Circuits. Fawwaz T. Ulaby, Michel M. Maharbiz, Cynthia M. Furse. Tables

Transcription

1 Circuits by Fawwaz T. Ulaby, Michel M. Maharbiz, Cynthia M. Furse Tables

2 Chapter 1: Circuit Terminology Chapter 2: Resisitive Circuits Chapter 3: Analysis Techniques Chapter 4: Operational Amplifiers Chapter 5: RC and RL First-Order Circuits Chapter 6: RLC Circuits Chapter 7: ac Analysis Chapter 8: ac Power Chapter 9: Frequency Response of Circuits and Filters Chapter 10: Three-Phase Circuits Chapter 11: Magnetically Coupled Circuits Chapter 12: Circuit Analysis by Laplace Transform Chapter 13: Fourier Analysis Technique

3 Chapter 1 Circuit Terminology Tables Table 1-1: Fundamental and electrical SI units. Table 1-2: Multiple and submultiple prefixes. Table 1-3: Symbols for common circuit elements. Table 1-4: Circuit terminology. Table 1-5: Voltage and current sources.

4 Chapter 2 Resistive Circuits Tables Table 2-1: Conductivity and resistivity of some common materials at 20 C. Table 2-2: Diameter d of wires, according to the American Wire Gauge (AWG) system. Table 2-3: Common resistor terminology. Table 2-4: Equally valid, multiple statements of Kirchhoff s Current Law (KCL) and Kirchhoff s Voltage Law (KVL). Table 2-5: Equivalent circuits.

5 Chapter 3 Analysis Techniques Tables Table 3-1: Properties of Thévenin/Norton analysis techniques. Table 3-2: Summary of circuit analysis methods.

6 Chapter 4 Operational Amplifiers Tables Table 4-1: Characteristics and typical ranges of op-amp parameters. The rightmost column represents the values assumed by the ideal op-amp model. Table 4-2: Characteristics of the ideal op-amp model. Table 4-3: Summary of op-amp circuits. Table 4-4: Correspondence between binary sequence and decimal value for a 4-bit digital signal and output of a DAC with G = 0.5. Table 4-5: List of Multisim components for the circuit in Fig Table 4-6: Components for the circuit in Fig

7 Chapter 5 RC and RL First-Order Circuits Tables Table 5-1: Common nonperiodic waveforms. Table 5-2: Relative electrical permittivity of common insulators: ε r = ε/ε 0 and ε 0 = F/m. Table 5-3: Relative magnetic permeability of materials, µ r = µ/µ 0 and µ 0 = 4π 10 7 H/m. Table 5-4: Basic properties of R, L, and C. Table 5-5: Response forms of basic first-order circuits. Table 5-6: Multisim component list for the circuit in Fig

8 Chapter 6 RLC Circuits Tables Table 6-1: Step response of RLC circuits for t 0. Table 6-2: General solution for second-order circuits for t 0. Table 6-3: Component values for the circuit in Fig Table 6-4: Parts for the Multisim circuit in Fig

9 Chapter 7 ac Analysis Tables Table 7-1: Useful trigonometric identities (additional relations are given in Appendix D). Table 7-2: Properties of complex numbers. Table 7-3: Time-domain sinusoidal functions x(t) and their cosine-reference phasor-domain counterparts X, where x(t) = Re[Xe jωt ]. Table 7-4: Summary of υ i properties for R, L, and C. Table 7-5: Inverting amplifier gain G as a function of oscillation frequency f. G ideal = 5.

10 Chapter 8 ac Power Tables Table 8-1: Summary of power-related quantities. Table 8-2: Power factor leading and lagging relationships for a load Z = R + jx.

11 Chapter 9 Frequency Response of Circuits and Filters Tables Table 9-1: Correspondence between power ratios in natural numbers and their db values (left table) and between voltage or current ratios and their db values (right table). Table 9-2: Bode straight-line approximations for magnitude and phase. Table 9-3: Attributes of series and parallel RLC bandpass circuits.

12 Chapter 10 Three-Phase Circuits Tables Table 10-1: Balanced networks.

13 Chapter 11 Magnetically Coupled Circuits Tables There are no tables in this chapter.

14 Chapter 12 Circuit Analysis by Laplace Transform Tables Table 12-1: Properties of the Laplace transform ( f (t) = 0 for t < 0 ). Table 12-2: Examples of Laplace transform pairs for T 0. Note that multiplication by u(t) guarantees that f (t) = 0 for t < 0. Table 12-3: Transform pairs for four types of poles. Table 12-4: Circuit models for R, L, and C in the s-domain.

