EE 230. Lecture 4. Background Materials

Transcription

1 EE 230 Lecture 4 Background Materials Transfer Functions Test Equipment in the Laboratory

2 Quiz 3 If the input to a system is a sinusoid at KHz and if the output is given by the following expression, what is the THD? o o V = 2sin(2000 πt) + 0.sin(4000πt+ 45 ) sin(0000πt+ 20 ) OUT

3 And the number is? 3 8 5?

4 Quiz 3 If the input to a system is a sinusoid at KHz and if the output is given by the following expression, what is the THD in % (based upon power)? o o V = 2sin(2000 πt) + 0.sin(4000πt+ 45 ) sin(0000πt+ 20 ) OUT THD = A 2 k k = 2 00% 2 A THD = % 0.3%

5 Test Equipment in the EE 230 Laboratory (Plus computer, oven, software)

6 Whats inside/on this equipment? Computer (except maybe dc power supply) Some analog circuitry Software Knobs/Buttons Computer Interface Test equipment is becoming very powerful Seldom need most of the capabilities of the equipment Versatility and flexibility makes basic (and most used) operation a little more difficult to learn

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20 When is the voltage reading on the Signal Generator Accurate? Almost Never! When it is reasonably close, (i.e. when not affected by the effects of the output impedance on the actual output) how does the accuracy compare with that of the scope or the digital multimeter measuring the same voltage? Two to three orders of magnitude worse!

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24 Test Equipment in the EE 230 Laboratory 984 Pages! The documentation for the operation of this equipment is extensive Critical that user always know what equipment is doing Consult the users manuals and specifications whenever unsure

25 Whats inside/on this equipment? Computer (except maybe dc power supply) Some analog circuitry Software Knobs/Buttons Computer Interface Why is this not the standard interface with a computer?

26 Review from Last Time Properties of Linear Networks T(s) frequency domain ( ) ( s) XOUT s = T(s) X is termed the transfer function IN This is often termed the s-domain or Laplace-domain representation ( ) P T s s=jω = T (jω) Will discuss the frequency domain representations and the more general concept of transfer functions in more detail later

27 Review from Last Time Distortion A system has Harmonic Distortion (often just termed Distortion ) if when a pure sinusoidal input is applied, the Fourier Series representation of the output contains one or more terms at frequencies different than the input frequency A linear system has Frequency Distortion if for any two sinusoidal inputs of magnitude X and X 2, the ratio of the corresponding sinusoidal outputs is not equal to X /X 2. Harmonic distortion is characterized by several different metrics including the Total Harmonic Distortion, Spurious Free Dynamic Range (SFDR) Frequency distortion is characterized by the transfer function, T(s), of the system

28 Review from Last Time Total Harmonic Distortion t+t 2 PAVG = f () t dt T t It can be shown that P = AVG k = 2 A 2 k f() t = Aksin(kωt+θ k) k = Define P to be the power in the fundamental A 2 P= 2 P = Harmonics k = 2 2 A 2 k P THD= THD = HARMONICS k = 2 P A A 2 THD often expressed in db or in % THD =0log THD 2 k db 0 ( ) Can also be expressed relative to signal instead of power

29 Amplifiers: Review from Last Time Amplifiers are circuits that scale a signal by a constant amount Ideally V OUT =AV IN where A is a constant (termed the gain) The dependent sources discussed in EE 20 are amplifiers S V S S M S S I S S M S

30 Amplifiers, Frequency Response, and Transfer Functions IN OUT The frequency dependent gain of a linear circuit or system is often termed the transfer function Sometimes linear circuits are termed Amplifiers or Filters when some specific properties of the relationship between the input and output are of particular interest

31 Example: X IN Linear System X OUT Will go through the mechanics first, then formalize the concepts R V OUT V IN C Obtain the phasor-domain transfer function Obtain the s-domain transfer function Plot the magnitude of the transfer function Plot the phase of the transfer function Obtain the sinusoidal steady state response if V IN =V M sin(2π f t) Do a time-domain analysis of this circuit

32 Example: R V OUT Obtain the phasor-domain transfer function V IN C Phasor-Domain Circuit IN jωc OUT V OUT jωc = V R+ jωc IN V OUT = V +jωrc IN V jω ( ) ( jω) OUT T(jω) P = = VIN +jωrc

33 Example: R V OUT Obtain the s-domain transfer function V IN C s-domain Circuit ( ) ( ) sc V s V s +src OUT T(s) = = IN V sc OUT s VIN s R+ sc ( ) = ( ) VOUT s VIN s +src ( ) = ( )

34 Example: R V OUT V IN C sc jωc T(s) = T(jω) P = +src +jωrc Observe: T(s) = s=jω +jωrc Observe: T(s) = T(jω) This property holds for any linear system! s=jω P

35 Example: R V OUT Plot the transfer function magnitude T(s) = +src T(jω) = +jωrc V IN C T(jω) = ( ) 2 + ωrc T( jω) 2 ω RC

36 Example: R V OUT Plot the phase of the transfer function T(s) = +src T(jω) = +jωrc V IN C T (jω) = -tan ωrc - ( ) T( jω) 0 o -45 o -90 o RC ω

37 Example: R V OUT Obtain the sinusoidal steady-state response if V IN =V M sin(2π f t) V IN C Need a theorem that expresses the sinusoidal steady-state response

38 Key Theorem: Theorem: The steady-state response of a linear network to a sinusoidal excitation of V IN =V M sin(ωt+γ) is given by OUT ( ) V t = V T jω sin ωt+γ+ T jω () ( ) ( ) m This is a very important theorem and is one of the major reasons phasor analysis was studied in EE 20 The sinusoidal steady state response is completely determined by T(jω) The sinusoidal steady state response can be written by inspection from the T jω T( jω ) ( ) and plots T(s) s=jω = T(jω) P

