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FREQUENCY RESPONSE & RESONANT CIRCUITS Filters, frequency response, time domain connection, bode plots, resonant circuits. OUTLINE AND TOPICS Reading 1. Boylestad Ch 21.1-21.11 2. Boylestad Ch 20.1-20.8 Low-pass filters High-pass filters Other filters Resonance (Ch 20) Ideal op-amps and active filters Decibels & log scales Linear systems and transfer functions Bode plots FILTERS FILTERS Any combination of passive (R, L, and C) and/or active (transistors or operational amplifiers) elements designed to select or reject a band of frequencies is called a filter. In communication systems, filters are used to pass those frequencies containing the desired information and to reject the remaining frequencies. FILTERS In general, there are two classifications of filters: Passive filters-gain always<1 Active filters-gain can be >1 Gain=Vo/Vi Can also be represented By a differential eqn. Vi Circuit or system T(jω)=T(s) Vo Gain works for DC always For AC, only for amplitude/phase of sine/cosine FILTERS Couple of definitions: Impedance of a circuit element is the AC analog of resistance. Only works for sines and cosines! Z = R + jX Z-impedance R-real part of impedance-resistance X-imag part of impedance-REACTANCE Resistance is always positive Reactance can be positive or negative Positive reactance-inductive Negative reactance-capacitive FIG. 21.7 Defining the four broad categories of FILTERS Impedance of a circuit element is the AC analog of resistance. Only works for sines and cosines! WHY? capacitor : dv dt if input and output are sine or cosine, we can represent them as i=C v = v0 e jwt i = i0 e jwt e jwt = cos(w t) + j sin(w t) cos(Why? w t) = 1 / 2[e jwt + e- jwt ] sin(w t) = 1 / 2 j[e jwt - e- jwt ] Then, the “resistance-like” ratio of the amplitudes of current and voltage FIG. 21.7 Defining the four broad categories of ZC = v0 1 1 = = j() i0 jwC wC R-C LOW-PASS FILTER FIG. 21.8 Lowpass filter. ZC = 1 / jwC = j(-1 / wC) ZC (w = 0) = - j¥ ZC (w ® ¥) = 0 FIG. 21.9 R-C lowpass filter at low frequencies. R-C LOW-PASS FILTER FIG. 21.10 R-C lowpass filter at high frequencies. Applying voltage divider Vo ZC gain = = Vi ZC + R ZC = 1 = jXC jwC Zc-impedance Xc-reactance FIG. 21.11 Vo versus frequency for a low-pass R-C filter. We must take the magnitude To get the size of the gain. This magnitude gives rise to the square root we see in the textbook. We take the “argument” or angle To get the phase of the gain. R-C LOW-PASS FILTER Links to help with sketching sines and cosines http://www.youtube.com/watch?v=ijTIraykUk&feature=relmfu http://www.youtube.com/watch?feature=e ndscreen&v=RzZyyIu9IvA&NR=1 FIG. 21.12 Normalized plot of Fig. 21.11. Length is amplitude gain At a given frequency, gain is a complex number that can be drawn as Imaginary/”reactive” θ-phase change Real R-C LOW-PASS FILTER-PHASE FIG. 21.13 Angle by which Vo leads Vi. R-C LOW-PASS FILTER FIG. 21.14 Angle by which Vo lags Vi. R-C LOW-PASS FILTER FIG. 21.15 Low-pass RL filter. FIG. 21.16 Example 21.5. R-C LOW-PASS FILTER Remember, w (rad / s) = 2p f (Hz) FIG. 21.17 Frequency response for the low-pass R-C network in Fig. 21.16. R-C LOW-PASS FILTER FIG. 21.18 Normalized plot of Fig. 21.17. R-C HIGH-PASS FILTER FIG. 21.19 Highpass filter. R-C HIGH-PASS FILTER FIG. 21.20 R-C high-pass filter at very high frequencies. Vo R gain = = Vi ZC + R jw RC = jw RC +1 FIG. 21.21 R-C highpass filter at f = 0 Hz. jw RC w RC | gain |=| |= jw RC +1 1+ (w RC)2 As w=0, |gain|=0 As winf, |gain|1 R-C HIGH-PASS FILTER FIG. 21.22 Vo versus frequency for a highpass R-C filter. R-C HIGH-PASS FILTER FIG. 21.23 Normalized plot of Fig. 21.22. R-C HIGH-PASS FILTER FIG. 21.24 Phase-angle response for the highpass R-C filter. Same as Boylestad result jw RC Ðgain = Ð jw RC +1 = Ðjw RC - Ð( jw RC +1) = 90° - arctan(wRC) R 1 / jwC + R = ÐR - Ð(1 / jw C + R) Ðgain = Ð = 0° - arctan(-(1 / wC) / R) = arctan(XC / R) R-L HIGH-PASS FILTER Z L = jw L Vo ZL gain = = Vi Z L + R jw L jw (L / R) = = jw L + R jw (L / R) +1 What is L/R? w=0, |gain|=0 winf., |gain|1 