15 Chapter 13 Fourier Analysis Technique Tables Table 13-1: Trigonometric integral properties for any integers m and n. The integration period T = 2π/ω 0. Table 13-2: Fourier series expressions for a select set of periodic waveforms. Table 13-3: Fourier series representations for a periodic function f (t). Table 13-4: Examples of Fourier transform pairs. Note that constant a 0. Table 13-5: Major properties of the Fourier transform. Table 13-6: Methods of solution. Table 13-7: Multisim circuits of the Σ modulator.

16 Table 1-1 Fundamental and electrical SI units. Dimension Unit Symbol Fundamental: Length meter m Mass kilogram kg Time second s Electric charge coulomb C Temperature kelvin K Amount of substance mole mol Luminous intensity candela cd Electrical: Current ampere A Voltage volt V Resistance ohm Ω Capacitance farad F Inductance henry H Power watt W Frequency hertz Hz

17 Table 1-2 Multiple and submultiple prefixes. Prefix Symbol Magnitude exa E peta P tera T giga G 10 9 mega M 10 6 kilo k 10 3 milli m 10 3 micro µ 10 6 nano n 10 9 pico p femto f atto a 10 18

18 Table 1-3 Symbols for common circuit elements.

19 Table 1-4 Circuit terminology. Node: An electrical connection between two or more elements. Ordinary node: An electrical connection node that connects to only two elements. Extraordinary node: An electrical connection node that connects to three or more elements. Branch: Trace between two consecutive nodes with only one element between them. Path: Continuous sequence of branches with no node encountered more than once. Extraordinary path: extraordinary nodes. Path between two adjacent Loop: Closed path with the same start and end node. Independent loop: Loop containing one or more branches not contained in any other independent loop. Mesh: Loop that encloses no other loops. In series: Elements that share the same current. They have only ordinary nodes between them. In parallel: Elements that share the same voltage. They share two extraordinary nodes.

20 Table 1-5 Voltage and current sources. Ideal Voltage Source Independent Sources Realistic Voltage Source Ideal Current Source Realistic Current Source Voltage-Controlled Voltage Source (VCVS) Dependent Sources Voltage-Controlled Current Source (VCCS) Current-Controlled Voltage Source (CCVS) Current-Controlled Current Source (CCCS) Note: α, g, r, and β are constants; υ x and i x are a specific voltage and a specific current elsewhere in the circuit. Lowercase υ and i represent voltage and current sources that may or may not be time-varying, whereas uppercase V and I denote dc sources.

21 Table 2-1 Conductivity and resistivity of some common materials at 20 C. Material Conductivity σ Resistivity ρ (S/m) (Ω-m) Conductors Silver Copper Gold Aluminum Iron Mercury (liquid) Semiconductors Carbon (graphite) Pure germanium Pure silicon Insulators Paper Glass Teflon Porcelain Mica Polystyrene Fused quartz Common materials Distilled water Drinking water Sea water Graphite Rubber Biological tissues Blood Muscle Fat

22 Table 2-2 Diameter d of wires, according to the American Wire Gauge (AWG) system. AWG Size Designation Diameter d (mm)

23 Table 2-3 Common resistor terminology. Thermistor Piezoresistor Light-dependent R (LDR) Rheostat Potentiometer R sensitive to temperature R sensitive to pressure R sensitive to light intensity 2-terminal variable resistor 3-terminal variable resistor

24 Table 2-4 Equally valid, multiple statements of Kirchhoff s Current Law (KCL) and Kirchhoff s Voltage Law (KVL). Sum of all currents entering a node = 0 [i = + if entering; i = if leaving] Sum of all currents leaving a node = 0 KCL [i = + if leaving; i = if entering] Total of currents entering = Total of currents leaving Sum of voltages around closed loop = 0 [υ = + if + side encountered first KVL in clockwise direction] Total voltage rise = Total voltage drop

25 Table 2-5 Equivalent circuits.

26 Table 3-1 Properties of Thévenin/Norton analysis techniques. To Determine Method Can Circuit Contain Dependent Sources? Relationship υ Th Open-circuit υ Yes υ Th = υ oc υ Th Short-circuit i (if R Th is known) Yes υ Th = R Th i sc R Th Open/short Yes R Th = υ oc /i sc R Th Equivalent R No R Th = R eq R Th External source Yes R Th = υ ex /i ex i N = υ Th /R Th ; R N = R Th

