Frequency Response part 2 (I&N Chap 12) Introduction & TFs Decibel Scale & Bode Plots Resonance Scaling Filter Networks Applications/Design Frequency response; based on slides by J. Yan Slide 3.1
Example Draw the Bode plot for H ( s ) = 50s ( s + 4)( s + ) 2 Magnitude (db) 20 0-20 -40-1 0 1 2 3 90 Phase (deg) 45 0-45 -90-135 -180-1 J.Yan, EECE 253: Variable 0 1 2 3 Slide 3.2
Draw the Bode plot for H ( s) = 2 s 20 Example s + 1 + 60s + 0 Magnitude (db) 0-20 -40-1 0 1 2 3 Phase (deg) 90 45 0-45 -90-135 -180-1 0 1 2 3 Slide 3.3
Example Draw the Bode plot for H ( s) = 4 ( s + 2) ( s + )( s + 0) Slide 3.4
Example: Magnitude Plot TF What is H(s) for the following magnitude sketch? Slide 3.5
Example: Magnitude Plot TF What is H(s) for the following magnitude sketch? Slide 3.6
Matlab Bode Plots Matlab can conveniently generate bode plots. 5( s + 2) Example : Recall the TF from slide 3.22 : H ( s) = s( s + ) Sample session: >> num=[5 ]; >> den=conv([1 0],[1 ]) den = 1 0 >> H=tf(num,den) Transfer function: 5 s + ---------- s^2 + s >> bode(h) >> grid on >> Magnitude (db) Phase (deg) 40 20 0-20 -40-60 -30-60 Bode Diagram (purple text typed by user) -90-1 0 1 2 3 Frequency (rad/sec) Slide 3.7
Bode Plots Q&A Q :On the Bode magnitude plot, for large values of ω, what is the slope if the transfer function is H ( s) = 1 (5 + s) 2? Q : What can you say about the poles and zeros of a TF if the magnitude plot starts with a slope of 40dB/decade and ends with - 40dB/decade? Q : If m + n = 3 (i.e., # of zeros+ # of poles), what can you say about the number of slope changes in the magnitude plot approximation? Slide 3.8
Resonance in RLC Circuits Resonant frequency: the frequency, w 0, for which capacitive and inductive reactances are equal in magnitude, resulting in a purely resistive impedance. When the circuit is operated at w 0, it is said to be in resonance. The series and parallel RLC circuits are duals. Let s analyse resonance for the series case. Slide 3.9
Series RLC Circuits For the series RLC circuit let s look for the resonant frequency and see what we can say about the characteristics. Slide 3.
Series RLC and Quality Factor 1 Resonance occurs at ω0 = ωn = for which Z( jωn) = R is purely resistive. LC WS For resonant circuits, the quality factor can be defined by: Q = 2π W where W = maximum stored energy and W = energy dissipated per cycle. S Let's express Q in terms of the component values. D D Slide 3.11
Series RLC Bandwidth 1 Define"bandwidth" using - power frequencies (magnitude 2 1 across resistor drops to of the maximum value). 2 Slide 3.12
More on the Q Factor Q is often used as a measure of frequency selectivity, especially for filters. Slide 3.13
Series vs Parallel RLC Circuits Series RLC + v r (t) R v s (t) + L C v c (t) + + v L (t) Parallel RLC i s (t) R L C Slide 3.14
Example In a series RLC circuit with R=4Ω and L=25 mh, calculate the value of C to provide Q=50. Find the resulting BW, ω Lo and ω Hi. Finally, taking V m =0 V, determine the average power dissipated at ω o, ω Lo, and ω Hi. Slide 3.15
Compute the circuit resonant frequency: Example Slide 3.16
Q&A Q: Does resonance always occur when the impedance magnitude is a minimum? Q: What are applications where high-q electrical resonance is desired? Q: What are applications where high-q electrical resonance is avoided? Slide 3.17
Caveat (for other fields) Unfortunately, there are at least 3 commonly accepted definitions of resonance that can lead to subtly different results: 1) In keeping with the text, we say circuit resonance occurs when the driving point impedance is purely real. 2) Resonance is the phenomenon of a peak in the TF magnitude plot. This is the def n used in many other fields (e.g., see how Wikipedia defines this phenomenon). Some analogies break down unless the specific type of TF is specified. 3) Circuit resonance occurs when the inductive and capacitive components of any 2 nd order term in the driving point impedance cancel out. This is simply the natural frequency of each 2 nd order term. Slide 3.18