Lab Aignment 6: Tranfer Function Analyi Reviion: May 6, 007 Produced in cooperation with www.digilentinc.com Overview In thi lab, we will employ tranfer function to determine the frequency repone and tranient repone of firt and econd-order paive circuit. We will then ue imilar technique to deign a Wien Bridge ocillator circuit and a econd order Butterworth low pa filter. Simplitically peaking, the tranfer function of a ytem reult from taking the Laplace Tranform of the differential equation relating the ytem output to the input. The ratio of the output to the input in the Laplace domain i the tranfer function. We will only conider ytem for which the tranfer function can be expreed a a ratio of two polynomial, a hown below: T ( ) b( ) b b m m 1 m m 1 = = n n 1 a( ) an an 1 L b L a 0 0 (1) In the above equation, =σjω. For ytem of interet to u in thi lab aignment, n>m. The root of the denominator of thi function are identical to the root of the characteritic polynomial governing the ytem; thee root are called the ytem pole. The root of the numerator of thi function are called the ytem zero. The tranfer function i a valuable analyi tool in that the differential equation governing the ytem ha now been converted to an algebraic relationhip. Multiplication of the tranfer function by the Laplace tranform of the input forcing function reult in the Laplace tranform of the output repone. Determining the time function correponding to thi output repone can be omewhat problematic, but often the ytem repone can be interpreted directly from the location of the ytem pole and zero. The tranfer function of a ytem appear to be very imilar to the ytem frequency repone. It hould be noted, however, that a ytem frequency repone can be ued only to characterize the ytem teady-tate repone to a inuoidal input, while the tranfer function can provide the tranient repone to an arbitrary input function. Repone of firt and econd order ytem can often be readily interpreted in term of their pole location. In thi lab aignment, we will determine the pole location for ome firt and econd order circuit. We will meaure the repone of thee ytem to variou input function and compare the meaured repone to our expectation baed on the ytem pole location. Contain material Digilent, Inc. 9 page
Lab Aignment 6:Tranfer Function Analyi Page of 9 Before beginning thi lab, you hould be able to: Determine input-output differential equation governing electrical circuit (Module.) Determine frequency repone of firt order ytem (amplitude and phae). Determine time contant of firt order ytem (Module 1.1) Determine tranfer function governing electrical circuit (Module 6.4) Determine pole, zero of tranfer function (Module 6.4) Determine ytem repone via Laplace tranform (Module 6.5) State the Initial and Final Value Theorem (Module 6.6) Ue the Initial and Final Value Theorem to tet tranfer function for conitency with time domain repone (Module 6.6) Infer the time-domain behavior of a ignal from the location of the pole of the Laplace tranform of the ignal (Module 6.1) Qualitatively relate the tranfer function pole to the ytem natural repone(module 6.4) State the relationhip between pole location, natural frequency and damping ratio for econd order ytem (Module 6.8) After completing thi lab, you hould be able to: Experimentally verify expected time-domain circuit repone determined via Laplace tranform analyi Experimentally verify expected frequency domain circuit repone determined via Laplace Tranform analyi Ue Initial and Final Value Theorem to interpret experimental data Deign active filter to provide deired ytem pole Thi lab exercie require: EE 35 Analog Part Kit Breadboard Function Generator, ocillocope Symbol Key: Demontrate circuit operation to teaching aitant; teaching aitant hould initial lab notebook and grade heet, indicating that circuit operation i acceptable. Analyi; include principle reult of analyi in laboratory report. Numerical imulation (uing PSPICE or MATLAB a indicated); include reult of Matlab numerical analyi and/or imulation in laboratory report. Record data in your lab notebook.
Lab Aignment 6:Tranfer Function Analyi Page 3 of 9 I. Firt order RC Circuit Firt order circuit are governed by a firt order differential equation whoe characteritic equation ha a ingle root. Thu, the tranfer function for thee ytem have a ingle pole. Lab 1 provided u with ome expoure to the time and frequency repone of firt order circuit. In that lab, we characterized the circuit by a cutoff frequency (ω c ) or a time contant (τ). In thi lab aignment, we will determine the pole location for two RC circuit and compare the circuit meaured repone to our expectation baed on the pole location. We will alo compare the ytem pole location to the time contant and the cutoff frequency. Pre-lab: (a) Determine the differential equation, the tranfer function, and the frequency repone for the circuit hown in Figure 1. Your reult will all be function of R and C. Determine the pole location for the ytem. What i the relationhip between the time contant, the pole location, and the cutoff frequency of the circuit? Uing partial fraction expanion technique, determine an analytical expreion for the ytem tep repone. Sketch the tep repone of the circuit, indicating the teady-tate repone and the time contant on your ketch. Sketch the frequency repone (gain and phae) of the circuit, uing a emi-log cale. Unit of phae hould be degree and unit of gain hould be db. Indicate the DC value and the cutoff frequency on your ketch. What i the phae at the cutoff frequency? (b) Repeat part (a) for the circuit of Figure. (c) What i the relationhip between the pole location for the ytem of Figure 1 and? v i (t) R C V o (t) - - Figure 1. Low pa RC network. v i(t) C R V o (t) - - Figure. High pa RC network.
