BSC Modul 4: Advanced Circuit Analysis. Fourier Series Fourier Transform Laplace Transform Applications of Laplace Transform Z-Transform

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1 BSC Modul 4: Advanced Circuit Analyi Fourier Serie Fourier Tranform Laplace Tranform Application of Laplace Tranform Z-Tranform

3 Trigonometric Fourier Serie () The Fourier erie of a periodic function f(t) i a repreentation that reolve f(t) into a dc component and an ac component compriing an infinite erie of harmonic inuoid. Given a periodic function f(t) f(t+nt) where n i an integer and T i the period of the function. f t a ( ) + ( a co nωt + bn in nωt) dc n where ω π/t i called the fundamental frequency in radian per econd. ac

4 Trigonometric Fourier Serie () and a n and b n are a follow a n T T f ( t)co( nω t) dt o b n T T f ( t)in( nω t) dt o in alternative form of f(t) f ( t) a + ( cn co( nωt + φn) dc n ac where c n a + b, φ n n n tan b ( a n n ) (Invere tangent or arctangent)

5 Fourier Serie Example Determine the Fourier erie of the waveform hown right. Obtain the amplitude and phae pectra., < t < f ( t) and f ( t) f ( t + ), < t < a n b n T T T T f ( t)co( nω t) dt f ( t)in( nω t) dt and / nπ, n odd, n even A φ n n / nπ, n odd, 9, n, n even odd n even a) Amplitude and b) Phae pectrum f ( t) + π k n in( nπt), n k Truncating the erie at N

6 Symmetry Conideration () Three type of ymmetry. Even Symmetry : a function f(t) if it plot i ymmetrical about the vertical axi. f ( t) f ( t) In thi cae, a a b n n T 4 T T T / / f ( t) dt f ( t)co( nω t) dt Typical example of even periodic function

7 Symmetry Conideration (). Odd Symmetry : a function f(t) if it plot i anti-ymmetrical about the vertical axi. f ( t) f ( t) In thi cae, a b n 4 T T / f ( t)in( nω t) dt Typical example of odd periodic function

8 Symmetry Conideration (3) 3. Half-wave Symmetry : a function f(t) if f T ( t ) f ( t) a a b n n 4 T 4 T T T / / f f ( t)co( nω t) dt, for n odd, for an even ( t)in( nω t) dt, for n odd, for an even Typical example of half-wave odd periodic function

9 Symmetry Conideration (4) Example Find the Fourier erie expanion of f(t) given below. An: f ( t) nπ nπ co in t π n n

10 Symmetry Conideration (5) Example Determine the Fourier erie for the half-wave coine function a hown below. An: f ( t) 4 π co nt, n k k n

11 Circuit Application () Step for Applying Fourier Serie. Expre the excitation a a Fourier erie.. Tranform the circuit from the time domain to the frequency domain. 3. Find the repone of the dc and ac component in the Fourier erie. 4. Add the individual dc and ac repone uing the uperpoition principle.

12 Example Baic in Sytem and Circuit Theory Circuit Application () Find the repone v (t) of the circuit below when the voltage ource v (t) i given by v ( t) + π n n in ( nπωt), n k

13 Circuit Application (3) Solution Phaor of the circuit V jnπ 5 + jnπ V For dc component, (ω n or n), V ½ > V o For n th harmonic, V S 9, nπ V 4 tan nπ / n π V In time domain, v ( t) k n π co( nπt tan nπ ) 5 Amplitude pectrum of the output voltage

14 Average Power and RMS Value () Given: v( t) V dc + Vn co( nωt φvn) and i( t) Idc + Im co( mωt φim) n m The average power i P V dc Idc + Vn In co( θ n φ n ) n The rm value i F rm a + n ( a n + b n )

15 Average Power and RMS Value () Example Determine the average power upplied to the circuit hown below if i(t)+co(t+ )+6co(3t+35 ) A Anwer: 4.5W

16 Exponential Fourier Serie () The exponential Fourier erie of a periodic function f(t) decribe the pectrum of f(t) in term of the amplitude and phae angle of ac component at poitive and negative harmonic. f ( t) n c n e jnω t o c n T jnω t f ( t) e dt, whereω T π / T The plot of magnitude and phae of c n veru nω are called the complex amplitude pectrum and complex phae pectrum of f(t) repectively.

