Voltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response

Transcription

1 Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure #5 OUTLINE Chap 4 RC and RL Circuis wih General Sources Paricular and complemenary soluions Time consan Second Order Circuis The differenial equaion Paricular and complemenary soluions The naural frequency and he damping raio Chap 5 Types of Circui Exciaion Why Sinusoidal Exciaion? Phasors Complex Impedances Reading Chap 4, Chap 5 (skip 5.7) EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu

2 Firs Order Circuis v s () v r () R C i c () v c () i s () R L i L () v L () KVL around he loop: v r () v c () = v s () dvc () RC vc() = vs() d KCL a he node: v( ) v( x) dx = i R L L dil () il() = is() R d s ( ) EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 3 Complee Soluion Volages and currens in a s order circui saisfy a differenial equaion of he form dx() x() τ = f() d f() is called he forcing funcion. The complee soluion is he sum of paricular soluion (forced response) and complemenary soluion (naural response). x() = x () x () Paricular soluion saisfies he forcing funcion Complemenary soluion is used o saisfy he iniial condiions. The iniial condiions deermine he value of K. dxp () x p () τ = f () d p c dxc () xc () τ = d = / xc () Ke τ Homogeneous equaion EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 4

3 The Time Consan The complemenary soluion for any s order circui is / xc () = Ke τ For an RC circui, τ = RC For an RL circui, τ = L/R EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 5 Wha Does X c () Look Like? x () = e c / τ τ = 4 τ is he amoun of ime necessary for an exponenial o decay o 36.7% of is iniial value. /τ is he iniial slope of an exponenial wih an iniial value of. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 6 3

4 The Paricular Soluion The paricular soluion x p () is usually a weighed sum of f() and is firs derivaive. If f() is consan, hen x p () is consan. If f() is sinusoidal, hen x p () is sinusoidal. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 7 nd Order Circuis Any circui wih a single capacior, a single inducor, an arbirary number of sources, and an arbirary number of resisors is a circui of order. Any volage or curren in such a circui is he soluion o a nd order differenial equaion. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 8 4

5 A nd Order RLC Circui i() v s () R L C Applicaion: Filers A bandpass filer such as he IF amp for he AM radio. A lowpass filer wih a sharper cuoff han can be obained wih an RC circui. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 9 The Differenial Equaion i() v r () R v s () C v l () KVL around he loop: v r () v c () v l () = v s () di( ) Ri() i( x) dx L vs () C = d Rdi () di () dvs () i () = L d LC d L d L v c () EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 5

6 The Differenial Equaion The volage and curren in a second order circui is he soluion o a differenial equaion of he following form: d x () dx() α ω x( ) = f( ) d d x() = x () x () p c X p () is he paricular soluion (forced response) and X c () is he complemenary soluion (naural response). EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu The Paricular Soluion The paricular soluion x p () is usually a weighed sum of f() and is firs and second derivaives. If f() is consan, hen x p () is consan. If f() is sinusoidal, hen x p () is sinusoidal. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 6

7 The Complemenary Soluion The complemenary soluion has he following form: s x () = Ke c K is a consan deermined by iniial condiions. s is a consan deermined by he coefficiens of he differenial equaion. s s d Ke dke s α ω Ke = d d s Ke αske ω Ke = s s s s αs ω = EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 3 Characerisic Equaion To find he complemenary soluion, we need o solve he characerisic equaion: s ζω s ω = α = ζω The characerisic equaion has wo rooscall hem s and s. x () = K e K e c s s s ζω ω ζ = s ζω ω ζ = EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 4 7

8 Damping Raio and Naural Frequency α ζ = ω damping raio s ζω ω ζ = s ζω ω ζ = The damping raio deermines wha ype of soluion we will ge: Exponenially decreasing (ζ >) Exponenially decreasing sinusoid (ζ < ) The naural frequency is ω I deermines how fas sinusoids wiggle. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 5 Overdamped : Real Unequal Roos If ζ >, s and s are real and no equal. i c ( ) = K e ςω ω ς K e ςω ω ς i().6.4..e6 i().4..e6. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 6 8

