Frequency Response (I&N Chap 12)

Transcription

1 Frequency Response (I&N Chap 12) Introduction & TFs Decibel Scale & Bode Plots Resonance Scaling Filter Networks Applications/Design Slide 3.1

2 Network Scaling Considering a Bode plot, there are two ways to scale. 1. For magnitude scaling, the magnitude plot is shifted up or down while the phase plot is unchanged. 2. For frequency scaling, both magnitude and phase plots are shifted left or right. Let s see how to change component values for resonant circuits to achieve these types of scaling. Slide 3.2

3 Magnitude/Impedance Scaling To scale the equivalent impedance by a factor K, simply scale each component impedance by K. Resistors: ZR = R R ' = KM R 1 C Capacitors: ZC = C ' = sc K Inductors: Z = sl L ' = K L L For a series RLC circuit, what happens to Q and ω with magnitude scaling? 0 M M M M Slide 3.3

4 Frequency Scaling To shift the plots by a frequency factor, we wish the impedance at the scaled frequency ω = K ω to be the same as at the original frequency ω. Capacitors: Inductors: Resistors: Z C Z = R R ' = R R K 1 1 C = = C ' = jωc jω C K Z = jωl = jω L L ' = L For a series RLC circuit, what happens to Q and ω with frequency scaling? 0 F L K F F F Slide 3.4

5 Magnitude and Frequency Scaling To simultaneously scale impedance in both magnitude and frequency: KmL C R ' = KmR, L' =, C' =, ω' = K K K K f m f E.g., a 3 rd order Butterworth filter normalized to ω c =1rad/s is shown. Scale the circuit to a cutoff frequency of 10kHz and use 15 nf capacitors. f ω Slide 3.5

6 Filter Networks Filter E.g., Low-Pass Filter Filters are used to allow a particular range of frequencies to pass through while rejecting others. The type of filter can readily be determined by the magnitude plot of the TF relating the filter input to the filter output. Filters may be classified as passive or active, depending on whether the filter has any internal sources of energy. Slide 3.6

7 Common Filter Networks We ll examine 4 common filters with self-explanatory names. Low-pass filter High-pass filter Band-pass filter Band-rejection filter We first focus on passive filters. We need op-amps for active ones. Slide 3.7

8 Ideal Filter Characteristics Q: Consider the ideal low-pass filter characteristic. Why do you suppose this isn t realizable in practice? (Hint: What TF would yield this characteristic?) Slide 3.8

9 Passive Low-Pass Filter (LPF) A simple low-pass filter is the RC series circuit where capacitor voltage is taken as the output. Slide 3.9

10 Passive High-Pass Filter (HPF) A simple high-pass filter is the RC series circuit where resistor voltage is taken as the output. Slide 3.10

11 Passive Band-Pass Filter (BPF) This RLC circuit gives a band-pass filter if v R is taken as the output. Q: Give an intuitive reason for why both low and high frequencies are rejected. Transfer Function : G G v V = V 0 1 v s= jω R = R + sl + 1 sc R = R + j( ωl = 1 ωc ) s 2 scr LC + scr + 1 Cut-off Frequencies: ω lo, hi = ( ) m( R / L) + R / L + 4ω 2 Center Freq: 1 ω 0 = Bandwidth: BW ωhi ωlo LC = = R L Slide 3.11

12 Band-Rejection Filter This RLC circuit gives a band-rejection (aka bandstop or notch ) filter if v L +v C is taken as the output. Transfer Function: G 2 V0 s LC + 1 v = = 2 V1 s LC src Cut-off Frequencies: ω lo, hi = ( ) m( R / L) + R / L + 4ω 2 Center Freq: ω = 0 1 LC Slide 3.12

13 Example (efts) By now, you should realize that the RLC circuit can be used for any of our four filter types, depending on where the output is taken. Slide 3.13

14 Examples Design an RL lowpass filter that uses a 40 mh coil and has a cutoff frequency of 5 khz. Design an RC highpass filter that uses a 20 µf capacitor and has a cutoff frequency of 3 khz. Slide 3.14

15 Example: Connecting Passive Filters What is the overall result of attaching the lowpass filter output to the highpass filter input of the previous slide? Slide 3.15

16 Example Determine the center frequency and BW of these bandpass filters. Slide 3.16

17 Passive Filter Limitations Passive filters have some drawbacks: The output voltage driving a load cannot be larger than the input (i.e., gains are no greater than unity). Loading effects mean that they must be reanalyzed when interconnected. Designs sometimes require inductors which are expensive and inherently lossy. Component values may need to be large to achieve desired specs. Active filters, using op-amps or transistors, can overcome these limitations. Slide 3.17

18 Active Filter Networks Using op-amps in filters, we can: achieve gains greater than unity buffer the input signal to avoid loading effects (especially helpful in cascading ) design all filters with only resistors and capacitors (no need for inductors) Slide 3.18

19 Inverting Op-Amp Configuration Many filters use the inverting op-amp configuration. e.g., 1 st Order Lowpass filter e.g., 1 st Order Highpass filter Slide 3.19

20 Active Band-Pass Filter One way to design a BPF is to cascade HPF and LPF as shown below. NB: cf slide 3.15, cascading passive filters required system reanalysis of the entire circuit. Using active filters, the design can be more modular so the poles of each stage carry into the final T.F. Slide 3.20

21 Active Band-Rejection Filter One way to design a notch filter is to sum HPF and LPF outputs as shown. Slide 3.21

22 Example Find the voltage gain TF and identify the filter type for the circuit shown. C 2 R 2 + R 1 C 1 + v i (t) v o (t) Slide 3.22

23 Example (Cascaded Filters) kΩ Consider the single-stage filter above for radio tuning. You want to listen to the 100 MHz station with little interference by the 98 MHz station. By cascading filters (connect end-to-end so the output of one is the input of another), greater steepness in response can be achieved resulting in increased selectivity. Slide 3.23

24 Cascading Low-Q Filters Bode Diagram Magnitude (db) Frequency (rad/sec) # of Stages BW (MHz) Q Single % Double % Triple % Quadruple % Slide 3.24

25 Take Home Message Passive filters can be simple to design and easy to implement. We could design four basic filter types using resistors, inductors and capacitors. Active filters utilize op-amps (or OTA) to permit gains greater than unity, isolation for better cascading effects and designs which avoid the necessity of inductors. Depending on the application, different filter types have been designed to optimise for passband flatness (Butterworth filters), for immediate passband-to-stopband transition (Tschebyscheff filters), and for phase response linearity (Bessel filters). Active filters are typically used for frequencies below 100 khz. Slide 3.25