17 Exponential Fourier Serie () The complex frequency pectrum of the function f(t)e t, <t<π with f(t+π)f(t) (a) Amplitude pectrum; (b) phae pectrum

18 Application Filter () Filter are an important component of electronic and communication ytem. Thi filtering proce cannot be accomplihed without the Fourier erie expanion of the input ignal. For example, (a) Input and output pectra of a lowpa filter, (b) the lowpa filter pae only the dc component when ω c << ω

19 Application Filter () (a) Input and output pectra of a bandpa filter, (b) the bandpa filter pae only the dc component when Β << ω

21 Definition of Fourier Tranform () It i an integral tranformation of f(t) from the time domain to the frequency domain F(ω) F(ω) i a complex function; it magnitude i called the amplitude pectrum, while it phae i called the phae pectrum. Given a function f(t), it Fourier tranform denoted by F(ω), i defined by F ω ( ) f ( t) e dt jωt

22 Example : Baic in Sytem and Circuit Theory Definition of Fourier Tranform () Determine the Fourier tranform of a ingle rectangular pule of wide τ and height A, a hown below. Solution: F( ω) τ / τ / Ae A e jω jωt jωt dt τ / τ / jωτ / A e e ω j ωτ Aτ in c jωτ / Amplitude pectrum of the rectangular pule

23 Example : Baic in Sytem and Circuit Theory Definition of Fourier Tranform (3) Obtain the Fourier tranform of the witched-on exponential function a hown. Solution: f ( t) e Hence, F( ω) at u( t) e f ( t) e ( a+ jω) t a + jω e, jωt dt at dt, t > t < e jat e jωt dt

24 [ ] ) ( ) ( ) ( ) ( ω ω F a F a t f a t f a F + + Linearity: If F (ω) and F (ω) are, repectively, the Fourier Tranform of f (t) and f (t) Example: [ ] ( ) ( ) [ ] [ ] ) ( ) ( ) in( ω ω δ ω ω δ π ω ω ω + j e F e F j t F t j t j Propertie of Fourier Tranform ()

25 Propertie of Fourier Tranform () Time Scaling: If F (ω) i the Fourier Tranform of f (t), then F a ω a [ f ( at) ] F( ), a i a contant If a >, frequency compreion, or time expanion If a <, frequency expanion, or time compreion

26 Propertie of Fourier Tranform (3) Time Shifting: If F (ω) i the Fourier Tranform of f (t), then F [ ] j ( ) ω t f t t e F( ) ω Example: F [ e u t ] ( t ) ( ) e + jω jω

27 Propertie of Fourier Tranform (4) Frequency Shifting (Amplitude Modulation): If F (ω) i the Fourier Tranform of f (t), then F [ ] jω t f t) e F( ω ) ( ω Example: F [ f t)co( ω t) ] F( ω ω ) + F( ω + ) ( ω

28 Propertie of Fourier Tranform (5) Time Differentiation: If F (ω) i the Fourier Tranform of f (t), then the Fourier Tranform of it derivative i df F dt u( t) jωf( ) Example: d ( at e u t ) F ( ) dt a + jω

29 Propertie of Fourier Tranform (6) Time Integration: If F (ω) i the Fourier Tranform of f (t), then the Fourier Tranform of it integral i Example: F( ω) F t f ( t) dt π F () δ ( ω) jω F [ u( t) ] ( ω) jω +πδ

30 Circuit Application () Fourier tranform can be applied to circuit with non-inuoidal excitation in exactly the ame way a phaor technique being applied to circuit with inuoidal excitation. Y(ω) H(ω)X(ω) By tranforming the function for the circuit element into the frequency domain and take the Fourier tranform of the excitation, conventional circuit analyi technique could be applied to determine unknown repone in frequency domain. Finally, apply the invere Fourier tranform to obtain the repone in the time domain.

