1. Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 4 Lecture 4: Resonance Prof. Niknejad

2. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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

3. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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?

4. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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! On a log scale, this term has a fixed slope of 20 dB/decade This term is a constant

5. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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… Which one is right?

6. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Bode Plot Question #3 I know what an arctan looks like and it looks nothing like what you showed us! Linear ScaleLog Scale

7. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Log Scales Log scales move forward non-uniformly…

8. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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! Why jω? LTI System H

9. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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, …)

10. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Series LCR Impedance With phasor analysis, this circuit is readily analyzed

11. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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 +VR−+VR− + V L – + V C – +Vs−+Vs− VLVL VCVC VRVR VsVs VLVL VCVC VRVR VsVs VCVC VLVL VRVR VsVs

12. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Series Resonance Voltage Gain Note that at resonance, the voltage across the inductor and capacitor can be larger than the voltage in the resistor: +VR−+VR− + V L –+ V C –

13. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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 +Vo−+Vo−

14. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Poles of 2 nd Order Transfer Function Denominator is a quadratic polynomial:

15. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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: Re Im

16. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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 Re Im

17. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Magnitude Response The response peakiness depends on Q

18. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley How Peaky is it? Let’s find the points when the transfer function squared has dropped in half:

19. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Half Power Frequencies (Bandwidth) We have the following: Four solutions! Take positive frequencies:

20. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley More “Notation” Often a second-order transfer function is characterized by the “damping” factor as opposed to the “Quality” factor

21. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Second Order Circuit Bode Plot Quadratic poles or zeros have the following form: The roots can be parameterized in terms of the damping ratio: damping ratio Two equal poles Two real poles

22. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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: Asymptotic Slope is 40 dB/dec Asymptotic Phase Shift is 180°

23. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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!

24. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Underdamped Mag Plot ζ=1 ζ=0.01 ζ=0.1 ζ=0.2 ζ=0.4 ζ=0.6 ζ=0.8

25. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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

26. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Phase Bode Plot ζ=0.01 0.1 0.2 0.4 0.6 0.8 ζ=1

27. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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!

28. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Energy Storage in “Tank” At resonance, the energy stored in the inductor and capacitor are

29. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley Energy Dissipation in Tank Energy dissipated per cycle: The ratio of the energy stored to the energy dissipated is thus:

30. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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

31. EECS 105 Fall 2003, Lecture 4Prof. A. Niknejad Department of EECS University of California, Berkeley 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 C0C0 CxCx RxRx LxLx C1C1 C2C2 R0R0 Pad Thin Piezoelectric Film