27 Table 3-2 Summary of circuit analysis methods. Method Common Use Ohm s law Relates V, I, R. Used with all other methods to convert V I. R, G in series and Y- or Π-T Voltage/current dividers Kirchhoff s laws (KVL/KCL) Node-voltage method Mesh-current method Node-voltage by-inspection method Mesh-current by-inspection method Combine to simplify circuits. R in series adds, and is most often used. G in adds, so may be used when much of the circuit is in parallel. Convert resistive networks that are not in series or in into forms that can often be combined in series or in. Also simplifies analysis of bridge circuits. Common circuit configurations used for many applications, as well as handy analysis tools. Dividers can also be used as combiners when used backwards. Solve for branch currents. Often used to derive other methods. Solves for node voltages. Probably the most commonly used method because (1) node voltages are easy to measure, and (2) there are usually fewer nodes than branches and therefore fewer unknowns (smaller matrix) than KVL/KCL. Solves for mesh currents. Fewer unknowns than KVL/KCL, approximately the same number of unknowns as node voltage method. Less commonly used, because mesh currents seem less intuitive, but useful when combining additional blocks in cascade. Quick, simplified way of analyzing circuits. Very commonly used for quick analysis in practice. Limited to circuits containing only independent current sources. Quick, simplified way of analyzing circuits. Very commonly used for quick analysis in practice. Limited to circuits containing only independent voltage sources. Superposition Simplifies circuits with multiple sources. Commonly used for both calculation and measurement. Source transformation Thévenin and Norton equivalents Input/output resistance (R in /R out ) Simplifies circuits with multiple sources. Commonly used for both calculation/design and measurement/test applications. Very often used to simplify circuits in both calculation and measurement applications. Also used to analyze cascaded systems. Thévenin is the more commonly used form, but Norton is often handy for analyzing parallel circuits. Source transformation allows easy conversion between Thévenin and Norton. Commonly used to evaluate when cascaded circuits can be analyzed individually or when full circuit analysis or a buffer is needed.

28 Table 4-1 Characteristics and typical ranges of op-amp parameters. The rightmost column represents the values assumed by the ideal op-amp model. Op-Amp Characteristics Parameter Typical Range Ideal Op Amp Linear input-output response Open-loop gain A 10 4 to 10 8 (V/V) High input resistance Input resistance R i 10 6 to Ω Ω Low output resistance Output resistance R o 1 to 100 Ω 0 Ω Very high gain Supply voltage V cc 5 to 24 V As specified by manufacturer

29 Table 4-2 Characteristics of the ideal op-amp model. Ideal Op Amp Current constraint i p = i n = 0 Voltage constraint υ p = υ n A = R i = R o = 0

30 Table 4-3 Summary of op-amp circuits.

31 Table 4-4 Correspondence between binary sequence and decimal value for a 4-bit digital signal and output of a DAC with G = 0.5. V 1 V 2 V 3 V 4 Decimal Value DAC Output (V)

32 Table 4-5 List of Multisim components for the circuit in Fig Component Group Family Quantity Description 1.5 k Basic Resistor kω resistor 15 k Basic Resistor 2 15 kω resistor 3 k Basic Variable resistor 1 3 kω resistor OP AMP 5T VIRTUAL Analog Analog Virtual 3 Ideal op amp with 5 terminals AC POWER Sources Power Sources 1 1 V ac source, 60 Hz VDD Sources Power Sources 1 15 V supply VSS Sources Power Sources 1 15 V supply

33 Table 4-6 Components for the circuit in Fig Component Group Family Quantity Description MOS N Transistors Transistors VIRTUAL 1 3-terminal N-MOSFET MOS P Transistors Transistors VIRTUAL 1 3-terminal P-MOSFET VDD Sources Power Sources V supply GND Sources Power Sources 2 Ground node

34 Table 5-1 Common nonperiodic waveforms. waveform Expression General Shape Step u(t T ) = { 0 for t < T 1 for t > T Ramp r(t T ) = (t T ) u(t T ) Rectangle ( t T rect τ ) = u(t T 1 ) u(t T 2 ) T 1 = T τ 2 ; T 2 = T + τ 2 Exponential exp[ (t T )/τ] u(t T )

35 Table 5-2 Relative electrical permittivity of common insulators: ε r = ε/ε 0 and ε 0 = F/m. Material Relative Permittivity ε r Air (at sea level) Teflon 2.1 Polystyrene 2.6 Paper 2 4 Glass Quartz Bakelite 5 Mica Porcelain 5.7