31 Circuit Application () Example: Find v (t) in the circuit hown below for v i (t)e -3t u(t) Solution: The Fourier tranform of the input ignal i V i ( ω) 3+ jω The tranfer function of the circuit i H ( ω) Hence, V ( ω) (3 + jω)(.5 + jω) V ( ω) V ( ω) i + jω Taking the invere Fourier tranform give v ( t).4( e.5t e 3t ) u( t)

33 Definition of Laplace Tranform It i an integral tranformation of f(t) from the time domain to the complex frequency domain F() Given a function f(t), it Laplace tranform denoted by F(), i defined by [ ] f ( t) f ( t) e t F( ) L dt Where the parameter i a complex number σ + jω σ, ω real number

34 Bilateral Laplace Tranform When one ay "the Laplace tranform" without qualification, the unilateral or one-ided tranform i normally intended. The Laplace tranform can be alternatively defined a the bilateral Laplace tranform or two-ided Laplace tranform by extending the limit of integration to be the entire real axi. If that i done the common unilateral tranform imply become a pecial cae of the bilateral tranform. The bilateral Laplace tranform i defined a follow: F [ f ( t) ] t ( ) L f ( t) e dt

35 Example of Laplace Tranform () Determine the Laplace tranform of each of the following function hown: a) The Laplace Tranform of unit tep, u(t) i given by [ ] ( t) F( ) e L u t dt

36 Example of Laplace Tranform () b) The Laplace Tranform of exponential function, e -at u(t), a> i given by [ ] αt ( t) F( ) e e L u t dt + α c) The Laplace Tranform of impule function, δ(t) i given by [ ] t) F( ) δ ( t) e t L u( dt

37 Example of Laplace Tranform (3) F( ) F( ) F( ) +α

38 Table of Selected Laplace Tranform ()

39 [ ] ) ( ) ( ) ( ) ( F a F a t f a t f a L + + Linearity: If F () and F () are, repectively, the Laplace Tranform of f (t) and f (t) Example: [ ] ( ) ) ( ) ( ) co( ω ω ω ω + + t u e e L t u t L t j t j Propertie of Laplace Tranform ()

40 Propertie of Laplace Tranform () Scaling: If F () i the Laplace Tranform of f (t), then Example: a a [ f ( at) ] F( ) L L ω + 4ω [ in(ωt) u( t) ]

41 Propertie of Laplace Tranform (3) Time Shift: If F () i the Laplace Tranform of f (t), then L [ ] a f ( t a) u( t a) e F( ) Example: L + ω a [ co( ω( t a)) u( t a) ] e

42 Propertie of Laplace Tranform (4) Frequency Shift: If F () i the Laplace Tranform of f (t), then [ ] e f ( t) u( t) F( a) L at + Example: + a L[ e co( ω ) ( )] at t u t ( + a) + ω

43 Propertie of Laplace Tranform (5) Time Differentiation: If F () i the Laplace Tranform of f (t), then the Laplace Tranform of it derivative i Time Integration: df L dt u( t) F( ) If F () i the Laplace Tranform of f (t), then the Laplace Tranform of it integral i L t f ( t) dt f F( ) ( )

44 Propertie of Laplace Tranform (6) Initial and Final Value: The initial-value and final-value propertie allow u to find f() and f( ) of f(t) directly from it Laplace tranform F(). f ( ) lim F( ) Initial-value theorem f ( ) lim F( ) Final-value theorem

45 The Invere Laplace Tranform () In principle we could recover f(t) from F() via f σ + j ( t) F( ) e j π F σ j x t d But, thi formula in t really ueful.

46 The Invere Laplace Tranform () Suppoe F() ha the general form of F ( ) N( ) D( ) numerator polynomial denominator polynomial The finding the invere Laplace tranform of F() involve two tep:. Decompoe F() into imple term uing partial fraction expanion.. Find the invere of each term by matching entrie in Laplace Tranform Table.