39 Example: Obtain the sinusoidal steady-state response if V IN =V M sin(2π f t) T(jω) = ( ) 2 + ωrc T (jω) = -tan ωrc - ( ) Thus, from the previous theorem with γ=0 OUT ( ) V t = V T jω sin ωt+γ+ T jω () ( ) ( ) m V () OUT t = Vm sin ωt - tan ωrc 2 + ωrc ( ) ( - ( ))

40 Observations: Authors of current electronics textbooks do not talk about phasors or T P (jω) This is consistent with the industry when discussing electronic circuits and systems The sinusoidal steady state response is of considerable concern in electronic circuits and is used extensively in the text for this cours Authors and industry use the concept of the transfer function T(s) when characterizing the frequency-dependent performance of linear circuits and systems

41 Questions Why is T(s) used instead of T P (jω) in the electronics field? What is T(s)? Why was T P (jω) emphasized in EE 20 instead of T(s) for characterizing the frequency dependence of linear networks?

42 Example: Do a time-domain analysis of this circuit ( ) ( ) VIN t -VOUT t it= () R () ( ) V it C d t OUT IN = dt () = n ( + ) V t V si ωt γ M Complete set of differential equations that can be solved to obtain V OUT (t) One way to solve this is to use Laplace Transforms I = scv OUT VIN -VOUT = IR ( sinγ ) s + ωcosγ VIN = V M 2 2 s + ω

43 Example: Do a time-domain analysis of this circuit I = scv OUT VIN -VOUT = IR ( sinγ ) s + ωcosγ VIN = V M 2 2 s + ω With some manipulations, can get expression for V OUT ( sinγ) s + ωcosγ V OUT = VM 2 2 s + ω + src With some more manipulations, we can take inverse Laplace transform to get sinγ ωrc t - () RC V OUT t = VM - e V M sin( ωt + γ tan 2 ( ωrc) ) + 2 ( RC) tanγ + ( ωrc )

44 Example: Do a time-domain analysis of this circuit sinγ ωrc t - () RC V OUT t = VM - e V M sin( ωt + γ tan 2 ( ωrc) ) + 2 ( RC) tanγ + ( ωrc ) Neglecting the natural response to obtain the sinusoidal steady state response, we obtain (with γ = 0) VOUT () t = V M sin ωt ωrc 2 + ( ωrc ) ( tan ( )) Note this is the same response as was obtained with the two previous solutons

45 Formalization of sinusoidal steady-state analysis phasor-domain approach L C jωl jωc All other components unchanged ( ) X = T jω X (this equation not needed) OUT P IN OUT ( ) ( ) = ( ) ( ) X t X T jω sin ωt + θ + T jω M

46 Formalization of sinusoidal steady-state analysis s-domain approach ( ) = ( ) X t X sin ωt + θ X X i (t) i (t) IN M s Transform X i (S) s-domain Circuit Circuit Analysis KVL, KCL L C sl sc All other components unchanged s Transform X i (S) s-domain Circuit Circuit Analysis KVL, KCL Set of Linear equations in s Set of Linear equations in s Solution of Linear Equations Solution of Linear Equations X OUT (S) T(s) Inverse s Transform X ( ) = T jω X OUT P IN (this equation not needed) X OUT (S) T(s) Inverse s Transform X OUT (t) general X OUT (t) ( ) XOUT ( t) = XM T( jω) sin ωt + θ + T ( jω) Sinusoidal steady state

47 Formalization of sinusoidal steady-state analysis X i (t) = X M sin(ωt+θ) X i (S) s Transform Phasor Transform X i (jω) s-domain Circuit Time Domain Circuit Phasor Domain Circuit Circuit Analysis KVL, KCL Circuit Analysis KVL, KCL Circuit Analysis KVL, KCL Set of Linear equations in s Set of Differential Equations Set of Linear equations in jω Solution of Linear Equations Solution of Differential Equations Solution of Linear Equations T(s) T P (jω) X OUT (S) Inverse s Transform Inverse Phasor Transform X OUT (jω) X OUT (t)

48 Which of the methods is most widely used? X i (t) = X M sin(ωt+θ) X i (S) s Transform Phasor Transform X i (jω) s-domain Circuit Time Domain Circuit Phasor Domain Circuit Circuit Analysis KVL, KCL Circuit Analysis KVL, KCL Circuit Analysis KVL, KCL Set of Linear equations in s Set of Differential Equations Set of Linear equations in jω Solution of Linear Equations Solution of Differential Equations Solution of Linear Equations T(s) T P (jω) X OUT (S) Inverse s Transform Inverse Phasor Transform X OUT (jω) X OUT (t) s-domain analysis almost totally dominates the electronics fields and most systems fields

49 Formalization of sinusoidal steady-state analysis - Summary s-domain The Preferred Approach ( ) = ( ) X t X sin ωt + θ IN M X i (t) s Transform L C sl sc All other components unchanged X i (S) s-domain Circuit Circuit Analysis KVL, KCL Set of Linear equations in s Solution of Linear Equations OUT ( ) () = ( ) ( ) X t X T jω sin ωt + θ + T jω M X OUT (S) T(s) Inverse s Transform X OUT (t)

50 Filters: A filter is an amplifier that ideally has a frequency dependent gain Simply a different name for an amplifier that typically has an ideal magnitude or phase response that is not flat Some standard filter responses with accepted nomenclature T( jω) T( jω) T( jω) T( jω) T( jω) T( jω)

51 Summary of frequency response appears on posted notes