Circuits Containing Inductances

Until now, only circuits containing resistances and capacitances have been discussed. Inductive effects in electrical circuits appear when alternative electrical current flow creates a magnetic field interacting with the flowing current of course, in a strait wire the inductance is very small, but in looped wires or a coil it becomes larger. The inductive effects always lead to positive imaginary impedances, as will be shown in what follows. Let us first consider the circuit in Fig. 2.40, which contains inductance L in series with resistance Rq and a nested coimection of two (RQ circuits, i.e., LRo(Ci(Ri(R2C2))). The complex plane and Bode plots for this circuit without inductance were presented in Fig. 2.39. 

Finally, let us consider a simple circuit containing an RLC connection in series. The impedance of such a circuit is 

The real part of the impedance is constant and equal to R, while the imaginary impedance may be positive at high frequencies or negative at low frequencies. The imaginary impedance becomes zero when cOkz = 1 / this is the so-called 

Impedance of an electrical circuit containing linear electrical elements R, C, and L can be calculated using the impedance of these elements and Ohm s and Kirchhoff s laws. The complex plane and Bode plots can be easily produced using programming in Excel, Zplot, Maple, Mathematica, etc., which are readily available. It should be stressed that these electrical elements are linear, that is, their impedance is independent of the applied ac amplitude. In subsequent chapters, we will see how the impedance of electrochemical systems can be described. 

Exercise 2.1 Generate a digitalized function E(i) = co ,(27ttlT containing 64 points from 0 to 63 for the sampling time 0.01 s and wave period = 0.32 s 

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Rectification — is defined as the conversion of -> alternating current, AC, into pulsating -> direct current, DC, by any means other than the use of a motor-generator. That is, the AC is primarily converted by use of a -> rectifier into (unfiltered) unidirectional half-cycles. The percentage ratio of the DC output voltage to the peak AC input voltage of a certain rectifier device is called rectification efficiency. A circuit containing parallel -> capacitance, sometimes in combination with series inductance, is afterwards frequently applied as rectification filter in order to smooth the ripple in the DC current output. A combined device of rectifier and filter is called rectifier-filter system. 

The use of inductors to model low-frequency inductive loops in an impedance response is somewhat controversial. Demonstrate that the circuit presented as Figure 4.6(b) with negative R and C in the nested element can be mathematically equivalent to a circuit containing an inductor... 

The quantity = L w is called the inductive reactance. If a circuit contains all three elements, R, L, C, it can be shown (see Prob. 10.3) that the phase difference and impedance are given by... 

The rotor currents are limited by the short-circuit impedance of the rotor circuit. This circuit contains resistance and reactance. The inductive reactance is directly proportional to the frequency of the induced emfs in the rotor. As the rotor accelerates two effects take place -... 

All circuits which contain inductive reactance and resistance have an X-to-R ratio, in practice between 2.0 and 100.00. In short-circuit analysis it is usually necessary to relate the asymmetrical current to the symmetrical current. This can only be done if the short-circuit power factor of the circuit and hence the X-to-R ratio is known. Table 8.1 shows the relationship between these parameters and currents. Normally the short-circuit power factor is low, between 0.01 and 0.45. It is customary in short-circuit analysis to assume that one of the phases has the worst-case situation of fully asymmetrical current. Figure 8.1 shows an example, together with the various definitions of times and currents. 

Many power system networks can be reduced to a simple series-connected circuit containing a resistance R and an inductance L, for the purpose of calculating the transient fault current. Furthermore a... 

Worked Example for the Calculation of Volt-drop in a Circuit Containing an Induction Motor... 

CALCULATION OF VOLT-DROP IN A CIRCUIT CONTAINING AN INDUCTION MOTOR... 

A number of instructive applications of differential equations concern alternating-current circuits containing resistance, inductance, capacitance, and an oscillating voltage source, as represented in Fig. 8.1. In the simplest case, in a circuit with resistance R and voltage (or emf) E, the current / is determined by Ohm s law. 

Electrical conductivity cell. Impedance is a measure of the total opposition to the flow of a sinusoidal alternating current in a circuit containing resistance, inductance, and capacitance. Inductance and capacitance together are called the reactive part of the circuit. The changes in impedance that occur in a microbial culture can be measured by placing two metal electrodes into the culture medium and introducing an alternating potential into the circuit. 

Ammeters must be connected In series with the load, and voltmeters in parallel across the load, as shown in Fig. 4.17. The power in a resistive load may be calculated from the readings of voltage and current since P = VI. TTiis will give accurate calculations on both a.c. and d.c. supplies, but when measuring the power of an a.c. circuit which contains inductance or capacitance, a wattmeter must be used because the voltage and cument will be out of phase. 

Both circuits contain some inductive (electromagnelic) energy, which means that they each possess a property of inductance called self-inductance, notated here L and L22. The influence between the two circuits is translated into two mutual inductances /.,2 and L21 and one usually writes each induction quantity (flux) as the sum of each contribution, in assuming that all inductances are scalars ... 

To understand the impedance of electrochemical objects, it is necessary to understand the behavior of simple electrical circuits, first in steady state, then in transient conditions. Such circuits contain simple linear electrical elements resistance, capacitance, and inductance. Then the cmicept of electrical impedance will be introduced. It demands an understanding of the Laplace and Fourier transforms, which will also be presented. To understand impedance, it is necessary to thoroughly understand the complex plane and Bode plots, which will be presented for a few typical connections of the electrical elements. They can be computed using Excel, Maple, Mathematica, and specialized programs such as ZView. Several examples and exercises will be included. 

In Chap. 2 we saw the responses of electrical circuits containing the elements R, C, and L. Because these are linear elements, their impedance is independent of the ac amplitude used. However, in electrochemical systems, we do not have such elements we have solution-electrode interfaces, redox species, adsorption, etc. In this and the following chapters, we will learn how to express the electrochemical interfaces and reactions in terms of equations that, in particular cases, can be represented by the electrical equivalent circuits. Of comse, such circuits are only the electrical representations of physicochemical phenomena, and electrical elements such as resistance, capacitance, or inductance do not exist physically in cells. However, such a presentation is useful and helps in our understanding of the physicochemical phenomena taking place in electrochemical cells. Before presenting the case of electrochemical reactions, the case of an ideally polarizable electrode will be presented. 

In the case of blocking electrodes, the impedance increases to infinity as the frequency approaches zero. In such cases, approximation with the Voigt circuit is not appropriate. When a high-frequency impedance is finite, the easiest way to verify the Kramers-Kronig compliancy is to fit the impedances to the admittance representation of the circuit containing a ladder of (RC) element series (Fig. 13.4) [575]. In addition, capacitance, Cq, or inductance can be added in parallel. 

This means that a simple complex plot presenting one capacitive semicircle can be represented by other connections of R and C elements (as shown earlier) but also by circuits containing negative inductance and resistance. Of course, as has been proven in the literature, only the parameters of circuit (a) are more directly related... 

The second type of electric circuit makes use of a well-known analogy between alternating-current networks containing inductances and capacitances and coupled mechanical systems. The single junction network shown in Fig. 9-5, for example, has a resonant frequency v given by... 

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