47 Example Find the invere Laplace tranform of Solution: ) ( F t ), ( ) 3in( 5 ( ) ( t u t e L L L t f t The Invere Laplace Tranform (3)

48 Application to Integro-differential Equation () The Laplace tranform i ueful in olving linear integro-differential equation. Each term in the integro-differential equation i tranformed into -domain. Initial condition are automatically taken into account. The reulting algebraic equation in the -domain can then be olved eaily. The olution i then converted back to time domain.

49 Application to Integro-differential Equation () Example: Ue the Laplace tranform to olve the differential equation d v( t) dt + 6 dv( t) dt + 8v( t) u( t) Given: v() ; v () -

50 Solution: Taking the Laplace tranform of each term in the given differential equation and obtain [ ] [ ] ) ( ) ( 4 ) ( By the invere Laplace Tranform, 4 ) ( 4 4 ) ( 8) 6 ( we have, '() ; () Subtituting ) ( 8 () ) ( 6 '() () ) ( t u e e t v V V v v V v V v v V t t Application to Integro-differential Equation (3)

52 Circuit Element Model () Step in Applying the Laplace Tranform:. Tranform the circuit from the time domain to the -domain. Solve the circuit uing nodal analyi, meh analyi, ource tranformation, uperpoition, or any circuit analyi technique with which we are familiar 3. Take the invere tranform of the olution and thu obtain the olution in the time domain.

53 Circuit Element Model () Aume zero initial condition for the inductor and capacitor, Reitor : Inductor: Capacitor: V()RI() V()LI() V() I()/C The impedance in the -domain i defined a Z() V()/I() The admittance in the -domain i defined a Y() I()/V() Time-domain and -domain repreentation of paive element under zero initial condition.

54 Circuit Element Model (3) Non-zero initial condition for the inductor and capacitor, Reitor : Inductor: Capacitor: V()RI() V()LI() + LI() V() I()/C + v()/

55 Introductory Example Charging of a capacitor v V()

56 Circuit Element Model Example () Example : Find v (t) in the circuit hown below, auming zero initial condition.

57 Circuit Element Model Example () Solution: Tranform the circuit from the time domain to the -domain: u( t) H 3 F L C 3 Apply meh analyi, on olving for V (): V ( 3 ( + 4) 3 4t ) v ( ) in( ) V, t e t t + ( ) Invere tranform

58 Example : Baic in Sytem and Circuit Theory Circuit Element Model Example (3) Determine v (t) in the circuit hown below, auming zero initial condition. An : 8( e t te t ) u( t) V

59 Circuit Element Model Example (4) Example 3: Find v (t) in the circuit hown below. Aume v ()5V. An : t t v( t) (e + 5e ) u( t) V

60 Example 4: Baic in Sytem and Circuit Theory Circuit Element Model Example (5) The witch hown below ha been in poition b for a long time. It i moved to poition a at t. Determine v(t) for t >. t / τ An : v( t) (V IR) e + IR, t >, whereτ RC

61 Circuit Analyi Circuit analyi i relatively eay to do in the -domain. By tranforming a complicated et of mathematical relationhip in the time domain into the -domain where we convert operator (derivative and integral) into imple multiplier of and /. Thi allow u to ue algebra to et up and olve the circuit equation. In thi cae, all the circuit theorem and relationhip developed for dc circuit are perfectly valid in the -domain.

62 Circuit Analyi Example () Example: Conider the circuit below. Find the value of the voltage acro the capacitor auming that the value of v (t)u(t) V and aume that at t, -A flow through the inductor and +5V i acro the capacitor.

63 Circuit Analyi Example () Solution: Tranform the circuit from time-domain (a) into -domain (b) uing Laplace Tranform. On rearranging the term, we have V By taking the invere tranform, we get t t v( t) (35e 3e ) u( t) V

64 Circuit Analyi Example (3) Example: The initial energy in the circuit below i zero at t. Aume that v 5u(t) V. (a) Find V () uing the Thevenin theorem. (b) Apply the initial- and finalvalue theorem to find v () an v ( ). (c) Obtain v (t). An: (a) V () 4(+.5)/((+.3)) (b) 4,3.333V, (c) ( e-.3t)u(t) V.

