High-frequency Resistors

Perhaps the most common circuit element in low-frequency electronics is a resistor whose purpose is to produce a voltage drop by converting some of the electric energy into heat. We can differentiate among several types of resistors:

• Carbon-composite resistor of high-density dielectric granules
• Wire-wound resistors of nickel or other winding material.
• Metal-film resistors of temperature stable materials
• Thin-film chip resistors of aluminum or beryllium based materials

Of these types mainly the thin-film chip resistors find application nowadays in RF and MW circuits as surface mounted devices (SMDs). This is due to the fact that they can be produced in extremely small sizes.

As the previous section has shown, even a straight wire possesses an associated inductance. Consequently, the electric equivalent circuit representation of a high-frequency resistor of nominal value R is more complicated and has to be modified such that the finite lead dimensions as well as parasitic capacitances are taken into account. This situation is depicted in figure.

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 resistances 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.

In most RF circuits chip capacitors find widespread application for the tuning of filters and matching networks as well as for biasing active component such as transistors. It is therefore important to understand their high-frequency behavior. Elementary circuit analysis defines capacitance for a parallel plate capacitor whose plane dimensions are large compared to its separation as follows:

$$C=\frac{εA}{d}= ε_0ε_1\frac{A}{d}$$

Where A is the plate surface area and d denotes the plate separation. Ideally there is no current flow between the plates. However, at high frequencies the dielectric materials become lossy (i.e. there is a conduction current flow). The impedance of a capacitor must thus be written as a parallel conductance $G_e$ and susceptance $wC$.

$$Z = \frac{1}{G_e+jwC}........(1.15)$$

In this expression the current flow at DC is due to the conductance $G_e = σ_{diel}A/d$ , with $σ_{diel}$ being the conductivity of the dielectric. It is now customary to introduce the series loss tangent tan$\Delta=we/σ_{diel}$ and insert it into the expression for $G_e$ to yield $$G_e=\frac{σ_{diel}A}{d}=\frac{weA}{dtan \Delta_s}=\frac{wC}{tan \Delta_s}......1.16$$

Some practical values for the loss tangent are summarized. The corresponding electric equivalent circuit with parasitic lead inductance L, series resistance Rs describing losses in the lead conductors, and dielectric loss resistance Re = 1/Ge, is shown in fig 1.11

High-Frequency inductors Although not employed as often as resistors and capacitors, inductors generally are used in transistor basing networks, for instance as RF coils (RFCs) to short circuit the device to DC voltage conditions. Since a coil is generally formed by winding a straight wire on a cylindrical former, we know from our previous discussion that the windings represent an inductance in addition to the frequency-dependent wire resistance. Moreover, adjacently positioned wires constitute separated moving charges, thus giving rise to a parasitic capacitance effect as shown in figure 1.14

The equivalent circuit model of the inductor is shown in figure 1.15. The parasitic shunt capacitance Cs and series resistance Rs represent composite effects of distributed capacitance Cd and resistance $R_d$ respectively.