Comer frequency

To help understand MATLAB results, a sketch of the low and high frequency asymptotes is provided in Fig. E8.9. A key step is to identify the comer frequencies. In this case, the comer frequency of the first order lead is at 1/5 or 0.2 rad/s, while the two first order lag terms have their comer frequencies at 1/10, and 1/2 rad/s. The final curve is a superimposition of the contributions from each term in the overall transfer function. 

Integral control adds additional phase lag (-90°) at low frequencies below the comer frequency l/x . 

Furthermore, we may want to choose Xj such that l/x is smaller than the comer frequency... 

By choosing xD < (i.e., comer frequencies l/xD > 1/Xj), the magnitude plot has a notch shape. How sharp it is will depend on the relative values of the comer frequencies. The low frequency asymptote below 1/Xj has a slope of-1. The high frequency asymptote above l/xD has a slope of +1. The phase angle plot starts at -90°, rises to 0° after the frequency l/xIs and finally reaches 90° at the high frequency limit. 

On the Bode plot, the comer frequencies are, in increasing order, l/xp, Zq, and p0. The frequency asymptotes meeting at co = l/xp and p0 are those of a first-order lag. The frequency asymptotes meeting at co = z0 are those of a first-order lead. The largest phase lag of the system is -90° at very high frequencies. The system is always stable as displayed by the root locus plot. 

Diodes D1, D2 limit the excursion of the amplifier outputs to being near the rails. Rfb, Cfb Set the comer frequency for roll-off of ripple rejection. 

The opamp gain is given by Av - (Fores/Freg) (Vout/Vbg) where Fores - output impedance comer frequency with 0=0 (typical value about 1MHz)... 

Freg - comer frequency in ripple rejection (typical value about 600 Hz)... 

Figure 7. Bilayer conductance induced by affinity-purified connexin-32. A, Sucrose-permeable liposomes formed with affinity-purified connexin-32 were fused with planar phospholipid bilayers as described. Highly filtered (5-Hz comer frequency) currents show unstable conductances, but large, rapid fluctuations that cluster around multiples of about 125 pS may be discerned (arrowheads). The bilayer voltage was 50 mV. B, Higher resolution recording of channels from affinity-purified connexin-32. Records show discrete gating conductance transitions, but with a high rate and amplitude of current fluctuations through the open channels. Unitary conductance is difficult to determine, but is near 200 pS. The bilayer voltage was 100 mV.

If an adjustable-gain high-pass filter has been used to eliminate the low-frequency contamination, the resulting spectrum would look like a band-pass filter spectrum. But, such a filter introduces a limit in the low range of the meter, and a nonlinearity at the lower end of the meter range. The comer frequencies or 3-dB points of such a filter will yield a frequency at the high end that can be related to the lowest possible measurable flow. 

To determine the comer frequency of the waveforms shown in Fig. 5-12 and relate it to flow, the property of a uniform spectrum is used. This property states that, for a uniform spectrum, the mean-square amplitude of the total signal energy is the product of the signal energy level and its bandwidth. In other words, the area under the curve is the product of its length and its height above the abscissa. The electronics circuit based on this concept was developed and successfully tested at the SRC-II pilot plant. 

During the SRC-II tests, the flow velocity in a vertical recycle line was varied by changing the pump speed from 300 to 800 rpm. The spectra of the Doppler signals are shown in Fig. 5-12 and the processed data are shown in Fig. 5-13. The voltage output of the flowmeter, which is related to the comer frequency of... 

The two constraints that make direct observation of weak absorption signals impracticable are the presence of pink noise, which contains a preponderance of low-frequency power compared with white noise, and the enhancement of this and other sources of noise by the rectification process through which MMW signals are detected. Both predicate the use of some kind of modulation at a frequency greater than the noise comer frequency and the use of a phase-coherent detector to convert the resulting modulated signal to a DC level suitable for display or for post-detection computer processing. 

To eliminate DC from an AC signal, a blocking capacitor is inserted. Together with a resistor they form a high-pass filter. Figure 8.10(a). The time constant is RC, and the so called 3 dB comer frequency fo is 1/2tcRC. At that frequency, the phase shift is 45°, and the amplitude has dropped to 63%. This is clear from the transfer function ... 

The phase shift is At fo cp = 45°, at lOfo cp = 5.7°, and at lOOfo cp = 0.57°. The phase shift in a filter is thus substantial even far away from die comer frequency in the passband, and the frequency must be much higher than the corner frequency to ensure negligible phase shift. 

The Doppler shift is used in ihe deep-level gold mining environment to discriminate between the fault plane and auxiliary plane of the nodal plane solution (Brawn 1989 Goldbach et al. 2006). In Fig. 5.23, graphs of l/ft are plotted versus the cosine of the angle 0 between Ihe slip vectors of the two fault plane solutions and each recording site, where ft is the comer frequency of the S-wave measured from the velocity spectra. It can be seen that plane 1 (shown in red) is the real fault plane, and that the mpture direction is in the opposite direction as the slip vector. 

Fig. 5.23. Example showing the Doppler shift in comer frequencies (plane 1, red) for an event recorded in a deep level gold mine allowing the ambiguity of the fault plane solution to be resolved.

Authors Monitored area Typical distances Seismic network Rate of located events Comer frequency Magni- tude... 

The ECG amplifier can readily be designed using the instrumentation amplifier as the principal building block. Active filters with a lower-comer frequency of 0.05 Hz and an upper-corner frequency of 100 Hz are also typically added [8]. 

Notii that tbrare is another comer frequency at 0.5 rad/sec and the sl< of the magnitude curve in tte loe-frequency range is -40 < /decsade, G(ja ) can be tentatively determined followss... 

The low-frequency asymptote of the amplitude ratio curve intersects the high frequency asymptote at 0)= O.l rad/s, at this point o)=1/t, where ris the process time constant. This frequency is often called the comer frequency. 

Figure 32.4 shows the Bode diagram of a PI eontroller with integral action = 10. Rather than plotting the amplitude ratio, the ratio between amphtude ratio and controller gain is plotted on the vertieal axis. As can be seen from Eqn. (32.25), plotting log(AR) versus log(phase shift ranges from -90° to 0°. 

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