36 Table 5-3 Relative magnetic permeability of materials, µ r = µ/µ 0 and µ 0 = 4π 10 7 H/m. Material Relative Permeability µ r All Dielectrics and Non-Ferromagnetic Metals 1.0 Ferromagnetic Metals Cobalt 250 Nickel 600 Mild steel 2,000 Iron (pure) 4,000 5,000 Silicon iron 7,000 Mumetal 100, 000 Purified iron 200, 000

37 Table 5-4 Basic properties of R, L, and C. Property R L C i υ relation i = υ i = 1 υ dt + i(t 0 ) R L t 0 υ-i relation υ = ir υ = L di dt p (power transfer in) p = i 2 R p = Li di dt t i = C dυ dt t υ = 1 i dt + υ(t 0 ) C t 0 p = Cυ dυ dt w (stored energy) 0 w = 1 2 Li2 w = 1 2 Cυ2 1 Series combination R eq = R 1 + R 2 L eq = L 1 + L 2 = C eq C 1 C 2 1 Parallel combination = = C eq = C 1 +C 2 R eq R 1 R 2 L eq R 1 R 2 dc behavior no change short circuit open circuit Can υ change instantaneously? yes yes no Can i change instantaneously? yes no yes

38 Table 5-5 Response forms of basic first-order circuits. Circuit Diagram Response RC { } υ C (t) = υ C ( ) + [υ C (T 0 ) υ C ( )]e (t T 0)/τ u(t T 0 ) (τ = RC) RL { } i L (t) = i L ( ) + [i L (T 0 ) i L ( )]e (t T 0)/τ u(t T 0 ) (τ = L/R) Ideal integrator υ out (t) = 1 t υ RC i dt + υ out (t 0 ) t 0 Ideal differentiator υ out (t) = RC dυ i dt

39 Table 5-6 Multisim component list for the circuit in Fig Component Group Family Quantity Description 1 k Basic Resistor 1 1 kω resistor 10 k Basic Resistor 1 10 kω resistor 5 f Basic Capacitor 1 5 ff capacitor VOLTAGE CONTROLLED SPST Basic Switch 1 Switch DC POWER Sources Power Sources V dc source PULSE VOLTAGE Sources Signal Voltage Source 1 Pulse-generating voltage source

40 Table 6-1 Step response of RLC circuits for t 0. Series RLC Parallel RLC Total Response Total Response Overdamped (α > ω 0 ) Overdamped (α > ω 0 ) υ C (t) = A 1 e s1t + A 2 e s2t + υ C ( ) i L (t) = A 1 e s1t + A 2 e s2t + i L ( ) 1 C A 1 = i C(0) s 2 [υ C (0) υ C ( )] s 1 s [ 2 ] 1C i C (0) s 1 [υ C (0) υ C ( )] A 2 = s 2 s 1 1 L A 1 = υ L(0) s 2 [i L (0) i L ( )] s 1 s [ 2 ] 1L υ L (0) s 1 [i L (0) i L ( )] A 2 = s 2 s 1 Critically Damped (α = ω 0 ) Critically Damped (α = ω 0 ) υ C (t) = (B 1 + B 2 t)e αt + υ C ( ) i L (t) = (B 1 + B 2 t)e αt + i L ( ) B 1 = υ C (0) υ C ( ) B 1 = i L (0) i L ( ) B 2 = C 1 i C(0) + α[υ C (0) υ C ( )] B 2 = L 1 υ L(0) + α[i L (0) i L ( )] Underdamped (α < ω 0 ) Underdamped (α < ω 0 ) υ C (t) = e αt (D 1 cosω d t + D 2 sinω d t) + υ C ( ) i L (t) = e αt (D 1 cosω d t + D 2 sinω d t) + i L ( ) D 1 = υ C (0) υ C ( ) D 1 = i L (0) i L ( ) D 2 = R 2L α = 1 2RC s 1 = α + 1 C i C(0) + α[υ C (0) υ C ( )] D 2 = ω d Auxiliary Relations Series RLC Parallel RLC α 2 ω 2 0 ω 0 = 1 LC ω d = s 2 = α ω 2 0 α2 1 L υ L(0) + α[i L (0) i L ( )] ω d α 2 ω 2 0

41 Table 6-2 General solution for second-order circuits for t 0. x(t) = unknown variable (voltage or current) Differential equation: x + ax + bx = c Initial conditions: x(0) and x (0) Final condition: α = a 2 x( ) = c b ω 0 = b Overdamped Response α > ω 0 x(t) = [A 1 e s1t + A 2 e s2t + x( )] u(t) s 1 = α + α 2 ω0 2 s 2 = α α 2 ω0 2 [ A 1 = x (0) s 2 [x(0) x( )] x ] (0) s 1 [x(0) x( )] A 2 = s 1 s 2 s 1 s 2 Critically Damped α = ω 0 x(t) = [(B 1 + B 2 t)e αt + x( )] u(t) B 1 = x(0) x( ) B 2 = x (0) + α[x(0) x( )] Underdamped α < ω 0 x(t) = [D 1 cosω d t + D 2 sinω d t + x( )]e αt u(t) D 1 = x(0) x( ) D 2 = x (0) + α[x(0) x( )] ω d ω d = ω0 2 α2