FIG. 21.25 High-pass RL filter. What kind of filter is this? R-L HIGH-PASS FILTER FIG. 21.26 Normalized plots for a low-pass and a high-pass filter using the same elements. R-L HIGH-PASS FILTER FIG. 21.27 Phase plots for a low-pass and a high-pass filter using the same elements. PASS-BAND FILTERS FIG. 21.28 Series resonant passband filter. CR gives you high pass, LR gives you low pass. RLC CIRCUITS-RESONANCE! The resonant electrical circuit must have both inductance and capacitance. In addition, resistance will always be present due either to the lack of ideal elements or to the control offered on the shape of the resonance curve. When resonance occurs due to the application of the proper frequency ( fr), the energy absorbed by one reactive element is the same as that released by another reactive element within the system. SERIES RESONANT CIRCUIT A resonant circuit (series or parallel) must have an inductive and a capacitive element. A resistive element is always present due to the internal resistance of the source (Rs), the internal resistance of the inductor (Rl), and any added resistance to control the shape of the response curve (Rdesign). SERIES RESONANT CIRCUIT At resonance, ZC=-ZL, or jXC=j(-XL) reactance Therefore, reactances cancel! Only resistive components remain C XC = -1/ wC XC = X L XL = w L w r =1 / LC = 2p fr FIG. 21.28 Series resonant passband filter. L fr, resonance ω PASS-BAND FILTERS FIG. 21.29 Parallel resonant passband filter. PASS-BAND FILTERS FIG. 21.30 Series resonant pass-band filter for Example 21.7. PASS-BAND FILTERS FIG. 21.31 Pass-band response for the network. PASS-BAND FILTERS FIG. 21.32 Normalized plots for the pass-band filter in Fig. 21.30. SELECTIVITY Q is the “sharpness” or selectivity of the resonance We define it as: Q= fr (location) BW (width) Q –Quality factor Q high, narrow BW, sharp, High quality resonance. Q small, wide BW Poor quality resonance FIG. 20.15 Effect of R, L, and C on the selectivity curve for the series resonant circuit. SELECTIVITY FIG. 20.16 Approximate series resonance curve for Qs ≥ 10. PASS-BAND FILTERS T(s) = T( jw ) This is a common shorthand notation T(s) is the ‘transfer’ function, or the ‘gain’ function Describes how voltage is ‘transferred’ from in to out T1(s) FIG. 21.33 Passband filter. T2(s) Total Gain=T(s)=T1(s)xT2(s) PASS-BAND FILTERS FIG. 21.34 Pass-band characteristics. PASS-BAND FILTERS FIG. 21.35 Passband filter. FIG. 21.36 Pass-band characteristics for the filter in Fig. 21.35. PASS-BAND FILTERS FIG. 21.37 Network of Fig. 21.35 at f = 994.72 kHz. BAND-REJECT FILTERS Since the characteristics of a band-reject filter (also called stop-band or notch filter) are the inverse of the pattern obtained for the band-pass filter, a band-reject filter can be designed by simply applying Kirchhoff’s voltage law to each circuit. BAND-REJECT FILTERS FIG. 21.38 Demonstrating how an applied signal of fixed magnitude can be broken down into a pass-band and band-reject response curve. BAND-REJECT FILTERS FIG. 21.39 Band-reject filter using a series resonant circuit. BAND-REJECT FILTERS FIG. 21.40 Band-reject filter using a parallel resonant network. BAND-REJECT FILTERS FIG. 21.41 Bandreject filter. BAND-REJECT FILTERS FIG. 21.42 Band-reject characteristics. OPERATIONAL AMPLIFIERS Active filters AMPLIFIERS GIVE GAIN Simple amp-1 input and 1 output Gain, A=Vout/Vin EXAMPLE If the amplifier above gives an output voltage of 1000V with an input voltage of 50V, what is the gain? IDEAL OPERATIONAL-AMPLIFIER (OP -AMP) http://www.youtube.com/watch?v=TQB1VlLBgJE Inputs draw no current-infinite input impedace Vout=A(Vplus-Vminus) A-open loop gain. A is ideally infinity-How is this useful? Output can provide as much voltage/current as needed-zero output impedance NEGATIVE FEEDBACK Negative feedback (NF) tries to reduce the difference with NF, Vplus=Vminus ALWAYS summing point constraints virtual ground. INVERTING AMPLIFIER Input goes into Vminus input-INVERTING input Gain, Ainv=-R2/R1, gain is negative because inverting INVERTING AMPLIFIER Vplus=Vminus Inputs draw no current NON-INVERTING AMPLIFIER Input goes into Vplus input-NON-INVERTING input Gain, Ainv=1+R2/R1, gain is positive UNITY GAIN BUFFER Gain is 1 i.e. Vin=Vout Used to isolate one side from the other REAL OP-AMPS http://www.national.com/mpf/LM/LM324.html#Overview Single LM741 Quad LM324 Output voltage determined by rails (power supply) Open loop gain not infinity Inputs draw small amount of current-nA’s or less BANDPASS FILTER AMPLIFIER f1=0.3Hz, f2=10Hz High freq., cap is short, low freq., cap is open FREQUENCY<F1 all caps are open. What is the gain? F1<FREQUENCY<F2 C1 is short. C2 is open. What is the gain?