A.J. Bard, L.R. Faulkner - Electrochemical methods - Fundamentals and Applications (794273), страница 95
Текст из файла (страница 95)
A bigadvantage common to all techniques based on nonlinearity is comparative freedomfrom charging currents. The double-layer capacitance is generally much more linearthan the faradaic impedance; hence charging currents are largely restricted to the excitation frequencies.10.1.2Review of ac CircuitsA purely sinusoidal voltage can be expressed ase = E sin cot(10.1.1)10.1 Introduction371Figure 10.1.3 Phasordiagram for an alternatingvoltage, e = E sin Ш.7l/C02тг/о)-E-7C/2where со is the angular frequency, which is 2тг times the conventional frequency in Hz.
Itis convenient to think of this voltage as a rotating vector (or phasor) quantity like that pictured in Figure 10.1.3. Its length is the amplitude E and its frequency of rotation is со. Theobserved voltage at any time, e, is the component of the phasor projected on some particular axis (normally that at 0°).One frequently wishes to consider the relationship between two related sinusoidalsignals, such as the current, /, and the voltage, e. Each is then represented as a separatephasor, / or Ё, rotating at the same frequency. As shown in Figure 10.1.4, they generallywill not be in phase; thus their phasors will be separated by a phase angle, ф. One of thephasors, usually E, is taken as a reference signal, and ф is measured with respect to it.
Inthe figure, the current lags the voltage. It can be expressed generally asi = I sm((ot + ф)(10.1.2)where ф is a signed quantity, which is negative in this case.The relationship between two phasors at the same frequency remains constant as theyrotate; hence the phase angle is constant. Consequently, we can usually drop the references to rotation in the phasor diagrams and study the relationships between phasors simply by plotting them as vectors having a common origin and separated by the appropriateangles.Let us apply these concepts to the analysis of some simple circuits.
Consider first apure resistance, R, across which a sinusoidal voltage, e = E sin cot, is applied. SinceOhm's law always holds, the current is (E/R)sin со t or, in phasor notation,(10.1.3)(10.1.4)E= IRThe phase angle is zero, and the vector diagram is that of Figure 10.1.5.еОП2л/со (2тг + ф)/со0/Figure 10.1.4 Phasor diagram showing the relationship between alternating current and voltagesignals at frequency со.372 • Chapter 10.
Techniques Based on Concepts of ImpedanceFigure 10.1.5Relationship between thevoltage across a resistorand current through theresistor.e or iSuppose we now substitute a pure capacitance, C, for the resistor. The fundamentalrelation of interest is then q = Ce, or / = C{deldt)\ thus/ = (oCE cos cot(10.1.5)i = x— sinf Ш + ~zc\(10.1.6)where Xc is the capacitive reactance, l/o)C.The phase angle is тг/2, and the current leads the voltage, as shown in Figure 10.1.6.Since the vector diagram has now expanded to a plane, it is convenient to represent phasors in terms of complex notation. Components along the ordinate are assigned as imaginary and are multiplied by j = V — 1 .
Components along the abscissa are real. Introducingcomplex notation here is only a bookkeeping measure to help keep the vector componentsstraight. We handle them mathematically as "real" or "imaginary," but both types are realin the sense of being measurable by phase angle. In circuit analysis, it turns out to be advantageous to plot the current phasor along the abscissa as shown in Figure 10.1.6, eventhough the current's phase angle is measured experimentally with respect to the voltage.If that is done, it is clear thatE =-jXcI(10.1.7)Of course, this relation must hold regardless of where / is plotted with respect to the abscissa, since only the relationship between E and / is significant.
A comparison of equations 10.1.4 and 10.1.7 shows that Xc must carry dimensions of resistance, but, unlike R,its magnitude falls with increasing frequency.Now consider a resistance, R, and a capacitance, C, in series. A voltage, E, is appliedacross them, and at all times it must equal the sum of the individual voltage drops acrossthe resistor and the capacitor; thusE = ER+ Ec(10.1.8)E= i(R-jxc)(10.1.9)E=/Z(10.1.10)Figure 10.1.6Relationship between analternating voltage acrossa capacitor and thealternating current throughthe capacitor.10.1 Introduction373In this way we find that the voltage is linked to the current through a vector Z =R - jXc called the impedance.
