Phase plane with control

Show that in the absence of control and feed disturbances (u = v = 0), the system has a singular, stable, steady-state solution of C = 0.1654 and T = 550. This can best be done by carrying out runs with different initial conditions (CO and TEMPO) and plotting the results as a phase-plane, TEMP versus C. 

Cables via the powder-in-tube (PIT) method The approach is especially suited to processing Bi-2223 into leads and cables for power applications. A silver or silver alloy tube, filled with the partially reacted precursor powders formulated to yield Bi-2223, is drawn down to a wire 1-2mm diameter. The wire is rolled into a tape if that is the required form, usually with a width-to-thickness ratio of approximately 10 1. The composite is then heated to 800-900 °C when the powder partially melts. The recrystallization process is controlled and the pure Bi-2223 phase develops with large grains oriented so that the Cu-O planes lie parallel to the silver surface to optimize Jc. 

In the model, the appearance of birhythmidty is closely linked to the existence of multiple oscillatory domains as a function of the substrate injection rate, which is taken as the control parameter. In these conditions, the increase or decrease of this parameter from a value corresponding to a stable steady state gives rise to either one of two stable rhythms which markedly differ in period and amplitude. In a very different context, a similar property characterizes neurons of the thalamus. Thalamic neurons are indeed capable of oscillating with a frequency of 6 Hz or 10 Hz when the membrane, initially in a stable resting state, is slightly depolarized or hyperpolarized (Jahnsen LUnas, 1984a,b Llinas, 1988). The phase plane analysis of the biochemical model provides a clue for this behaviour and for the existence of multiple excitability thresholds, which are also observed in these neurons. 

Phase plane analysis thus readily accounts for the main experimental observation on the control of glycolytic oscillations by the substrate injection rate. Below the lower critical value of the substrate injection rate v, a stable steady state is estabUshed, corresponding to a low level of reaction product and to an enzyme predominantly in the inactive T state. Above the higher critical value of v, the stable steady state is associated with a higher level of product and with an enzyme predominantly in the active state R. Sustained oscillations, in the course of which the enzyme switches back and forth between the R and T states, occur in the range delimited by the two critical values of the substrate input. 

Fig. 1 shows the block diagram of the vibrometer, in which the most sensible to small phase variations interferometric scheme is employed. It consists of the microwave and the display units. The display unit consists of the power supply 1, controller 2 of the phase modulator 3, microprocessor unit 9 and low-frequency amplifier 10. The microwave unit contains the electromechanical phase modulator 3, a solid-state microwave oscillator 4, an attenuator 5, a bidirectional coupler 6, a horn antenna 7 and a microwave detector 11. The horn antenna is used for transmitting the microwave and receiving the reflected signal, which is mixed with the reference signal in the bidirectional coupler. In the reference channel the electromechanical phase modulator is used to provide automatic calibration of the instrument. To adjust the antenna beam to the object under test, the microwave unit is placed on the platform which can be shifted in vertical and horizontal planes. 

In polarization modulated ENDOR spectroscopy (PM-ENDOR)45, discussed in Sect. 4.7, the linearly polarized rf field B2 rotates in the laboratory xy-plane at a frequency fr fm, where fm denotes the modulation frequency of the rf carrier. In a PM-ENDOR experiment the same type of cavity, with two rf fields perpendicular to each other, and the same rf level and phase control units used in CP-ENDOR can be utilized. To obtain a rotating, linearly polarized rf field with a constant magnitude B2 and a constant angular velocity Q = 2 fr (fr typically 30-100 Hz), double sideband modulation with a suppressed carrier is applied to both rf signals. With this kind of modulation the phase of the carrier in each channel is switched by 180° for sinQt = 0. In addition, the phases of the two low-frequency envelopes have to be shifted by 90° with respect to each other. The coding of the two rf signals is shown in Fig. 8. 

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