From conventional AC circuit analysis we know that a resistance R is frequency independent and that a capacitor C and an inductor L can simply be satisfied by their reactances XC and XL as follows:

$$X_c=\frac {1}{wC}......(1.5a)$$ $$X_L=wL........(1.5b)$$

The implications of (1), for example, are such that a capacitor of C = 1pF and an inductor of L = 1nH at low frequencies of 60Hz represent, respectively, either an open or short circuit condition because

$$X_c(60Hz)=\frac{1}{2π.60..10^{-12}}≅2.65 × 10^9 Ω≈∞ ...... (1.6a)$$ $$X_L(60Hz)=2π.60..10^{-9}≅3.77× 10^{-7} Ω≈∞ ...... (1.6b)$$

It is important to point out that resistance, inductances, and capacitances are not only created by wires, coils, and plates as typically encountered in conventional low frequency electronics. Even a single straight wire or a copper segment of a printed circuit board (PCB) layout possesses frequency dependent resistance and inductance. For instance, a cylindrical copper conductor of radius a, length l, and conductivity $σ_{cond}$ has a DC resistance of

$$R_{DC}=\frac{a}{πa^2σ_{cond}}..... (1.7)$$

For a DC signal the entire conductor cross-sectional area is utilized for the current flow. At AC the situation is complicated by the fact that the alternating charge carrier flow establishes a magnetic field that induces an electric field (according to Faraday’s law) whose associated current density opposes the initial current flow. The effect is strongest at the center r = 0, therefore significantly increasing the resistance in th center of the conductor. The result is a current flow that tends to reside at the outer perimeter with increasing frequency. As derived in Appendix B, the z-directed current density Jz can be represented by

$$J_z=\frac{P^IJ_0(Pr)}{2πaJ_1(Pa)}......(1.8)$$

Where $P^2 = -jwμσ_{cond}$, and $J_0 (pr)$, $J_1 (pa)$ are Bessel functions of zeroth and first order and I is the total current flow in the conductor. Further calculations reveal that the normalized resistance and inductance under high-frequency conditions (f> 500MHz) can be put in the form

$$R/R_{DC}≅a/(2δ)........(1.9)$$ $$(wL)/R_{DC}≅a/(2δ)........(1.10)$$

In these expressions δ is the so-called skin depth

$$δ= (πfμσ_{cond})^{-1/2}......(1.11)$$

Which describes the spatial drop-off in resistance and reactance as a function of frequency f, permeability µ and conductivity $σ_{cond}$. For the equations (1.9) and (1.10) to be valid it is assumed that δ << a. In most cases, the relative permeability of the conductor is equal to unity (i.e., $µ_r = 1$). Because of the inverse square root frequency behavior, the skin depth is large for low frequencies and rapidly decreases for increasing frequencies. Figure 1.4 exemplifies the skin depth behavior as a function of frequency for material conductivities of copper, aluminum and gold.

We notice the significant increase in current flow at the outer perimeter of the wire ven for moderate frequencies of less than 1 MHz. At frequencies around 1 GHz, the current flow is almost completely confined to the surface of the wire with negligible radial penetration. An often used high-frequency approximation for the z-directed current density is

$$J_z=\frac{I_P}{j2πa\sqrt{r}}e^{-(1+j)\frac{a-r}{δ}}...... (1.12)$$

As seen in (1.12), the skin depth has a simple physical meaning. It denotes the reduction in the current density to the e-1 factor (approximately 37%) of its original DC value. If we rewrite (1.9) slightly, we find

$$R=R_{DC}\frac{a}{2δ}=R_{DC}\frac{πa^2}{2πaδ}......(1.13)$$

This equation shows that the resistance increases inverse proportionally with the cross-sectional skin area, see Figure 1.6

Fig. Increase in resistance over the cross sectional surface area. The current flow is confined to a small area defined by the skin depth δ

To standardize the sizes of wires, the American Wire Gauge (AWG) system is commonly used in the United States. For instance, the diameter of the wire can be determined by its AWG value.

Electrical Equivalent circuit of Resistor

The two inductances L model the leads, while the capacitances are needed to account for the actual wire arrangement, which always represents a certain charge

Separation effect modeled by capacitance $C_a$, and interlead capacitance $C_b$. The lead resistance is generally neglected when compared with the nominal resistance R. For a wire wound resistor the model is more complex, as figure 1.9 shows.

Electrical Equivalent circuit of capacitor

Electric equivalent circuit with parasitic lead inductance L, series resistance $R_1$ describing losses in the lead conductors, and dielectric loss resistance $R_e=1/G_e$ is shown in figure 1.11.

Electrical Equivalent circuit of inductor: The equivalent circuit model of the inductor is shown in figure1.15. The parasitic shunt capacitance $C_s$ and series resistance Rs represent composite effects of distributed capacitance $C_d$ and resistance $R_d$, respectively.