle of the impedance: X M R X tan = + φ= R Since the sinusoidal impedance gives the amplitude ratio and phase angle of voltage with respect to current, if the impedance of any circuit (no matter how complex) is known (from either theory or measurement), and either voltage or current is given, we can quickly calculate the other. 40

41 The rules for combining series or parallel impedances are extensions to the dynamic case of the rules governing series and parallel static resistance elements. If the same flow passes through two or more impedances, those impedances are said to be in series, and they are equivalent to a single impedance whose impedance is the sum of the individual impedances. If the same effort difference exists across two or more impedances, those impedances are said to be in parallel and they are equivalent to a single impedance whose reciprocal is equal to the sum of the reciprocals of the individual impedances. 41

42 Electromechanical Analogies A signal, element, or system which exhibits mathematical behavior identical to that of another, but physically different, signal, element, or system is called an analogous quantity or analog. Analogous quantities: force voltage velocity damper spring mass current resistor capacitor inductor 42

43 Force causes velocity, just as voltage causes current. A damper dissipates mechanical energy into heat, just as a resistor dissipates electrical energy into heat. Springs and masses store energy in two different ways, just as capacitors and inductors store energy in two different ways. The product (f)(v) represents instantaneous mechanical power, just as (e)(i) represents instantaneous electrical power. 43

44 Recommendation: Model systems directly rather than try to force mixed-media systems into, say, an allmechanical or all-electrical form. 44

45 φ= ( edt ) e f e q = C R e = ir C L φ= Li q i q = ( ) idt General Model Structure for Electrical Systems 45

46 p = ( ) f dt f p f K = Kx f = Bv B M p = Mv x v x = ( ) vdt General Model Structure for Mechanical Systems 46

47 The Operational Amplifier Op-Amps are possibly the most versatile linear integrated circuits used in analog electronics. The Op-Amp is not strictly an element; it contains elements, such as resistors and transistors. However, it is a basic building block, just like R, L, and C. Uses include: Constant gain multiplication Impedance buffering Active filters Analog-digital interfacing 47

48 The op-amp has has two inputs, an inverting input (-) and a non-inverting input (+), and one output. The output goes positive when the non-inverting input (+) goes more positive than the inverting (-) input, and vice versa. The symbols + and do not mean that that you have to keep one positive with respect to the other; they tell you the relative phase of the output. A fraction of a millivolt between the input terminals will swing the output over its full range. Inverting Input Non-Inverting Input - + +V -V Output 48

49 Operational amplifiers have enormous voltage gain (10 6 or so), and they are never used without negative feedback. Negative feedback is the process of coupling the output back in such a way as to cancel some of the input. This does lower the amplifier s gain, but in exchange it also improves other characteristics, such as: Freedom from distortion and nonlinearity Flatness of frequency response or conformity to some desired frequency response Predictability 49

50 As more negative feedback is used, the resultant amplifier characteristics become less dependent on the characteristics of the open-loop (no feedback) amplifier and finally depend only on the properties of the feedback network itself. Basic Inverting Op-Amp R F Gain = R R F IN V in R in - + +V -V V out R F VOUT = VIN R IN 50

51 A properly designed op-amp allows us to use certain simplifying assumptions when analyzing a circuit which uses op-amps; we accept these assumptions on faith. They make op-amp circuit analysis quite simple. The so-called golden rules for op-amps with negative feedback are: The output attempts to do whatever is necessary to make the voltage difference between the inputs zero. The op-amp looks at its input terminals and swings its output terminal around so that the external feedback network brings the input differential to zero. The inputs draw no current (actually < 1 na). 51

52 Simplifying Assumptions: The op-amp s gain A is infinite Z i is infinite; thus no current is drawn at the input terminals Z o is zero; thus e o = A(e i2 e i1 ) The time response is instantaneous The output voltage has a definite design range, such as ± 10 volts. Proper operation is possible only for output voltages within these limits. 52

