Saturday, April 9, 2011

rise time, settling time, bandwidth, slew rate, quality factor, octave, natural frequency, damping ratio, butterworth, bessel, Chebyshev, PID, Ideal op-amp

rise time refers to the time required for a signal to change from a specified low value to a specified high value. Typically, these values are 10% and 90% of the step height. =2.2/(2*pi*fc)

settling time error band = 0.04*ks


The output slew-rate of an amplifier or other electronic circuit is defined as the maximum rate of change of the output voltage for all possible input signals.
\mathrm{SR} = \max\left(\left|\frac{dv_\mathrm{out}(t)}{dt}\right|\right)
SR = SRI/sqrt((SRI/SRSS)^2+Avcl^2)     Avcl:close loop gain  SRI:input slew  SRSS:output slew
sine wave srs=2*i*1e-6*Vp        Vp:peak voltage

time domain
• The time constant occurs at 63% of the steady state value for a first order system
• The time constant occurs at 26.24% of the steady state value for a second order
system

frequency domain
fc -3db*max, time constant=1/(2*pi*fc)
bandwidth


quality factor
The physical meaning of the quality factor $Q$ of a circuit is the ratio between the energy stored in the circuit (in $C$ and $L$) and the energy dissipated (by $R$):



\begin{displaymath} Q=2\pi\frac{\mbox{maximum energy stored}}{\mbox{energy dissipated per cycle}} \end{displaymath}

factors Q, damping ratio ζ, and attenuation α
 f0 is the resonant frequency, and Δf, the bandwidth, is the width of the range of frequencies for which the energy is at least half its peak value





  • A system with low quality factor (Q < ½) is said to be overdamped. Such a system doesn't oscillate at all, but when displaced from its equilibrium steady-state output it returns to it by exponential decay, approaching the steady state value asymptotically. It has an impulse response that is the sum of two decaying exponential functions with different rates of decay. As the quality factor decreases the slower decay mode becomes stronger relative to the faster mode and dominates the system's response resulting in a slower system. A second-order low-pass filter with a very low quality factor has a nearly first-order step response; the system's output responds to a step input by slowly rising toward an asymptote.









    • A system with high quality factor (Q > ½) is said to be underdamped. Underdamped systems combine oscillation at a specific frequency with a decay of the amplitude of the signal. Underdamped systems with a low quality factor (a little above Q = ½) may oscillate only once or a few times before dying out. As the quality factor increases, the relative amount of damping decreases. A high-quality bell rings with a single pure tone for a very long time after being struck. A purely oscillatory system, such as a bell that rings forever, has an infinite quality factor. More generally, the output of a second-order low-pass filter with a very high quality factor responds to a step input by quickly rising above, oscillating around, and eventually converging to a steady-state value.
    • A system with an intermediate quality factor (Q = ½) is said to be critically damped. Like an overdamped system, the output does not oscillate, and does not overshoot its steady-state output (i.e., it approaches a steady-state asymptote). Like an underdamped response, the output of such a system responds quickly to a unit step input. Critical damping results in the fastest response (approach to the final value) possible without overshoot. Real system specifications usually allow some overshoot for a faster initial response or require a slower initial response to provide a safety margin against overshoot.
    Sinusoidally driven resonators having higher Q factors resonate with greater amplitudes (at the resonant frequency) but have a smaller range of frequencies around that frequency for which they resonate; the range of frequencies for which the oscillator resonates is called the bandwidth. Thus, a high Q tuned circuit in a radio receiver would be more difficult to tune, but would have more selectivity; it would do a better job of filtering out signals from other stations that lie nearby on the spectrum. High Q oscillators oscillate with a smaller range of frequencies and are more stable. (See oscillator phase noise.)