42 Table 6-3 Component values for the circuit in Fig Component Group Family Quantity Description 1 Basic Resistor 1 1 Ω resistor 300 m Basic Inductor mh inductor 5.33 m Basic Capacitor mf capacitor SPDT Basic Switch 1 Single-pole double-throw (SPDT) switch DC POWER Sources Power Sources 1 1 V dc source

43 Table 6-4 Parts for the Multisim circuit in Fig Component Group Family Quantity Description TS IDEAL Basic Transformer 1 1 mh:1 mh ideal transformer 1 k Basic Resistor 1 1 kω resistor 1 µ Basic Capacitor 1 1 µf capacitor SPDT Basic Switch 1 SPDT switch AC CURRENT Sources Signal Current Source 1 1 ma, khz

44 Table 7-1 Useful trigonometric identities (additional relations are given in Appendix D). sinx = ±cos(x 90 ) cosx = ±sin(x ± 90 ) sinx = sin(x ± 180 ) cosx = cos(x ± 180 ) sin( x) = sinx cos( x) = cos x sin(x ± y) = sinxcosy ± cosxsiny cos(x ± y) = cosxcosy sinxsiny 2sinxsiny = cos(x y) cos(x + y) 2sinxcosy = sin(x + y) + sin(x y) 2cosxcosy = cos(x + y) + cos(x y) (7.7a) (7.7b) (7.7c) (7.7d) (7.7e) (7.7f) (7.7g) (7.7h) (7.7i) (7.7j) (7.7k)

45 sinθ = e jθ e jθ 2 j z = x + jy = z e jθ Table 7-2 Properties of complex numbers. Euler s Identity: e jθ = cosθ + j sinθ cosθ = e jθ + e jθ 2 z = x jy = z e jθ x = Re(z) = z cosθ z = zz = x 2 + y 2 tan 1 (y/x) if x > 0, tan 1 (y/x) ± π if x < 0, y = Im(z) = z sinθ θ = π/2 if x = 0 and y > 0, π/2 if x = 0 and y < 0. z n = z n e jnθ z 1/2 = ± z 1/2 e jθ/2 z 1 = x 1 + jy 1 z 2 = x 2 + jy 2 z 1 = z 2 iff x 1 = x 2 and y 1 = y 2 z 1 + z 2 = (x 1 + x 2 ) + j(y 1 + y 2 ) z 1 z 2 = z 1 z 2 e j(θ 1+θ 2 ) z 1 z 2 = z 1 z 2 e j(θ 1 θ 2 ) 1 = e jπ = e jπ = 1 ±180 j = e jπ/2 = 1 90 j = e jπ/2 = 1 90 j = ±e jπ/4 (1 + j) = ± 2 j = ±e jπ/4 (1 j) = ± 2

46 Table 7-3 Time-domain sinusoidal functions x(t) and their cosine-reference phasor-domain counterparts X, where x(t) = Re[Xe jωt ]. x(t) Acosωt Acos(ωt + φ) Acos(ωt + φ) Asinωt Asin(ωt + φ) Asin(ωt + φ) d dt (x(t)) d [Acos(ωt + φ)] dt x(t) dt Acos(ωt + φ) dt X A Ae jφ Ae j(φ±π) Ae jπ/2 = ja Ae j(φ π/2) Ae j(φ+π/2) jωx jωae jφ 1 jω X 1 jω Ae jφ

47 Table 7-4 Summary of υ i properties for R, L, and C. Property R L C υ i υ = Ri υ = L di i = C dυ dt dt V I V = RI V = jωli V = I jωc Z R jωl 1 jωc dc equivalent R High-frequency equivalent R Frequency response

48 Table 7-5 Inverting amplifier gain G as a function of oscillation frequency f. G ideal = 5. f (Hz) A G Error 0 (dc) % % 1 k % 10 k % 100 k % 1 M % The error is defined as ( ) Gideal G % error = 100. G ideal