Lab Aignment 6:Tranfer Function Analyi Page 4 of 9 Lab Procedure: (a) Contruct the circuit hown in Figure 1 and, uing R=100 Ω, and C = 0.1µF. (Notice that the two circuit hown are actually the ame circuit you are jut meauring a different output. Thi i an important obervation. Keep it in mind for later.) (b) Apply a 10V peak-to peak quare wave input with period = m a hown in Figure 3 to each of the circuit in part (a) and ketch the repone in your lab notebook. Determine the time contant and teady-tate repone of the output voltage, v o (t). Compare the experimental reult with your expectation from the pre-lab. m 5V - 5V Figure 3. Square wave input ignal, v i (t) for part (b). (c) Apply a inuoidal voltage input, v i (t) to each of the circuit in part (a). Vary the input frequency over a range of 100 Hz to 1MHz. Record the amplitude of the input and output voltage and the time hift between the two. Calculate the gain and phae difference between the input and output. Plot the gain in db and the phae in degree a a function of frequency. Compare your experimental reult with your expectation from the pre-lab. (d) Demontrate operation of either the low or high pa circuit to the Teaching Aitant for both the inuoidal input and tep input cae. Have the TA initial the appropriate page() of your lab notebook and the lab checklit. Pot Lab Exercie: Uing your meaured value for R and C, imulate the tep and frequency repone for both the high and low pa filter circuit uing either PSPICE or MatLab. Compare your imulation reult with your meaured repone. II. Second order RLC Circuit Second order ytem will have characteritic equation with two root. Thu, the ytem will have two pole. Second order ytem are generally categorized by the location of thee pole. Underdamped ytem will have pole with complex value; their value will be complex conjugate of one another. Overdamped ytem will have two real valued pole; one of the pole will be cloer to the origin of the -plane than the other. Critically damped ytem have two pole at the ame location on the real axi. Note: ytem pole are alway located ymmetrically with repect to the real axi of the -plane thi i a requirement for the ytem to provide a real-valued time repone to a real-valued input. The denominator of the tranfer function of a econd order ytem i often written in the form: ζω ωn n ()
Lab Aignment 6:Tranfer Function Analyi Page 5 of 9 where ζ i the damping ratio and ω n i the natural frequency. Overdamped ytem have a damping ratio greater than one, underdamped ytem have a damping ratio le than one, and critically damped ytem have a damping ratio of exactly one. Pre-lab: (a) Write the tranfer function for the ytem hown in Figure 4. Deign the circuit o that ω n i approximately 3 KHz and ζ i approximately 0.1. (You may aume that the inductor reitance, R L, i zero for the pre-lab analyi). (b) From the tranfer function, determine the aymptotic behavior of the ytem frequency repone for very low (ω=0) and very high (ω ) frequencie. The frequency repone can be determined from the tranfer function by replacing with jω. What i the lope of the amplitude repone (in db/decade) at high and low frequencie? What i the phae repone of the ytem at high and low frequencie? What i the phae repone at the natural frequency? Sketch the aymptotic behavior of the frequency repone (gain and phae) of the circuit, uing a emi-log cale. Unit of phae hould be degree and unit of gain hould be db. (c) Ue either PSPICE or Matlab to imulate your deign with frequency repone and tep repone plot. (d) Compare the imulation reult to your hand calculation of part (b). R R L L v i - C v o - Figure 4. Serie RLC circuit. Lab Procedure: Contruct the circuit you deigned above and experimentally determine: (a) the frequency repone of your circuit (magnitude and phae). Meaure the amplitude and phae repone for at leat 10 different frequencie, from a frequency range of at leat an order of magnitude below the natural frequency to an order of magnitude above the natural frequency. Plot your reult in your lab notebook, on our typical emi-log cale with unit of gain a db and unit of phae a degree. (b) the tep repone of your circuit. Sketch the time repone in your lab notebook or ave the data to a flah drive for later proceing. Determine the period of any ocillation in the repone and compare it to the natural frequency of the circuit. (c) Demontrate operation of your circuit to the Teaching Aitant for both the inuoidal input and tep input cae. Have the TA initial the appropriate page() of your lab notebook and the lab checklit.