-midband gain. FREQUENCY>F2 All caps are shorts What is the gain? FILTER OP-AMP What is T(s)? sC1R2 T (s) = (sC2 R2 +1)(sC1R1 +1) FILTER OP-AMP zero at s=0 poles at 1/R1C1 and 1/R2C2 What happens at the zero? At the poles? DECIBELS & BODE PLOTS The key to amplifiers and control systems. INTRODUCTION The unit decibel (dB), defined by a logarithmic expression, is used throughout the industry to define levels of audio, voltage gain, energy, field strength, and so on. Vo Po gain(dB) = 20 log10 (| |) = 20 log10 (| T(s) |) =10 log10 ( ) Vi Pi Disambiguate now: Vo/Vi-refers to |T(s)| i.e. ratio of amplitudes T(s) is the full “gain” equation before taking the absolute value INTRODUCTION LOGARITHMS Basic Relationships Let us first examine the relationship between the variables of the logarithmic function. The mathematical expression: INTRODUCTION LOGARITHMS Some Areas of Application The following are some of the most common applications of the logarithmic function: 1. The response of a system can be plotted for a range of values that may otherwise be impossible or unwieldy with a linear scale. 2. Levels of power, voltage, and the like can be compared without dealing with very large or very small numbers that often cloud the true impact of the difference in magnitudes. 3. A number of systems respond to outside stimuli in a nonlinear logarithmic manner. 4. The response of a cascaded or compound system can be rapidly determined using logarithms if the gain of each stage is known on a logarithmic basis. INTRODUCTION LOGARITHMS FIG. 21.1 Semilog graph paper. INTRODUCTION LOGARITHMS FIG. 21.2 Frequency log scale. INTRODUCTION LOGARITHMS FIG. 21.3 Finding a value on a log plot. FIG. 21.4 Example 21.1. PROPERTIES OF LOGARITHMS There are a few characteristics of logarithms that should be emphasized: The common or natural logarithm of the number 1 is 0 The log of any number less than 1 is a negative number The log of the product of two numbers is the sum of the logs of the numbers The log of the quotient of two numbers is the log of the numerator minus the log of the denominator The log of a number taken to a power is equal to the product of the power and the log of the number PROPERTIES OF LOGARITHMS CALCULATOR FUNCTIONS Using the TI-89 calculator, the common logarithm of a number is determined by first selecting the CATALOG key and then scrolling to find the common logarithm function. The time involved in scrolling through the options can be reduced by first selecting the key with the first letter of the desired function—in this case, L, as shown below, to find the common logarithm of the number 80. DECIBELS Power Gain Voltage Gain Human Auditory Response DECIBELS TABLE 21.1 DECIBELS TABLE 21.2 Typical sound levels and their decibel levels. DECIBELS FIG. 21.5 LRAD (Long Range Acoustic Device) 1000X. (Courtesy of the American Technology Corporation.) DECIBELS INSTRUMENTATION FIG. 21.6 Defining the relationship between a dB scale referenced to 1 mW, 600Ω and a 3 V rms voltage scale. LINEAR SYSTEMS RLC circuits, op-amps are linear circuit elements i.e. a differential equation can describe them. You can add solutions at a given ω i.e. if exp(jωt) and exp(-jωt) are solutions, exp(jωt)+exp(-jωt)=2cos(ωt) is a solution. t t t t LINEAR SYSTEMS Any voltage signal can be represented by a sum of sinusoidal voltage signalsFourier/Laplace theorems If s=jω, the input voltage is represented by: V0exp(jωt)= V0exp(st) Allows us to use algebra instead of