49 Table 8-1 Summary of power-related quantities. Time Domain Phasor Domain υ(t) = V m cos(ωt + φ υ ) i(t) = I m cos(ωt + φ i ) V rms = V m / 2 I rms = I m / 2 Real Average Power P av = Re[S] = V rms I rms cos(φ υ φ i ) = IrmsR 2 = VrmsR/ Z 2 2 Apparent Power S = S = P 2 av + Q 2 = V rms I rms = I 2 rms Z = V 2 rms/ Z Complex Power S = 1 2 VI = V rms I rms = P av + jq V = V m e jφ υ I = I m e jφ i V rms = V rms e jφ υ I rms = I rms e jφ i S = Se j(φ υ φ i ) = Se jφ pf = z φ z = φ υ φ i Reactive Power Q = Im[S] = V rms I rms sin(φ υ φ i ) = IrmsX 2 = VrmsX/ Z 2 2 Power Factor P av S = cos(φ υ φ i ) = cosφ z

50 Table 8-2 Power factor leading and lagging relationships for a load Z = R + jx. Load Type φ z = φ υ φ i I-V Relationship pf Purely Resistive (X = 0) φ z = 0 I in phase with V 1 Inductive (X > 0) 0 < φ z 90 I lags V lagging Purely Inductive φ z = 90 I lags V by 90 lagging (X > 0 and R = 0) Capacitive (X < 0) 90 φ z < 0 I leads V leading Purely Capacitive φ z = 90 I leads V by 90 leading (X < 0 and R = 0)

51 Table 9-1 Correspondence between power ratios in natural numbers and their db values (left table) and between voltage or current ratios and their db values (right table). V V 0 P P 0 db 10 N 10N db db db db 4 6 db 2 3 db 1 0 db db db db 10 N 10N db or I db I 0 10 N 20N db db db db 4 12 db 2 6 db 1 0 db db db db 10 N 20N db

52 Table 9-2 Bode straight-line approximations for magnitude and phase.

53 Table 9-3 Attributes of series and parallel RLC bandpass circuits. RLC Circuit Transfer Function H = V R V s H = V R I s Resonant Frequency, ω 0 1 LC 1 LC Bandwidth, B ω 0 Quality Factor, Q B = ω 0L R [ Lower Half-Power Frequency, ω c1 1 2Q Q 2 Upper Half-Power Frequency, ω c2 1 2Q Q 2 [ R L ]ω 0 [ ]ω 0 [ ω 0 1 2Q + 1 2Q + 1 RC B = R ω 0 L Q 2 ] ] Q 2 Notes: (1) The expression for Q of the series RLC circuit is the inverse of that for Q of the parallel circuit. (2) For Q 10, ω c1 ω 0 B 2, and ω c 2 ω 0 + B 2. ω 0 ω 0

54 Table 10-1 Balanced networks. V 1 = V Ys φ 1 V 2 = V Ys φ V 3 = V Ys φ V 12 = V V 23 = V V 31 = V Z = 3Z Y V an = (V L / 3) 0 V ab = V L 30 V bn = (V L / 3) 120 V bc = V L 90 V cn = (V L / 3) 240 V ca = V L 150 V L = rms magnitude of line-to-line voltage I L = 3 V L / Z Y I L = 3 V L / Z S T = 3 V L I L φ Y S T = 3 V L I L φ Phase relative to V an of Y-load

55 Table 12-1 Properties of the Laplace transform ( f (t) = 0 for t < 0 ). Property f (t) F(s) = L [ f (t)] 1. Multiplication by constant K f (t) K F(s) 2. Linearity K 1 f 1 (t) + K 2 f 2 (t) K 1 F 1 (s) + K 2 F 2 (s) 3. Time scaling f (at), a > 0 1 ( s ) a F a 4. Time shift f (t T ) u(t T ) e T s F(s), T 0 5. Frequency shift e at f (t) F(s + a) 6. Time 1st derivative f = d f dt s F(s) f (0 ) 7. Time 2nd derivative f = d2 f dt 2 s 2 F(s) s f (0 ) f (0 ) t 8. Time integral f (τ) dτ 1 0 s F(s) 9. Frequency derivative t f (t) d ds F(s) f (t) 10. Frequency integral F(s ) ds t s