65 Tranfer Function The tranfer function H() i the ratio of the output repone Y() to the input repone X(), auming all the initial condition are zero. H ( ) Y ( ) X ( ) h(t) i the impule repone function. Four type of gain:. H() voltage gain V ()/V i (). H() Current gain I ()/I i () 3. H() Impedance V()/I() 4. H() Admittance I()/V()

66 Example: The output of a linear ytem i y(t)e -t co4t when the input i x(t)e -t u(t). Find the tranfer function of the ytem and it impule repone. Solution: Tranfer Function Example () Tranform y(t) and x(t) into -domain and apply H()Y()/X(), we get Y ( ) ( + ) 4 H ( ) 4 X ( ) ( + ) + 6 ( + ) Apply invere tranform for H(), we get + 6 t h( t) δ ( t) 4e in(4t) u( t)

67 Example: Tranfer Function Example () The tranfer function of a linear ytem i H ( ) + 6 Find the output y(t) due to the input e -3t u(t) and it impule repone. An : e 3t + 4e 6t, t ; δ (t) -e -6t u( t)

69 Introduction In continuou ytem Laplace tranform play a unique role. They allow ytem and circuit deigner to analyze ytem and predict performance, and to think in different term - like frequency repone - to help undertand linear continuou ytem. Z-tranform play the role in ampled ytem that Laplace tranform play in continuou ytem. In continuou ytem, input and output are related by differential equation and Laplace tranform technique are ued to olve thoe differential equation. In ampled ytem, input and output are related by difference equation and Z-tranform technique are ued to olve thoe differential equation.

70 Fourier, Laplace and Z-Tranform For right-ided ignal (zero-valued for negative time index) the Laplace tranform i a generalization of the Fourier tranform of a continuou-time ignal, and the z-tranform i a generalization of the Fourier tranform of a dicrete-time ignal. Fourier Tranform generalization Laplace Tranform Z-Tranform continuou-time ignal dicrete-time ignal

71 The Z-tranform convert a dicrete time-domain ignal, which i a equence of real or complex number, into a complex frequency-domain repreentation. It can be conidered a a dicrete-time equivalent of the Laplace tranform. There are numerou ampled ytem that look like the one hown below. dicrete-time ignal

72 Definition of the Z-Tranform Let u aume that we have a equence, y k. The ubcript "k" indicate a ampled time interval and that y k i the value of y(t) at the k th ample intant. y k could be generated from a ample of a time function. For example: y k y(kt), where y(t) i a continuou time function, and T i the ampling interval. We will focu on the index variable k, rather than the exact time kt, in all that we do in the following. Z k [ y ] y z k k k

73 k k a y y Z-Tranform Example Given the following ampled ignal: [ ] a z z z a z a z a a Z k k k k k k We get the Z-Tranform for y

74 Z-Tranform of Unit Impule and Unit Step Given the following ampled ignal D k : D k i zero for k>, o all thoe term are zero. D k i one for k, o that Z[ D k ] Given the following ampled ignal u k : u k i one for all k. Z [ u ] k + z + z + z 3... z z

75 More Complex Example of Z-Tranform Given the following ampled ignal f k : f k f akt ( kt ) e in(bkt ) Z k k k k akt k [ f ] f z e in( bkt ) z k Finally: j z z c z z c [ f ] + Z at + jbt k * where c e

76 S- and Z-Plane Preentation

77 Invere Z-Tranform The invere z-tranform can be obtained uing one of two method: a) the inpection method, b) the partial fraction method. In the inpection method each imple term of a polynomial in z, H(z), i ubtituted by it time-domain equivalent. For the more complicated function of z, the partial fraction method i ued to decribe the polynomial in term of impler term, and then each imple term i ubtituted by it time-domain equivalent term.