differential eqns. RLC circuit, for example. t t t t BODE PLOTS There is a technique for sketching the frequency response of such factors as filters, amplifiers, and systems on a decibel scale that can save a great deal of time and effort and provide an excellent way to compare decibel levels at different frequencies. The curves obtained for the magnitude and/or phase angle versus frequency are called Bode plots (Fig. 21.44). Through the use of straight-line segments called idealized Bode plots, the frequency response of a system can be found efficiently and accurately. TRANSFER FUNCTION Transfer function T(s), or H(s) describes how the output is affected by the input. i.e. T(s)=Vo/Vi s=jω, so ZC=1/sC and ZL=sL The ‘s’ notation is convenient shorthand, but is also important in the context of Laplace Transforms, which you will see later in the class. Transfer because it describes how voltage is “transferred” from the input to output. LINEAR SYSTEMS T(s) has zeros when T(s)=0 T(s) has poles when T(s)=infinity POLES & ZEROS Write T(s) in the form A(s)/B(s), where A and B do not have any fractions in them. They should look like factored polynomials. All transfer functions have poles and zeros. Zeros are when T(s)=0 i.e. A(s)=0 Poles are when 1/T(s)=0 or T(s)=∞ i.e. B(s)=0 These contribute very distinct behaviors to the frequency response of a system. They tell us that these are the critical frequencies in the system. TRANSFER FUNCTION FOR LOW-PASS Again we go to our good friend, the low-pass filter. 0.16uF FIG. 21.16 Example 21.5. Now, we will redo this in the language of “transfer function” LOW PASS FILTER TRANSFER FUNCTION Vo ZC T (s) = = Vi ZC + R 1 / sC 1 = = 1 / sC + R 1+ sRC 0.16uF Zeros: NONE FIG. 21.16 3 Poles: s=-1/RC6.28x10 rad/s or 1kHz Example 21.5. (We can ignore the minus signs) Why? T(s)1 as w0, so gain is 1 (0dB) and phase is 0 T(s)-j/wC as winf., so gain0 and phase-90deg BODE PLOTS There is a technique for sketching the frequency response of such factors as filters, amplifiers, and systems on a decibel scale that can save a great deal of time and effort and provide an excellent way to compare decibel levels at different frequencies. The curves obtained for the magnitude and/or phase angle versus frequency are called Bode plots (Fig. 21.44). Through the use of straight-line segments called idealized Bode plots, the frequency response of a system can be found efficiently and accurately. WHAT DOES A BODE PLOT TELL YOU? Only works for sine/cosine If input is , cos(wt) output becomes |T(s)|cos(wt+phase(T(s))) Vi=Cos(wt) Circuit or system T(jω)=T(s) Vo=|T(s)|cos(wt+phase(T(s))) BODE PLOTS Once you have the transfer function, there is a clear recipe that lets you sketch the frequency response by hand very accurately. There is a surprisingly easy way to do this. List the poles and zeros from low to high. Go to w=0 and w=infinity to get start/end points. At each pole/zero, apply the 20dB/dec add/subtract At each pole/zero apply the 90deg phase shift These contributions are additive. Because of the simplicity, there are some errors, but only small ones near the critical cutoff frequencies. LOW-PASS RESPONSE-MAGNITUDE |T(S)| FIG. 21.52 Bode plot for the high-frequency region of a low-pass R-C filter. LOW PASS RESPONSE ANGLE(T(S)) MORE RULES Phase starts at 0deg UNLESS -sign, starts at +/-180 deg. Pole/zero at 0 (see next slide). Gain starts with slope of 0dB/dec UNLESS Pole/zero at 0 (see next slide) (Remember, the starting value of gain could be any value and must be determined from the transfer function) SPECIAL CASE POLES AND ZEROS AT W=0 Zero Pole [email protected]/s Gain Magnitude +20dB/dec -20dB/dec +20dB/dec [email protected]/s -20dB/dec Phase +45deg at zero, +90 deg thru zero. -45deg at pole, -90 deg thru pole. Phase starts at +90 deg (in addition to minus signs which give 180 deg) Phase starts at -90 deg (in addition to minus signs which give 180 deg) We will now work some examples that have been posted ‘REVERSE’ BODE PLOTS If I give you the Bode plot, and tell you what the input is, can you sketch the output?