Module 5: Non-Ideal Behavior of Circuit Components 5.0 Introduction Most engineers are introduced to common circuit elements such as resistors, capacitors, and inductors through a course in basic circuit analysis. In this context, these common devices are often presented in the ideal sense, i.e. as being purely resistive, capacitive, or inductive, respectively. The effects due to wires, leads, and connectors are also usually neglected in circuit analysis. While this approach is clearly necessary to impart the fundamental concepts of circuit behavior, this ideal presentation often leads to misconceptions about how actual devices function. In reality, each type of circuit element exhibits some combination of resistive, capacitive, and inductive behaviors when operated at any frequency other than zero. The types of materials and construction techniques employed may also affect the performance of circuit elements. These factors can cause the impedance, capacitance, or inductance of these devices to differ greatly from the expected ideal values. As was demonstrated in previous chapters, even simple devices and circuits may require relatively complex models in order to correctly predict behavior over a wide frequency range. Through the application of certain approximations, these complex models may be simplified somewhat, while still providing the needed physical insight into device behavior. In this chapter, equivalent circuit models of basic circuit elements will be developed and analyzed. The response of these circuit elements to a broad range of operational inputs will be discussed and related to the topics presented in the earlier sections. From this it will become clear that what is usually referred to as "non-ideal" behavior of a circuit element is, in actuality, perfectly natural behavior in a regime that lies outside the range of validity of commonly accepted approximations. 5.1 Internal Impedance of Electric Circuit Elements circular wires Effects due to wires are often overlooked when considering the behavior of electric circuits. For most low-frequency applications, wires are modeled as having no effect on circuit performance. However, the impedance of wires can become significant under certain conditions. In this section a general expression for the internal impedance of a long cylindrical conductor (i.e., a wire) will be developed. Consider an infinitely-long, thin conductor, with cylindrical cross-section of radius a, having £ ¡ permittivity , permeability µ, and conductivity ¢ . It will be assumed that a current with density J £ £ flows through the conductor, supported by electric and magnetic fields E and H. Because the conductor is infinite in length, and has rotational symmetry, longitudinal and axial invariance will be assumed, therefore £ £ ¤ ¤ ¥ E ¥ H ¤ ¤ 0 z z and 5-1 z z^ r ¦ E ^ ¦ H p(r, ,z) r^ E z zo y ¦ J 2a x Figure 1. Infinitely-long cylindrical conductor. £ £ ¤ ¤ ¥ E ¥ H ¤¨§ ¤¨§ 0 . It will also be assumed that only an axially directed electric field exists within the conductor £ ¥ E zˆEz(r) which corresponds to the current density within the conductor. Also let the electric field maintained at the conductor surface by the external sources which excite the system be represented by ¥ ¥ Ezo Ez (r a) . A system of time-harmonic Maxwell's equations appropriately specialized for this structure is £ © ¥ E ¥ 0 ¡ £ £ © ¥ ×E j B 5-2 £ £ © ¡ ¥ ×B µ ( ¢ j )E £ © B ¥ 0 . Expanding the curl equations yields ¤ § § E ¥ ˆ z ˆ ¤ j (rˆB B zˆB ) r r z and B z 1 ˆ zˆ (rB ) µ ( j )zˆE . r r r z = = From these it is apparent that Br Bz 0 with Ez j B r and 1 (rB ) µ( j )E . (*) r r z Substitution of the expression 1 Ez B j r into (*) yields 1 r Ez µ ( j )Ez . r r j r which may be rewritten 5-3 2E E z 1 z 2 k E 0 z (**) 2r r r where 2 2 k µ 1 j . Equation (**) is known as Bessel's equation of order zero, with parameter k. This second order partial differential equation has solution Ez(r) AJo (kr) BNo (kr) where Jo (kr) is a Bessel function of the first kind of order zero, and No (kr) is a Bessel function of the second kind of order zero. A review of Bessel functions appears at the end of this chapter. The region under consideration is