Modeling and Analysis of Systems Lecture #9 - Frequency Response Guillaume Drion Academic year 2015-2016 1
Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 2
Transmission of complex exponentials through LTI systems Continuous case: LTI system where is the transfer function of the LTI system. 3
Transmission of complex exponentials through LTI systems Discrete case: LTI system where is the transfer function of the LTI system. 4
How do we characterize the response of a LTI systems to Continuous case: an oscillatory signal at a specific frequency? Using the polar representation, we have 5
How do we characterize the response of a LTI systems to an oscillatory signal at a specific frequency? Using the polar representation, we have Change in amplitude Change in phase When an oscillatory signal goes through a LTI systems, his amplitude (amplification/attenuation) and phase (advance, delay) are affected. Not his frequency! 6
Frequency response of LTI systems The frequency response of a LTI system can be fully characterize by, and in particular: : GAIN (change in amplitude) : PHASE (change in phase) A change in phase in the frequency domain corresponds to a time delay in the time domain: which gives The slope of the phase curve corresponds to a delay in the time domain. 7
Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 8
Frequency response of LTI systems The frequency response of a LTI system can be fully characterize by, and in particular: : GAIN (change in amplitude) : PHASE (change in phase) A plot of and for all frequencies gives all the informations about the frequency response of a LTI system: the BODE plots. In practice, we use a logarithmic scale for such that becomes 9
The Bode plots The Bode plots graphically represent the frequency response of a LTI system. They are composed of two plots: The amplitude plot (in db):. The phase plot:. For discrete time systems, we use a linear scale for the frequencies, ranging from to. 10
The Bode plots Examples of Bode plots of continuous (left) and discrete (right) LTI systems. 11
Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 12
Bandwidth The bandwidth of a system is the range of frequencies that transmits faithfully through the system. We can define the bandwidth of a system the same way we defined its timeconstant: 13
Bandwidth We define the bandwidth of a system the same way we defined its time-constant: There is a tradeoff between the bandwidth of a system and its time-constant. Indeed, let s consider, which gives If we now consider and, we have 14
Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 15
Time and frequency responses of 1st order systems We consider the general 1st order system of the form The transfer function of the system is given by The frequency response ( ), impulse response ( ) and step response ( ) writes H(j!) = 1 j! +1, h(t) = 1 e t/ I(t), s(t) =(1 e t/ )I(t) 16
Time and frequency responses of 1st order systems 17
Bode plots of 1st order systems: amplitude Amplitude plot: If : If : Low frequencies: constant frequency response ( ) High frequencies: frequency response linear decays by -20dB/dec. Cutoff frequency:. 18
Bode plots of 1st order systems: amplitude Amplitude plot: first order systems are low-pass filters! (but the slope at HF might be too low to achieve good filtering properties...). 19
Bode plots of 1st order systems: phase Phase plot: Low frequencies: no phase shift. Mid frequencies: phase response decays linearly (slope = time-delay = ). High frequencies: phase-delay of. 20
Bode plots of 1st order systems: amplitude and phase 21
Time and frequency responses of 2nd order systems We consider the general 2nd order system of the form = natural frequency = damping factor The transfer function of the system is given by The frequency response writes 22
Time and frequency responses of 2nd order systems The transfer function of a 2nd order system is given by The transfer function has two poles: Case 1 ( ) : two real poles cascade of two first order systems. Case 2 ( ) : two complex conjugates poles. New behaviors (oscillations, overshoot, etc.) 23
Time and frequency responses of 2nd order systems 24
Bode plots of 2nd order systems Amplitude plot: Phase plot: 25
Bode plots of 2nd order systems Amplitude plot: second order systems are low-pass filters! (higher slope, possible resonant frequency with overshoot in the frequency response). 26
Bode plots of 2nd order systems: resonant frequency If, there is an overshoot in the frequency response at a resonant frequency. The amplitude of the peak is given by For, there is no peak in the frequency response. 27
Time and frequency responses of 2nd order systems 28
Outline Frequency response of LTI systems Bode plots Bandwidth and time-constant 1st order and 2nd order systems (continuous) Rational transfer functions 29
Frequency response of LTI systems Bode plots of first and second order systems are building blocks for the construction of Bode plots of any LTI systems. Indeed, the transfer function of LTI systems is rational, and the denominator terms can all be expressed as or In other terms, the Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function! 30
Frequency response of LTI systems: poles and zeros The Bode plots of LTI systems can be sketched from the poles and zeros of the transfer function! Each real pole induce a first order system response where. Each pair of complex conjugate poles system response where induce a second order Zeros induce the opposite behavior. 31
Frequency response of LTI systems: poles and zeros 32
Amplitude: Frequency response of LTI systems: Bode plots any real pole induces a decrease in the slope of -20dB/dec. any real zero induces an increase in the slope of 20dB/dec. any pair of complex conjugate poles induces a decrease in the slope of -40dB/dec. Phase: any real pole induces a decrease in the phase of. any real zero induces an increase in the phase of. any pair of complex conjugate poles induces a decrease in the phase of. 33
Frequency response of LTI systems: Bode plots Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz. 34
Frequency response of LTI systems: Bode plots Example: DC gain of -20dB, zero in 10 K Hz and pole in 100 K Hz. 35
Frequency response of LTI systems: poles and zeros 36