The output of our system is therefore defined by the expression shown in Eq. (8.48), b-t -m '

where b = Rv/ and m = /L. Substituting b = 2 2Em into Eq. (8.48), also yields p2 = Ae-2sE/mt + Bte-VsE>mt (8.49)

Comparing the definition for the natural frequency from Eq. (8.32), we find that we now have an expression for critically damped catheters that relates the output of the measuring system to the natural frequency of the catheter.

Critically damped systems are a very important case. This kind of system is on the borderline, but does not oscillate as it returns to equilibrium after perturbation. This disturbance is damped out as quickly as possible, as in the case of a door closer designed to close a screen door as quickly as possible with no back and forth motion.

The critical time constant for the system is 1/Acriticaj as shown in Eq. (8.51).

2m 2/L 2L cur2 Critical time constant = —— = —— = —— =- (8.51)

It is useful to point out that the Eqs. (8.52) and (8.53) are general expressions for the catheter measuring system and not a special case for the critically damped system.

8.3.7 Pop test—measurement of transient step response

One way to determine some characteristics of a second order system is to create a step input to the system and measure the response. One straightforward and simple method of measuring the system response is to input a step change into the system by popping a balloon or surgical glove and measuring the response. Imagine a catheter with one end connected to a pressure transducer and the other end inserted into a container covered with a balloon or surgical glove. If the system is pressurized and the balloon or glove is popped, the resulting pressure output from the system looks like that shown in Fig. 8.8. The original steady state pressure was Y1 and the pressure output of the transducer oscillates back and forth around zero, depending on the characteristics of the pressure measuring system.

The period of the oscillation is labeled as Td in Fig. 8.8. The damped natural frequency of the system, measured in rad/s, is given by the equation vD = 2p/TD.

The logarithmic decrement is given by 8 and is defined by the equation 8 = ln[P2(0)/P2(1)].

The damping ratio of the pressure measuring system can now be calculated from the logarithmic decrement as shown in Eq. (8.54).

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