Angular Capacitor

A major use of high-quality pot-core inductors is in combination with capacitors in filter circuits. They were once extensively used in telephone systems but solid state switched systems have replaced them. The soft MnZn and NiZn ferrite systems are dominant for pot-core manufacture, although metal dust cores are used for certain applications. The important design principles can be understood by reference to a simple series LCR circuit to which a sinusoidal voltage U of angular frequency co is applied (Fig. 9.46). The circuit impedance Z = R+](coL - 1/coC) is a minimum when ojZ, = /<<>C. Thus the resonant frequency o0 is given by... 

Constant phase angle — The phase angle is the angle between two phasors that rotate with the same -> angular velocity. The constant phase angle is independent of angular velocity. For a solitary -> resistor the phase angle is zero, and for a solitary -> capacitor it is ir/2. 

The reciprocal of the specimen resistance in the equivalent parallel circuit for a given frequency is sometimes called the specimen conductance GP. It is a combination of DC conductance, by which we mean any real flow of charge through the sample under the influence of the applied field, and the anomalous conductance due to any time-dependent polarisation processes. The contribution that a true DC conductivity dielectric loss at an angular frequency w can be readily calculated as follows for the material in a parallel-plate capacitor. If the capacitor plates have area A and separation s ... 

The coil with inductance L has complex reactance iu>L while the tuning capacitor with capacitance C has complex reactance -i/wC, when they are subjected to rf at angular speed u) (which is 271 times the frequency). The reactance of L and C in parallel, then, is... 

For a parallel plate capacitor with the sinusoidally modulated plate separation d(t) given by Eq. (2) (do=average distance between the parallel plates. Ad=vibration amplitude and ( =angular frequency of vibration), the time dependent capacitance C(t) is given by Eq. (3) with C(t) defined by Eq. (3). 

V can be represented on an Argand diagram (Fig. 12.13b) by a vector of length Vo rotating at an angular frequency co. For an ideal capacitor, of capacitance C, the current is given by... 

By replacing the imaginary frequency variable jw by an extended complex frequency variable s = a + jw (here a is not conductivity), it is possible to define, for example, impedance not only in the angular frequency w-domain, but also in the s-domain. The impedance of a capacitor of capacitance C is, for example, in the frequency domain ... 

While microelectronics can be described as the fabrication of electrical components like transistors, diodes, resistors and capacitors on a semiconductor substrate, mostly silicon, MEMS and MOEMS are using the manufacturing technologies of microelectronics to fabricate mechanical and optical structures as well as sensing or actuating devices. Typical examples are pressure sensors, microphones, acceleration and angular-rate sensors, magnetic compasses, inkjet heads, micro-scanners, micro-fluidic devices, biosensors, etc., to name some. 

As discussed earlier, the two-element series combinations such as RC and RL circuits showed the exponential fimctions for the growth or decay of charge, potential, and cmrent over time with a time scale measured by their respective time constants Tc and Tj. However, an inductor-capacitor (LC) circuit demonstrates different behavior from that of RC and RL circuits. There are two new parameters introduced to describe LC circuit behavior (1) a sinusoidal oscillation period T and (2) an angular frequency co. The LC circuit s capacitor is initially charged to a maximum potential V and the magnetic energy stored in the inductor is zero. 

When a capacitor of value C is connected to a generator of RMS voltage (U) at angular frequency (a) will draw a current (1) according Equation 7. 

CPE is like a non-ideal capacitor (a capacitor with a constant phase shift lower than 90°). T is a measure of the magnitude of Zcpe n is a. constant parameter (0 < n < 1) representing inhomogeneities in the surface and co is the angular frequency. In the case of n = 1, Zcpe equals a pure capacitor corresponding to the electrode-electrolyte interface capacitance. The parameters n and T depend on the electrode material [3]. 

The various resistors, capacitors, etc., which are connected together in series and parallel combinations to form the equivalent circuit that corresponds to the test-cell, contribute to the real and imaginary parts in a way that is quite easy to calculate. The frequency-dependent information can be displayed in the form of Z or Z" as functions of the angular frequency, (o, or of log (co). 

---