56 Table 12-2 Examples of Laplace transform pairs for T 0. Note that multiplication by u(t) guarantees that f (t) = 0 for t < 0. Laplace Transform Pairs f (t) 1 δ(t) 1 1a δ(t T ) e T s 2 1 or u(t) 2a u(t T ) 3 e at u(t) 3a e a(t T ) u(t T ) 4 t u(t) 4a (t T ) u(t T ) 5 t 2 u(t) 6 te at u(t) 7 t 2 e at u(t) 8 t n 1 e at u(t) 9 sinωt u(t) 10 sin(ωt + θ) u(t) 11 cos ωt u(t) 12 cos(ωt + θ) u(t) 13 e at sinωt u(t) 14 e at cosωt u(t) 15 2e at cos(bt θ) u(t) 16 2t n 1 (n 1)! e at cos(bt θ) u(t) Note: (n 1)! = (n 1)(n 2)...1. F(s) = L [ f (t)] 1 s e T s s 1 s + a e T s s + a 1 s 2 e T s s 2 2 s 3 1 (s + a) 2 2 (s + a) 3 (n 1)! (s + a) n ω s 2 + ω 2 ssinθ + ω cosθ s 2 + ω 2 s s 2 + ω 2 scosθ ω sinθ s 2 + ω 2 ω (s + a) 2 + ω 2 s + a (s + a) 2 + ω 2 e jθ s + a + jb + e jθ s + a jb e jθ (s + a + jb) n + e jθ (s + a jb) n

57 Table 12-3 Transform pairs for four types of poles. Pole F(s) f (t) 1. Distinct real 2. Repeated real 3. Distinct complex 4. Repeated complex [ [ A s + a Ae jθ s + a + jb + Ae at u(t) A (s + a) n A tn 1 (n 1)! e at u(t) ] Ae jθ s + a jb Ae jθ (s + a + jb) n + Ae jθ (s + a jb) n ] 2Ae at cos(bt θ) u(t) 2At n 1 (n 1)! e at cos(bt θ) u(t)

58 Table 12-4 Circuit models for R, L, and C in the s-domain. Time-Domain Resistor s-domain Inductor υ = Ri V = RI OR υ L = L di L dt i L = 1 L t Capacitor 0 υ L dt + i L (0 ) V L = sli L L i L (0 ) I L = V L sl + i L(0 ) s OR i C = C dυ C dt υ C = 1 C t 0 i C dt + υ C (0 ) V C = I C sc + υ C(0 ) s I C = scv C C υ C (0 )

59 Table 13-1 Trigonometric integral properties for any integers m and n. The integration period T = 2π/ω 0. Property Integral T 0 T 0 T 0 T 0 T 0 T 0 T 0 sinnω 0 t dt = 0 cosnω 0 t dt = 0 sinnω 0 t sinmω 0 t dt = 0, cosnω 0 t cosmω 0 t dt = 0, sinnω 0 t cosmω 0 t dt = 0 sin 2 nω 0 t dt = T /2 cos 2 nω 0 t dt = T /2 n m n m Note: All integral properties remain valid when the arguments nω 0 t and mω 0 t are phase shifted by a constant angle φ 0. Thus, Property 1, for example, becomes T 0 sin(nω 0t + φ 0 ) dt = 0, and Property 5 becomes T 0 sin(nω 0t + φ 0 )cos(mω 0 t + φ 0 ) dt = 0.

60 Table 13-2 Fourier series expressions for a select set of periodic waveforms. Waveform Fourier Series 1. Square Wave f (t) = n=1 4A ( nπ nπ sin 2 ) cos ( ) 2nπt T 2. Time-Shifted Square Wave f (t) = n=1 n=odd ( ) 4A 2nπt nπ sin T 3. Pulse Train f (t) = Aτ T + 2A ( nπτ n=1 nπ sin T 4. Triangular Wave f (t) = 5. Shifted Triangular Wave f (t) = 6. Sawtooth f (t) = n=1 n=odd n=1 n=odd n=1 ( ) 8A 2nπt n 2 π 2 cos T 8A ( nπ n 2 π 2 sin 2 ) sin ) cos ( ) n+1 2A 2nπt ( 1) nπ sin T 7. Backward Sawtooth f (t) = A ( ) 2 + A 2nπt n=1 nπ sin T ( ) 2nπt T ( ) 2nπt T 8. Full-Wave Rectified Sinusoid 9. Half-Wave Rectified Sinusoid f (t) = 2A ( ) π + 4A 2nπt n=1 π(1 4n 2 ) cos T f (t) = A π + A ( ) 2πt 2 sin + T n=2 n=even ( ) 2A 2nπt π(1 n 2 ) cos T

