Surface resistivity skin depth

Equation (5.4) is valid as long as the skin depth is large in comparison to the mean free path of the electrons in the metal. This holds true in the microwave range at room temperature, for cryogenic temperature the surface resistance lies above the values predicted by Equation (5.4) and exhibits a f2 3 rather than a f1 2 frequency dependence (anomalous skin effect [7]). 

As an example, for copper with a room temperature conductivity of 5.8 x 107(O m) 1 the surface resistance at 10 GHz is 26 mil, the skin depth is 0.66 pm. Therefore, the Q of a cavity resonator with a geometric factor of several hundred is in the 104 range. However, for planar resonators like the ones shown in Figure 5.8 the G values are only a few Ohms leading to Q values of only a few hundred. This is too small for many filter and oscillator applications. 

The skin depth Ss is inversely proportional to the square root of the frequency, and at 1 MHz is 0.064mm for silver, 0.066mm for copper and 0.19mm for a typical solder. A surface resistivity ps can be defined by... 

Therefore, when using round wires, if we choose the diameter as twice the skin depth, no point inside the conductor will be more than one skin depth away from the surface. So no part of the conductor is unutilized. In that case, we can consider this wire as having an ac resistance equal to its dc resistance — there is no need to continue to account for high-frequency effects so long as the wire thickness is chosen in this manner. 

At high frequencies, the surface of the insulator may have a different resistivity from the bulk of the material owing to impurities absorbed on the surface, external contamination, or water moisture hence, electric current is conducted chiefly near the surface of the conductor (i.e., skin effect). The depth, S, at which the current density falls to 1/e of its value at the surface is called the skin depth. The skin depth and the surface resistance are dependent upon the AC frequency. The surface resistivity, R, expressed in 2, is the DC sheet resistivity of a conductor having a thickness of one skin depth ... 

However, conductor loss depends on the resistance (surface resistance) of the conductor. As the frequency increases, there is a tendency for the current to concentrate in the surface parts of the conductor. The part where the current flows is known as skin depth (the depth where current density falls to 1/e = 0.37 of its value at the surface), and it decreases in inverse proportion to the square root of the frequency. Surface resistance Rs is determined by skin depth d and conductor conductivity o as in the formula below. It is inversely proportional to the square root of conductor conductivity, and increases proportional to the square root of the frequency. 

To reduce conductor loss in high frequency ranges, it is necessary to take an proach that reduces conductor resistance to the minimum (refer to Chapter 1). Since the inductance of the conductor inside increases at high frequencies, current flows only near the surface of the conductor layer. The thickness of the area where the current flows is called skin depth. Figure 10-1 shows the relationship between the frequency of each type of conductor and the skin depth. The relationship with skin depth ( ) is in accordance with the formula below, and there is a tendency for the skin depth to become shallower as the frequency increases with materials that are not magnetized. 

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