    The quality factor of oscillators vary substantially from system to system. Systems for which damping is important (such as dampers keeping a door from slamming shut) have Q = ½. Clocks, lasers, and other resonating systems that need either strong resonance or high frequency stability need high quality factors. Tuning forks have quality factors around Q = 1000. The quality factor of atomic clocks and some high-Q lasers can reach as high as 1011[10] and higher.[11]

     Beta = Rin / (Rin + Rf)

    1. The distance between the frequencies 20 Hz and 40 Hz is 1 octave.
    2. A magnitude of 400 (52 dB) at 4 kHz decreases as frequency increases at −2 dB/octave. What is the magnitude at 13 kHz?
    \text{number of octaves} = \log_2\left(\frac{13}{4}\right) = 1.7
    \text{Mag}_{13\text{ kHz}} = 52\text{ dB} + (1.7\text{ octaves} \times -2\text{ dB/octave}) = 48.6\text{ dB} = 269.\,
    natural frequency W0, Wn
    Resonances occur when a system is able to store and easily transfer energy between two or more different storage modes (such as kinetic energy and potential energy in the case of a pendulum). However, there are some losses from cycle to cycle, called damping. When damping is small, the resonant frequency is approximately equal to a natural frequency of the system, which is a frequency of unforced vibrations. Some systems have multiple, distinct, resonant frequencies.

    Increase of amplitude as damping decreases and frequency approaches resonant frequency of a damped simple harmonic oscillator.[1][2]


    The Butterworth filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband so that it is also termed a maximally flat magnitude filter.

    nFactors of Polynomial Bn(s)
    1(s + 1)
    2s2 + 1.4142s + 1
    3(s + 1)(s2 + s + 1)
    4(s2 + 0.7654s + 1)(s2 + 1.8478s + 1)
    5(s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
    6(s2 + 0.5176s + 1)(s2 + 1.4142s + 1)(s2 + 1.9319s + 1)
    7(s + 1)(s2 + 0.4450s + 1)(s2 + 1.2470s + 1)(s2 + 1.8019s + 1)
    8(s2 + 0.3902s + 1)(s2 + 1.1111s + 1)(s2 + 1.6629s + 1)(s2 + 1.9616s + 1)
    q
     Analog Bessel filters are characterized by almost constant group delay across the entire passband, thus preserving the wave shape of filtered signals in the passband.

                                                         n=1; s+1
                                                         n=2; s^2+3s+3

    q
    Chebyshev
                                                       n=1; s+4.108
                                                       n=2; s^2+1.748s+2.14

    q




    q
    The reset time is the time taken for the integrator term output to equal the proportional term
    output in response to a step change in input applied to a PI controller.

    The rate time is the time taken for the proportional term output to equal the derivative term output
    in response to a ramp change input applied to a PD controller.



    q

    q
    An ideal op-amp is usually considered to have the following properties, and they are considered to hold for all input voltages:
    • Infinite open-loop gain (when doing theoretical analysis, a limit may be taken as open loop gain AOL goes to infinity).
    • Infinite voltage range available at the output (vout) (in practice the voltages available from the output are limited by the supply voltages V_{\text{S}\!+} and V_{\text{S}\!-}). The power supply sources are called rails.
    • Infinite bandwidth (i.e., the frequency magnitude response is considered to be flat everywhere with zero phase shift).
    • Infinite input impedance (so, in the diagram, R_{\text{in}} = \infty, and zero current flows from v_{\!+} to v_{\!-}).
    • Zero input current (i.e., there is assumed to be no leakage or bias current into the device).
    • Zero input offset voltage (i.e., when the input terminals are shorted so that v_{\!+}=v_{\!-}, the output is a virtual ground or vout = 0).
    • Infinite slew rate (i.e., the rate of change of the output voltage is unbounded) and power bandwidth (full output voltage and current available at all frequencies).
    • Zero output impedance (i.e., Rout = 0, so that output voltage does not vary with output current).
    • Zero noise.
    • Infinite Common-mode rejection ratio (CMRR).
    • Infinite Power supply rejection ratio for both power supply rails.
    p
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