53 Differential Input Amplifier (if e i2 is connected to ground, it is called single-ended) A = amplifier gain Z i = input impedance Z o = output impedance Simplified Model (based on simplifying assumptions) (Amplifier requires connection to DC power) 53

54 Basic Op-Amp Cautions In all op-amp circuits, the golden rules will be obeyed only if the op-amp is in the active region, i.e., inputs and outputs are not saturated at one of the supply voltages. Note that the op-amp output cannot swing beyond the supply voltages. Typically it can swing only to within 2V of the supplies. The feedback must be arranged so that it is negative; you must not mix the inverting and non-inverting inputs. There must always be feedback at DC in the op-amp circuit. Otherwise, the op-amp is guaranteed to go into saturation. 54

55 Many op-amps have a relatively small maximum differential input voltage limit. The maximum voltage difference between the inverting and non-inverting inputs might be limited to as little as 5 volts in either polarity. Breaking this rule will cause large currents to flow, with degradation and destruction of the op-amp. Note that even though op-amps themselves have a high input impedance and a low output impedance, the input and output impedances of the op-amp circuits you will design are not the same as that of the op-amp. 55

56 Coefficient Multiplier or Inverter Rarely used by itself, the op-amp is usually combined with passive elements, mainly resistors and capacitors. Integrator Summer 56

57 To analyze op-amp circuits we use two basic electrical circuit laws, Kirchhoff s voltage loop and current node laws, together with the op-amp simplifying assumptions. Let s analyze the coefficient multiplier circuit: e R 1 i1 R i i e e i o = = i e R fb i1 R fb = Ae e i1 o e o o 1 + o R Of course A cannot be infinite, but it can be, say, 10 6 volts/volt, and then this is a good approximation. i e A = e A R i fb e R fb eo = e 1 if A = R 57

58 Why not use the op-amp directly as an amplifier since it has more than enough gain? The op-amp gain can be relied upon to be very large but cannot be relied upon to be an accurate stable value. The gain A is guaranteed to be, say, in the range 1 to 5 million V/V. As long as A is large enough, our approximation is valid. Also note that in the multiplier example, the accuracy and stability depends on the values of the two resistors and not on the value of A, as long as A is large enough. Using op-amps, we can construct circuits whose performance depends mainly on passive components selected to have accurate and stable values. 58

59 A is called the open-loop gain, while e o /e 1 in the example is called the closed-loop gain, which in this case due to the circuit configuration, is negative. Since A may be treated as infinite, the voltage e i1, the summing junction voltage, can always be treated as zero in those op-amps where the positive input is grounded. The summing junction is known as virtual ground, since its voltage is for all practical purposes zero, the same as true ground, whose voltage is exactly zero. 59

60 When op-amps don t ground the positive input (differential input), the difference (e i2 e i1 ) is taken to be practically zero. Using these assumptions, we can analyze the integrator circuit as follows: e 0 d i i C ( 0 e ) CDe R dt 1 R = = C = o = o 1 1 eo = e1 = edt 1 RCD RC 60

61 Similarly, for the summer circuit: i + i + i = i R fb e1 0 e2 0 e 0 0 e + + = R R R R 3 o fb R R R e e e e R1 R 2 R 3 fb fb fb o = ( ) e = e + e + e if R = R = R = R o fb While multiplier, integrator, and summer are fundamental operations for solving differential equations, op-amps have many other uses. 61

62 Examples are high-pass filters, low-pass filters, band-pass filters, band-reject filters, lead controllers, lag controllers, lead-lag controllers, approximate integrators and differentiators. i Z i and Z o represent arbitrary impedances e 0 0 eo = = i = Z D Z D 1 1 fb i ( ) ( ) 1 i ( D) ( ) e Z o fb ( D ) = e Z D e o R 2CD 1 ( D ) = e RCD 1 R C D 1 fb ( + )( + )