Lab Aignment 6:Tranfer Function Analyi Page 6 of 9 III. Second order Butterworth low pa filter Butterworth filter have no overhoot in the pa band of the filter. (A low pa Butterworth filter i aid to monotonically decreae with frequency. They are alo referred to a maximally flat.) Thi lack of overhoot in the pa band i highly deirable, though one pay a penalty in that the magnitude repone decreae relatively lowly in the top band. The tranfer function H() of a Butterworth filter i defined in term of the pole of H()H(-). If the order of H() i n, then H() ha n pole and H()H(-) ha n pole. Thee n pole are choen to lie on a circle with radiu ω n, where ω n i the cutoff frequency of the filter. Additionally, thee pole are equally paced angularly around the circle. Of thee pole, the n pole in the left half of the -plane are choen to create the tranfer function H() of the Butterworth filter. Figure 5 below how the pole location for H()H(-) and the correponding pole location for H() for Butterworth filter of order n = 1,, and 3. A Butterworth filter of order n ha a magnitude repone given by: H ( jω ) = 1 (3) n 1 ω ωn A econd order Butterworth filter can be implemented by the circuit hown in Figure 6. The tranfer function for the circuit of Figure 6 i: H ( ) V ( ) 1 R C C o 1 = = (4) V ( ) 1 i RC1 R C1C Thi tranfer function correpond to the tranfer function of a prototype econd order ytem, which i of the form: Y ( ) T ) = = U ( ) ( ω n ξω ω n n For the pole location to be a hown in Figure 5(b), ω n = ω c and ξ=in(45 )=0.7071. Thu, the denominator of the tranfer function of equation (4) become: ω ωc n Given the deired cutoff frequency will now et the coefficient in the denominator of equation (4) and R, C 1, and C can be elected.
Lab Aignment 6:Tranfer Function Analyi Page 7 of 9 (a) n = 1 (b) n = (c) n = 3 Figure 5. Example Butterworth filter pole location.
Lab Aignment 6:Tranfer Function Analyi Page 8 of 9 Figure 6. A circuit that implement a econd order Butterworth filter. Butterworth polynomial (the denominator of the tranfer function) for variou order Butterworth filter can be generated uing Matlab if you have acce to the ignal proceing toolbox. The butter command provide non-normalized tranfer function coefficient for a Butterworth filter of pecified order and cutoff frequency. Syntax for thi command i: [num,den] = butter(n,wn, ) Thi command return tranfer function numerator coefficient in the variable num, tranfer function denominator coefficient in the variable den, for a filter of order n and normalized cutoff frequency Wn in rad/. The pecifie that an analog, rather than a digital filter, i deired. A an example, if we wih to deign a econd order Butterworth filter with cutoff frequency of 500 Hz, the appropriate command i: [num,den] = butter(,500**pi, ) We can then equate the returned coefficient with the coefficient in equation (4) above to determine appropriate value for R, C 1, and C. Pre-lab: (a) Deign a circuit to implement a econd order low pa Butterworth filter with a cutoff frequency of 1000 Hz and a pa-band gain of 1. (Tip: there are really only two independent coefficient in equation (4), but there are three unknown parameter in Figure 5 R, C 1, and C. You get to chooe one of thee parameter omewhat arbitrarily and then et the other two to provide the correct tranfer function coefficient.) (b) Plot the theoretical frequency repone (gain and phae) of your filter. You may ue either PSPICE or Matlab to generate your plot. (Matlab freq or bode command, in the control ytem toolbox, can be handy for generating frequency repone if you wih to ue Matlab.) Do your reult indicate that your deign meet the pecification?
Lab Aignment 6:Tranfer Function Analyi Page 9 of 9 Lab Procedure: Implement the circuit you deigned in the pre-lab above. Meaure the frequency repone (gain and phae) of the circuit and record the data in your lab notebook. Compare the theoretical frequency repone of your filter with the meaured frequency repone. Ae the filter performance relative to the pecified cutoff frequency and pa band gain. Demontrate operation of your circuit to the Teaching Aitant for both the inuoidal input and tep input cae. Have the TA initial the appropriate page() of your lab notebook and the lab checklit. Lab Report: In your lab report, provide a ummary of the reult of thi lab aignment. You hould include, at a minimum, all item indicated on the lab checklit. Append the lab checklit heet with teaching aitant initial indicating completed lab demo to your report.