Vectors impedance

The easiest way to find out the characteristics of a passive L-C-R network in the rf range is to use a vector impedance meter which can tell you the magnitude and the phase of the impedance simultaneously. Just adjust the components to null the imaginary component and to set the real component to the desired value. The only disadvantage of this device is that it is expensive. It turns out that magic-tees can be used to accomplish some of the same goals and they are orders of magnitude cheaper. 

To go first class in building probes, you should have a vector impedance meter. As pointed out in Section V.C.ll., the magic-tee may be substituted at two orders of magnitude reduction in cost but with signficantly more fiddling. 

Vector Impedance Meters (Vector Volt Meters, Network Analyzers)... 

Membrane impedance measurements can be achieved by using various methods. One possible approach consists to apply a sinusoidal current or voltage signal of a given frequency and to determine the corresponding voltage or current response by means of a lock-in amplifier or a vector impedance meter. This procedure must be repeated for every frequency of interest, which makes the measurements quite time consuming. Two... 

Since the potential and current are sinusoidal, the impedance has a magnitude and a phase, which can be represented as a vector. A sinusoidal potential or current can be pictured as a rotating vec tor. For standard AC current, the rotation is at a constant angular velocity of 60 Hz. 

In maldug electrochemical impedance measurements, one vec tor is examined, using the others as the frame of reference. The voltage vector is divided by the current vec tor, as in Ohm s law. Electrochemical impedance measures the impedance of an electrochemical system and then mathematically models the response using simple circuit elements such as resistors, capacitors, and inductors. In some cases, the circuit elements are used to yield information about the kinetics of the corrosion process. 

The traction oj is the vector force per unit area acting on a surface. The unit of impedance is the rayl, where 1 rayl = 1 kg m-2 s 1 the usual multiple is the megarayl, abbreviated Mrayl. The impedance is equal to the product of the density p of a medium and the velocity v of a given wave propagating in it this is how impedance is usually calculated,... 

These reflection and transmission coefficients relate the pressure amplitude in the reflected wave, and the amplitude of the appropriate stress component in each transmitted wave, to the pressure amplitude in the incident wave. The pressure amplitude in the incident wave is a natural parameter to work with, because it is a scalar quantity, whereas the displacement amplitude is a vector. The displacement amplitude reflection coefficient has the opposite sign to (6.90) or (6.94) the displacement amplitude transmission coefficients can be obtained from (6.91) and (6.92) by dividing by the appropriate longitudinal or shear impedance in the solid and multiplying by the impedance in the fluid. The impedances actually relate force per unit area to displacement velocity, but displacement velocity is related to displacement by a factor to which is the same for each of the incident, reflected, and transmitted waves, and so it all comes to the same thing in the end. In some mathematical texts the reflection... 

Traditionally, the instrument of choice for accurate conductance measurements that are relatively free of capacitance effects has been the ac Wheatstone bridge illustrated in Figure 8.14. The details of operation and the derivation of the balance condition of the ac bridge are presented in considerable detail elsewhere [16,17], The balance condition is exactly analogous to that of the dc bridge except that impedance vectors must be substituted for resistances in the arms of the bridge when reactive circuit elements are present. 

This impedance can be presented as a vector in the complex plane with modulus Z =EJIm and argument o=a-( . As a consequence, it is expected to obtain a plane with axes having unit 1 for the real and j for the imaginary axis. However, mainly R2 is presented, so both axes are real33. The projection of the impedance vector at these axes results in the resistance Z and the reactance Z", also called the real and imaginary part of the impedance, respectively (Fig. 2.5) ... 

Equations 2.37-2.40 result in the commonly used presentation of the impedance, e.g. the Nyquist and the Bode plots. The first one shows the total impedance vector point for different values of co. The plane of this figure is a complex plane, as shown in the previous section. Electrochemical-related processes and effects result in resistive and capacitive behaviour, so it is common to present the impedance as ... 

Note that the quantities in bold are vectors represented in the complex plane. In Fig. 12.5 a transfer function spectrum obtained in 0.25 s is shown for 100 frequencies around 10 MHz. It can be used for real time evaluation of the quartz electro-acoustical impedance when the viscoelastic properties change. 

Figure 4 Cartesian coordinate system with imaginary j notation depicting an impedance vector Z and its real and imaginary components Z and Z" as well as phase angle 9.

The a.c. impedance technique [33,34] is used to study the response of the specimen electrode to perturbations in potential. Electrochemical processes occur at finite rates and may thus be out of phase with the oscillating voltage. The frequency response of the electrode may then be represented by an equivalent electrical circuit consisting of capacitances, resistances, and inductors arranged in series and parallel. A simplified circuit is shown in Fig. 16 together with a Nyquist plot which expresses the impedance of the system as a vector quantity. The pattern of such plots indicates the type and magnitude of the components in the equivalent electrical network [35]. 

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