63 Deviations of Real Op-Amps from Ideal Assumptions Effect of non-infinite gain A For the multiplier circuit e o o 1 + o R i e A = e A R fb e R fb 1 R eo = e R i A AR fb i The open-loop gain A may be in the range of 10 4 to 10 8, while R fb /R i rarely exceeds 10 3 ; thus the error upper limit is from about 10-5 to If one selects precision resistors so as to get a precise e o /e 1, if the gain A is too low, the ratio will be inaccurate. 63

64 Offset voltage Offset voltage refers to the fact that if e 1 is made zero by grounding it, e o will not be exactly zero, due to imperfections in the amplifier. The best values of offset voltage e os are the order of 30 µv over a temperature range 0f 25 to +85 C, with a temperature coefficient of about 0.2 µv/ C. Op-amps can be trimmed using some additional circuitry with adjustable resistors to eliminate this offset. Bias Current Bias current is the small current that flows in the amplifier input leads, even when no input voltage is applied. Values of i b1 can be as small as 75E-15 amps at 25 C, and would never exceed ± 4 pa. 64

65 Non-infinite Input Impedance and Nonzero Output Impedance Analysis shows that the effect of non-infinite input impedance is equivalent to a loss of open-loop gain A, the effective value being given by: A eff = 1 + ( + ) A similar effect is produced by nonzero output impedance (R L is a load resistance representing the input resistance of any device which would be connected to the op-amp circuit): A RR i fb R R R 1 i fb A eff = A R R 2 2 fb R R L 65

66 Input resistances are in the range of 10 6 to ohms while output resistances are the order of 100 ohms. Speed of Response The speed of response is specified in several different ways. One method considers the closed-loop frequency response when the op-amp is connected as coefficient multiplier. The fastest op-amps will have this frequency response flat to about 500 MHz when the input resistance and feedback resistance are set equal, i.e., a closed-loop gain of 1. For a closed-loop gain of 20, the flat range of amplitude drops to about 80 MHz. Another method uses settling time after a step input is applied. Times to settle within 1, 0.1, and 0.01% of the final value may be quoted. The settling time is typically a few nanoseconds. Power Limitations Most op-amps supply only limited electrical power at their output terminals, e.g., ± 10 volts and 0.05 amps. 66

67 Offset Voltage e = e 1+ o os R R fb i Bias Current eo = ib1r fb Non-infinite Input Impedance and Nonzero Output Impedance 67

68 General Circuit Laws We focus on basic analysis techniques which apply to all electrical circuits. Just as Newton s Law is basic to the analysis of mechanical systems, so are Kirchhoff s Laws basic to electrical circuits. One needs to know how to use these laws and combine this with knowledge of the current/voltage behavior of the basic circuit elements to analyze a circuit model. 68

69 Kirchhoff s Voltage Loop Law It is merely a statement of an intuitive truth; it requires no mathematical or physical proof. This law can be stated in several forms: The summation of voltage drops around a closed loop must be zero at every instant. The summation of voltage rises around a closed loop must be zero at every instant. The summation of the voltage drops around a closed loop must equal the summation of the voltage rises at every instant. 69

70 Kirchhoff s Current Node Law It is based on the physical fact that at any point (node) in a circuit there can be no accumulation of electric charge. In circuit diagrams we connect elements (R, L, C, etc.) with wires which are considered perfect conductors. This law can be stated in several forms: The summation of currents into a node must be zero. The summation of currents out of a node must be zero. The summation of currents into a node must equal the summation of currents out. 70

71 In mechanical systems we need sign conventions for forces and motions; in electrical systems we need them for voltages and currents. If the assumed positive direction of a current has not been specified at the beginning of a problem, an orderly analysis is quite impossible. For voltages, the sign conventions consist of + and signs at the terminals where the voltage exists. 71

72 Once sign conventions for all the voltages and currents have been chosen, combination of Kirchhoff s Laws with the known voltage/current relations which describe the circuit elements leads us directly to the system differential equations. 72

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