61 Table 13-3 Fourier series representations for a periodic function f (t). Cosine/Sine Amplitude/Phase Complex Exponential f (t) = a 0 + a 0 = 1 T a n = 2 T b n = 2 T T 0 T 0 T 0 n=1 (a n cosnω 0 t + b n sinnω 0 t) f (t) = a 0 + n=1 A n cos(nω 0 t + φ n ) f (t) = n= c n e jnω 0t f (t) dt A n e jφ n = a n jb n c n = c n e jφ n ; c n = c n f (t) cosnω 0 t dt A n = a 2 n + b 2 n c n = A n /2; c 0 = a 0 ( ) tan 1 bn, a n > 0 a f (t) sinnω 0 t dt φ n = ( n ) π tan 1 bn, a n < 0 a 0 = c 0 ; a n = A n cosφ n ; b n = A n sinφ n ; c n = 1 2 (a n jb n ) a n c n = 1 T T 0 f (t) e jnω 0t dt

62 Table 13-4 Examples of Fourier transform pairs. Note that constant a 0. f (t) F(ω) = F [ f (t)] F(ω) BASIC FUNCTIONS 1. δ(t) 1 1a. δ(t t 0 ) e jωt π δ(ω) 3. u(t) π δ(ω) + 1/ jω 4. sgn(t) 2/ jω 5. rect(t/τ) τ sinc(ωτ/2) 6. t 2/ω 2 7. e at u(t) 1/(a + jω) 8. cosω 0 t π[δ(ω ω 0 ) + δ(ω + ω 0 )] 9. sinω 0 t jπ[δ(ω + ω 0 ) δ(ω ω 0 )] ADDITIONAL FUNCTIONS 10. e jω 0t 2π δ(ω ω 0 ) 11. te at u(t) 1/(a + jω) [e at sinω 0 t] u(t) ω 0 /[(a + jω) 2 + ω0 2] 13. [e at cosω 0 t] u(t) (a + jω)/[(a + jω) 2 + ω0 2]

63 Table 13-5 Major properties of the Fourier transform. Property f (t) F(ω) = F [ f (t)] 1. Multiplication by a constant K f (t) K F(ω) 2. Linearity K 1 f 1 (t) + K 2 f 2 (t) K 1 F 1 (ω) + K 2 F 2 (ω) 3. Time scaling f (at) 1 ( ω ) a F a 4. Time shift f (t t 0 ) e jωt 0 F(ω) 5. Frequency shift e jω 0t f (t) F(ω ω 0 ) 6. Time 1st derivative f = d f dt 7. Time nth derivative 8. Time integral t d n f dt n f (t) dt jω F(ω) ( jω) n F(ω) F(ω) jω 9. Frequency derivative t n f (t) ( j) n dn F(ω) dω n + π F(0) δ(ω) 10. Modulation cosω 0 t f (t) 1 2 [F(ω ω 0 ) + F(ω + ω 0 )] 11. Convolution in t f 1 (t) f 2 (t) F 1 (ω) F 2 (ω) 12. Convolution in ω f 1 (t) f 2 (t) 1 2π F 1(ω) F 2 (ω)

64 Input x(t) Table 13-6 Methods of solution. Duration Waveform Solution Method Output y(t) Everlasting Sinusoid Phasor Domain Steady-State Component (no transient exists) Everlasting Periodic Phasor Domain and Fourier Series Steady-State Component (no transient exists) One-sided, Any Laplace Transform (unilateral) Complete Solution x(t) = 0, for t < 0 (can accommodate nonzero initial conditions) (transient + steady-state) Everlasting Any Bilateral Laplace Transform Complete Solution or Fourier Transform (transient + steady-state)

65 Table 13-7 Multisim circuits of the Σ modulator. Multisim Circuit Description and Notes Subtractor: This is a difference amplifier (following Table 4-3) with a voltage gain of 1. VPLUS and VMINUS are the extremes of the analog input (in the complete circuit, they are set to ±12 V). Integrator: This circuit consists of an inverting integrator amplifier (Section 5-6.1) and an inverting amplifier (following Table 4-3) with a voltage gain of 1 (to remove the integrator s negative sign). Comparator: The comparator is a simple op amp with no feedback (open loop). Since the internal voltage gain A of the op amp is so high (Section 4-1.2), any positive difference between the noninverting and the inverting inputs immediately drives the amplifier output to V DD ; a negative difference drives the amplifier output to 0 V. V DD is set to the desired digital voltage level (5 V, in the case of the complete circuit in Fig ). 1-Bit Digital-to-Analog Converter (DAC): The DAC is very similar to the comparator. The input voltage is compared to a voltage level halfway between 0 and V DD ; this has the effect of transforming an input of V DD into an output voltage of VPLUS/2 (+6 V in Fig ) and an input voltage of 0 V into an output voltage of VMINUS/2 ( 6 V in Fig ).