9 Underdamped: Complex Roos If ζ <, s and s are complex. Define he following consans: α = ζω ω = ω ζ d ( ω ω ) x () = e A cos A sin α c d d i() e5..e5 3.E EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 7 Criically damped: Real Equal Roos If ζ =, s and s are real and equal. x () = K e K e c ςω ςω EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 8 9

10 Example For he example, wha are ζ and ω? i() Ω 769pF 59μH d i( ) R di( ) dvs ( ) i( ) = d L d LC L d d xc() dxc() ζω ω d xc ( ) = d R R C ω =, ζω =, ζ = LC L L EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 9 Example ζ =. ω = π455 Is his sysem over damped, under damped, or criically damped? Wha will he curren look like? i() e5..e5 3.E EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu

11 Slighly Differen Example Increase he resisor o kω Wha are ζ and ω? i() kω v s () 769pF 59μH i() e6 ζ =. ω = π455 EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Types of Circui Exciaion Linear Time Invarian Circui SeadySae Exciaion (DC SeadySae) Linear Time Invarian Circui Sinusoidal (Single Frequency) Exciaion AC SeadySae Sep Exciaion Digial Pulse Source Linear Time Invarian Circui OR Linear Time Invarian Circui Transien Exciaions EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu

12 Signal (V) Relaive Ampliude signal Signal (V) signal Signal (V) Why is SingleFrequency Exciaion Imporan? Some circuis are driven by a singlefrequency sinusoidal source. Some circuis are driven by sinusoidal sources whose frequency changes slowly over ime. You can express any periodic elecrical signal as a sum of singlefrequency sinusoids so you can analyze he response of he (linear, imeinvarian) circui o each individual frequency componen and hen sum he responses o ge he oal response. This is known as Fourier Transform and is remendously imporan o all kinds of engineering disciplines! EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 3 a Represening a Square Wave as a Sum of Sinusoids b c d T i me (ms) Frequency (Hz) (a)square wave wih second period. (b) Fundamenal componen (doed) wih second period, hirdharmonic (solid black) wih/3second period, and heir sum (blue). (c) Sum of firs en componens. (d) Specrum wih erms. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 4

13 SeadySae Sinusoidal Analysis Also known as AC seadysae Any seady sae volage or curren in a linear circui wih a sinusoidal source is a sinusoid. This is a consequence of he naure of paricular soluions for sinusoidal forcing funcions. All AC seady sae volages and currens have he same frequency as he source. In order o find a seady sae volage or curren, all we need o know is is magniude and is phase relaive o he source We already know is frequency. Usually, an AC seady sae volage or curren is given by he paricular soluion o a differenial equaion. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 5 The Good News! We do no have o find his differenial equaion from he circui, nor do we have o solve i. Insead, we use he conceps of phasors and complex impedances. Phasors and complex impedances conver problems involving differenial equaions ino circui analysis problems. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 6 3

14 Phasors A phasor is a complex number ha represens he magniude and phase of a sinusoidal volage or curren. Remember, for AC seady sae analysis, his is all we need o compuewe already know he frequency of any volage or curren. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 7 Complex Impedance Complex impedance describes he relaionship beween he volage across an elemen (expressed as a phasor) and he curren hrough he elemen (expressed as a phasor). Impedance is a complex number. Impedance depends on frequency. Phasors and complex impedance allow us o use Ohm s law wih complex numbers o compue curren from volage and volage from curren. EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 8 4

15 Sinusoids Ampliude: V M Angular frequency: ω = π f Radians/sec Phase angle: θ Frequency: f = /T Uni: /sec or Hz Period: T v () = V cos( ω θ ) M Time necessary o go hrough one cycle EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 9 Phase Wha is he ampliude, period, frequency, and radian frequency of his sinusoid? EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 3 5

16 Phasors A phasor is a complex number ha represens he magniude and phase of a sinusoid: X M cos ( ω θ ) Time Domain X = X M θ Frequency Domain EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu 3 6