Effective Resistance

The resistance of a conductor as determined by the equation $$\bbox[10px,border:1px solid grey]{R = \rho({l \over A})}$$ is often called the dc, ohmic, or geometric resistance. It is a constant quantity determined only by the material used and its physical dimensions. In ac circuits, the actual resistance of a conductor (called the effective resistance) differs from the dc resistance because of the varying currents and voltages that introduce effects not present in dc circuits.
These effects include radiation losses, skin effect, eddy currents, and hysteresis losses. The first two effects apply to any network, while the latter two are concerned with the additional losses introduced by the presence of ferromagnetic materials in a changing magnetic field.

Experimental Procedure

The effective resistance of an ac circuit cannot be measured by the ratio V/I since this ratio is now the impedance of a circuit that may have both resistance and reactance. The effective resistance can be found, however, by using the power equation $P = I^2R$, where $$ \bbox[10px,border:1px solid grey]{R_{eff} = { P \over I^2}} \tag{1}$$ A wattmeter and an ammeter are therefore necessary for measuring the effective resistance of an ac circuit.

Radiation Losses

Let us now examine the various losses in greater detail. The radiation loss is the loss of energy in the form of electromagnetic waves during the transfer of energy from one element to another. This loss in energy requires that the input power be larger to establish the same current I, causing R to increase as determined by Eq. (1). At a frequency of $60 Hz$, the effects of radiation losses can be completely ignored. However, at radio frequencies, this is an important effect and may in fact become the main effect in an electromagnetic device such as an antenna.

Skin Effect

The explanation of skin effect requires the use of some basic concepts previously described. Remember from Chapter 11 that a magnetic field exists around every current-carrying conductor (Fig. 1). Since the amount of charge flowing in ac circuits changes with time, the magnetic field surrounding the moving charge (current) also changes. Recall also that a wire placed in a changing magnetic field will have an induced voltage across its terminals as determined by Faraday's law, $$e = N {d \Pi \over dt}$$ The higher the frequency of the changing flux as determined by an alternating current, the greater the induced voltage will be.
Demonstrating the skin effect on the effective
resistance of a conductor.
Fig. 1: Demonstrating the skin effect on the effective resistance of a conductor.
For a conductor carrying alternating current, the changing magnetic field surrounding the wire links the wire itself, thus developing within the wire an induced voltage that opposes the original flow of charge or current. These effects are more pronounced at the center of the conductor than at the surface because the center is linked by the changing flux inside the wire as well as that outside the wire. As the frequency of the applied signal increases, the flux linking the wire will change at a greater rate. An increase in frequency will therefore increase the counter-induced voltage at the center of the wire to the point where the current will, for all practical purposes, flow on the surface of the conductor. At 60 Hz, the skin effect is almost noticeable. However, at radio frequencies the skin effect is so pronounced that conductors are frequently made hollow because the center part is relatively ineffective. The skin effect, therefore, reduces the effective area through which the current can flow, and it causes the resistance of the conductor, given by the equation $R = \rho(l/A), to increase.

Hysteresis and Eddy Current Losses

As mentioned earlier, hysteresis and eddy current losses will appear when a ferromagnetic material is placed in the region of a changing magnetic field. To describe eddy current losses in greater detail, we will consider the effects of an alternating current passing through a coil wrapped around a ferromagnetic core. As the alternating current passes through the coil, it will develop a changing magnetic flux $\Pi$ linking both the coil and the core that will develop an induced voltage within the core as determined by Faraday's law. This induced voltage and the geometric resistance of the core $R_C = \rho(l/A)$ cause currents to be developed within the core, $i_{core} = (e_{ind} /R_C)$, called eddy currents. The currents flow in circular paths, as shown in Fig. 2, changing direction with the applied ac potential.
Defining the eddy current losses of a
ferromagnetic core.
Fig. 2: Defining the eddy current losses of a ferromagnetic core.
The eddy current losses are determined by $$P_{eddy} = i_{eddy}^2R_{core}$$ The magnitude of these losses is determined primarily by the type of core used. If the core is nonferromagnetic—and has a high resistivity like wood or air—the eddy current losses can be neglected. In terms of the frequency of the applied signal and the magnetic field strength produced, the eddy current loss is proportional to the square of the frequency times the square of the magnetic field strength: $$P_{eddy} \propto f^2B^2$$ Eddy current losses can be reduced if the core is constructed of thin, laminated sheets of ferromagnetic material insulated from one another and aligned parallel to the magnetic flux. Such construction reduces the magnitude of the eddy currents by placing more resistance in their path. Hysteresis losses were described in Section 10.6. You will recall that in terms of the frequency of the applied signal and the magnetic field strength produced, the hysteresis loss is proportional to the frequency to the 1st power times the magnetic field strength to the nth power: $$P_{hys} \propto f^1 B^n$$ where n can vary from 1.4 to 2.6, depending on the material under consideration.
Hysteresis losses can be effectively reduced by the injection of small amounts of silicon into the magnetic core, constituting some 2% or 3% of the total composition of the core. This must be done carefully, however, because too much silicon makes the core brittle and difficult to machine into the shape desired.