that lying inside the conductor. This region contains the point r =0, therefore the behavior of the Bessel functions for small arguments must be examined. As r approaches zero, the Bessel function of the second kind becomes infinitely large. In order for Ez to remain finite, the constant B must be equal to zero. The general expression for the electric field at any point inside the conductor is thus Ez (r) AJo (kr) . The unknown constant A is determined through application of boundary conditions. At the surface of the cylindrical conductor Ez(r a) Ezo A Jo(ka) thus Ezo A . Jo (ka) Substitution into the general expression for electric field within the conductor yields the electric field distribution over the cross-section of the wire. Jo(kr) E z(r) Ezo . Jo(ka) 5-4 Application of Ohm's law gives the distribution of current density in the cross-section of the conductor Jo(kr) Jz(r) Ezo . Jo(ka) The total axial current is found by integration of the current density over the cross-section of the conductor a a 2#$ Ezo " # I ! J (r)ds J (r)2 rdr rJ (kr)dr . z z z J (ka) o c.s. 0 o 0 A change of variables is now made by letting kr x. Then when r 0, x 0, and when r a, x ka, and dr 1 kdx. This results in ka 2#$ Ezo 1 I xJ (x)dx . z J (ka) 2 o o 0 k Application of a look-up integration formula for Bessel functions (see the Bessel function review section) leads to x & ka 2#$ Ezo 1 % I x J (x) z 2 1 Jo(ka) k x & 0 or 2# a Ezo J1(ka) Iz . k Jo(ka) From this expression for the total axial current, the internal impedance per unit length of a cylindrical conductor is determined E J (ka) ' k z i zo o . * I 2) a J (ka) z ( 1 This general expression is valid for all frequencies. Approximations to this general formula can 5-5 be made to obtain expressions which are valid in the limit that the frequency of operation is either very low, or very high. - internal impedance in the low frequency limit The frequency dependence of the impedance expression above occurs through the wavenumber term 2 2 * - k µ , 1 j . , (+ + From this, it can be seen that the wavenumber k approaches zero as approaches zero in the low frequency limit. This would suggest performing a power series + expansion to investigate the behavior of the Bessel function terms for small arguments. The Bessel function of the first kind may be expressed as a power series 3 .0/ m n 2m ( 2 1) z Jn(z) . n 3 2m 1 m 0 4 ( 2 m! (m n)! For small values of z z 2 z 2 2 4 6 2 J (z) 5 1 ... 1 o 4 4 and z z 3 z z 2 2 4 6 2 J (z) 5 ... 1 . 1 2 16 2 8 Substitution of these approximations into the general impedance expression results in 2 2 2 2 2 2 i k 1 2 k a /4 1 k a k a 6 2 4 z 6 1 1 2 2 2 7 8 7 8 2 a ka/2(1 2 k a /8) a 4 8 or 5-6 2 i 1 k 6 z 2 2 7$8 8 7 a 8 but 2 2 8 : 2 6!2 9 8 k 59 µ 1 j j µ 9: 9: because 8 ( ) » 1 at low frequencies. Substitution leads to the expression for the impedance per unit length of a cylindrical conductor in the low frequency limit i i i i i 1 µ 4 4 5 4 9 5 9 z 5 r jx r j l j 2 7 8 7 a 8 where i 1 r 5 2 8 7 a is the low frequency (dc) internal resistance per unit length, and i µ l 5 87 is the low frequency (dc) internal inductance per unit length. - internal impedance in the high frequency limit Once again, we begin with an examination of the wavenumber term 2 ;< 2 > ? k µ = 1 j . < = < For a good conductor, conductivity = is large, therefore it is assumed that / » 1. It is ? ? apparent that k approaches infinity as the frequency < approaches infinity, thus in the high frequency limit 5-7 < 2 @ k > j µ . ? or 1 A j A j ; < < ; < k ; j j µ j µ e 2 2 je 4 µ ? ? ? < 2 B j 2 > B < ; ; ; µ 1 j > B ? j µ ( 1 j) C 2 ? 2 where CED 2 ; 1 < µ F fµ ? ? is the skin depth or the depth that fields penetrate into a conductor, as discussed in Chapter 2. In this case, the behavior of the Bessel function terms for large arguments must be investigated. The asymptotic formula for large arguments for the Bessel function of the first kind is K K 2 K n L L lim J (x) J cos x . n K x GIH x 2 4 The internal impedance per unit length of the cylindrical conductor can now be expressed as K J (ka) cos ka L 1/4 i k o k z J K K K N M 2 a M J (ka) 2 a 1 cos ka L 3/4 or P Q S jO (ka 1/4) j (ka 1/4) R i k e e R z J .