Figure 10.1.7 is a display of the relationships betweenthese various quantities. In general the impedance can be represented as 2(10.1.11)where ZR C and Z\m are the real and imaginary parts of the impedance. For the examplehere, ZRC = R and Z\m = XQ = l/o)C. The magnitude of Z, written \Z\ or Z, is given by|Z| 2 = R2= (Z R e ) 2 + (Z I m ) 2(10.1.12)= XCIR = llojRC(10.1.13)and the phase angle, ф, is given bytan ф = Zim/ZReThe impedance is a kind of generalized resistance, and equation 10.1.10 is a generalized version of Ohm's law.
It embodies both (10.1.4) and (10.1.7) as special cases. Thephase angle expresses the balance between capacitive and resistive components in the series circuit. For a pure resistance, ф = 0; for a pure capacitance, ф = тг/2; and for mixtures, intermediate phase angles are observed.The variation of the impedance with frequency is often of interest and can be displayed in different ways. In a Bode plot, log \Z\ and ф are both plotted against log со.
Analternative representation, a Nyquist plot, displays Z I m vs. Z R e for different values of со.Plots for the series RC circuit are shown in Figures 10.1.8 and 10.1.9. Similar plots for aparallel RC circuit are shown in Figures 10.1.10 and 10.1.11.More complex circuits can be analyzed by combining impedances according to rulesanalogous to those applicable to resistors.
For impedances in series, the overall impedance is the sum of the individual values (expressed as complex vectors). For impedancesin parallel, the inverse of the overall impedance is the sum of the reciprocals of the individual vectors. Figure 10.1.12 shows a simple application.=-jXcI(a)(b)Figure 10.1.7 (a) Phasor diagram showing the relationship between the current and the voltagesin a series RC network. The voltage across the whole network is E, and ER and EQ are itscomponents across the resistance and the capacitance, (b) An impedance vector diagram derivedfrom the phasor diagram in (a).2In many treatments, the definition of impedance is taken as Z = Z R e + jZim, but we can simplify matters withthe definition in (10.1.11).
In electrochemistry, the imaginary impedance is almost always capacitive andtherefore negative. With our definition of Z, we can generally work with positive values and expressions forZ\m, and impedance plots appear naturally in the first quadrant. Our choice is in accord with the general,although sometimes implicit, practice in the field. In reviewing other literature, it is wise to take note of thedefinition of Z.Chapter 10. Techniques Based on Concepts of Impedance6-3iIi13log frequency-11i180gle37460CD40сCOwСб.CQ.I\—\20I-3\IVs^-1Figure 10.1.8 Bode plots for a series RC circuit withR = 100 ft and С = 1 fiF.7Sometimes it is advantageous to analyze ac circuits in terms of the admittance, Y,which is the inverse impedance, 1/Z, and therefore represents a kind of conductance.
Thegeneralized form of Ohm's law, (10.1.10), can then be rewritten as / = EY. These concepts are especially useful in the analysis of parallel circuits, because the overall admittance of parallel elements is simply the sum of the individual admittances.Later we will be interested in the vector relationship between Z and Y. If Z is writtenin its polar form (Section A.5):Z = Zej<i>(10.1.14)then the admittance isY = ±e->+(10.1.15)Here we see that Y is a vector with magnitude 1/Z and a phase angle equal to that of Z, butopposite in sign. Figure 10.1.13 is a picture of the arrangement.100.01 18о 6xКГ 4CO-—оо20.1 I01501100Figure 10.1.9 Nyquist plot for a series RCcircuit with R = 100 И and С = 1 fiF.37510.1 Introduction90*I_L-3--2 -1I1II2347055030-1i-2-i3-log frequencyFigure 10.1.10210-I10123456I7log frequencyBode plots for a parallel RC circuit with R = 100 П and С = 1100Figure 10.1.11 Nyquist plot for aparallel RC circuit with R = 100 пand С = 1 fiF.z2 + z3Figure 10.1.12i'" z2+z3Calculation of a total impedance from component impedances.Figure 10.1.13 Relationship between the impedance, Z, andthe admittance, Y.37610.13Chapter 10.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
