Lecture 4: Resonance

1. Lecture 4: Resonance Prof. Niknejad

2. Lecture Outline • Some comments/questions about Bode plots • Second order circuits: • Series impedance and resonance • Voltage transfer function (bandpass filter) • Bode plots for second order circuits University of California, Berkeley

3. Good Questions… • Why does the Bode plot of a simple pole or zero always have a slope of 20 dB/dec regardless of the break frequency? • You’ve been sloppy with signs, what’s the deal? • Why does the arctangent plot look so funny? • Why do we factor the transfer function into terms involving jω? University of California, Berkeley

4. On a log scale, this term has a fixed slope of 20 dB/decade This term is a constant Bode Plot Question #1 • Note that the slope of a pole or zero is independent of the break point: • On a log-log scale, all straight lines have the same slope … the slope gets translated into an intercept shift! University of California, Berkeley

5. Which one is right? Bode Plot Question #2 • Why do you sometimes use a positive sign and other times a negative sign in the transfer function? • The plus sign is right! For “passive” circuits, the poles are all in the LHP (left-half plane). A simple RC circuit has: • Otherwise the circuit has a negative resistor! • We can synthesize negative resistance with active circuits… University of California, Berkeley

6. Bode Plot Question #3 • I know what an arctan looks like and it looks nothing like what you showed us! Linear Scale Log Scale University of California, Berkeley

7. Log Scales • Log scales move forward non-uniformly… University of California, Berkeley

8. LTI System H Why jω? • When we factor our transfer functions, why do we always like to put things in terms of jω, as opposed to say ω? • Recall that we are trying to find the response of a system to an exponential with imaginary argument: • Real sinusoidal steady-state requires the argument to be imaginary. We must therefore only consider the transfer functions for such values … • If this doesn’t make sense, hang in there! University of California, Berkeley

9. Second Order Circuits • The series resonant circuit is one of the most important elementary circuits: • The physics describes not only physical LCR circuits, but also approximates mechanical resonance (mass-spring, pendulum, molecular resonance, microwave cavities, transmission lines, buildings, bridges, …) University of California, Berkeley

10. Series LCR Impedance • With phasor analysis, this circuit is readily analyzed University of California, Berkeley

11. + VL – VL + VC – VL VL Vs VR VR + VR − + Vs − VR Vs VC Vs VC VC Resonance • Resonance occurs when the circuit impedance is purely real • Imaginary components of impedance cancel out • For a series resonant circuit, the current is maximum at resonance University of California, Berkeley

12. + VL – + VC – + VR − Series Resonance Voltage Gain • Note that at resonance, the voltage across the inductor and capacitor can be larger than the voltage in the resistor: University of California, Berkeley

13. + Vo − Second Order Transfer Function • So we have: • To find the poles/zeros, let’s put the H in canonical form: • One zero at DC frequency  can’t conduct DC due to capacitor University of California, Berkeley

14. Poles of 2nd Order Transfer Function • Denominator is a quadratic polynomial: University of California, Berkeley

15. Im Re Finding the poles… • Let’s factor the denominator: • Poles are complex conjugate frequencies • The Q parameter is called the “quality-factor” or Q-factor • This is an important parameter: University of California, Berkeley

16. Im Re Resonance without Loss • The transfer function can be parameterized in terms of loss. First, take the lossless case, R=0: • When the circuit is lossless, the poles are at real frequencies, so the transfer function blows up! • At this resonance frequency, the circuit has zero imaginary impedance and thus zero total impedance • Even if we set the source equal to zero, the circuit can have a steady-state response University of California, Berkeley

17. Magnitude Response • The response peakiness depends on Q University of California, Berkeley

18. How Peaky is it? • Let’s find the points when the transfer function squared has dropped in half: University of California, Berkeley

19. Half Power Frequencies (Bandwidth) • We have the following: Four solutions! Take positive frequencies: University of California, Berkeley

20. More “Notation” • Often a second-order transfer function is characterized by the “damping” factor as opposed to the “Quality” factor University of California, Berkeley

21. damping ratio Two equal poles Two real poles Second Order Circuit Bode Plot • Quadratic poles or zeros have the following form: • The roots can be parameterized in terms of the damping ratio: University of California, Berkeley

22. Asymptotic Slope is 40 dB/dec Asymptotic Phase Shift is 180° Bode Plot: Damped Case • The case of ζ>1 and ζ =1 is a simple generalization of simple poles (zeros). In the case that ζ >1, the poles (zeros) are at distinct frequencies. For ζ =1, the poles are at the same real frequency: University of California, Berkeley

23. Underdamped Case • For ζ<1, the poles are complex conjugates: • For ωτ << 1, this quadratic is negligible (0dB) • For ωτ >> 1, we can simplify: • In the transition region ωτ ~ 1, things are tricky! University of California, Berkeley

24. Underdamped Mag Plot ζ=0.01 ζ=0.1 ζ=0.2 ζ=0.4 ζ=0.6 ζ=0.8 ζ=1 University of California, Berkeley

25. Underdamped Phase • The phase for the quadratic factor is given by: • For ωτ < 1, the phase shift is less than 90° • For ωτ = 1, the phase shift is exactly 90° • For ωτ > 1, the argument is negative so the phase shift is above 90° and approaches 180° • Key point: argument shifts sign around resonance University of California, Berkeley

26. Phase Bode Plot ζ=0.01 0.1 0.2 0.4 0.6 0.8 ζ=1 University of California, Berkeley

27. Bode Plot Guidelines • In the transition region, note that at the breakpoint: • From this you can estimate the peakiness in the magnitude response • Example: for ζ=0.1, the Bode magnitude plot peaks by 20 log(5) ~14 dB • The phase is much more difficult. Note for ζ=0, the phase response is a step function • For ζ=1, the phase is two real poles at a fixed frequency • For 0<ζ<1, the plot should go somewhere in between! University of California, Berkeley

28. Energy Storage in “Tank” • At resonance, the energy stored in the inductor and capacitor are University of California, Berkeley

29. Energy Dissipation in Tank • Energy dissipated per cycle: • The ratio of the energy stored to the energy dissipated is thus: University of California, Berkeley

30. Physical Interpretation of Q-Factor • For the series resonant circuit we have related the Q factor to very fundamental properties of the tank: • The tank quality factor relates how much energy is stored in a tank to how much energy loss is occurring. • If Q >> 1, then the tank pretty much runs itself … even if you turn off the source, the tank will continue to oscillate for several cycles (on the order of Q cycles) • Mechanical resonators can be fabricated with extremely high Q University of California, Berkeley

31. C0 R0 C1 C2 Cx Rx Lx thin-Film Bulk Acoustic Resonator (FBAR)RF MEMS • Agilent Technologies(IEEE ISSCC 2001) • Q > 1000 • Resonates at 1.9 GHz • Can use it to build low power oscillator Pad Thin Piezoelectric Film University of California, Berkeley