Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 62

Resistors are perhaps the simplest of electronic elements to consider, and Ohms law is often the first lesson of an electronic text.

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However, the model of an RF resistor becomes much more complex as frequencies rise and distributed effects and parasitic elements become dominant. In this discussion the focus will be on surface‐mount PC board components, as they are used almost exclusively today in modern circuits. Thin film or thick film hybrid resistors have similar effects, and although the parasitic and distributed effects tend to hold off until higher frequencies, much of this discussion applies to them as well.

A good model for a resistor consists of a resistive value in series with an inductance, both shunted by a capacitance. This is a reasonable model for an SMT resistor in isolation, but the values and effects of the model are modified greatly by the mounting scheme of the component. For example, if it is mounted in series with a microstrip transmission line and the impedance is such that the resistor is much narrower than the transmission line, then this model works well for predicting circuit behavior; on the other hand, if it is mounted on a narrow line, then the contact pads will provide additional shunt capacitance to ground, and the model must include some element to account for this effect. At lower frequencies, some shunt capacitance will do well, but at higher frequencies, a length of low impedance transmission line might be a better choice.

A resistor used in shunt mode to ground can have an entirely different model when it comes to parasitic effects from that of a series resistance. While the RF value of the resistive element may stay almost the same as the series value (close to the DC value), the effective inductance can be substantially higher as the inductance of the ground via adds to that of the resistor in a microstrip configuration. A larger pad on the ground via, surprisingly, can add even more effective inductance as it resonates with the inductance of the via to increase the apparent inductance of the pair. Meanwhile, the shunt capacitance of the resistor may be absorbed in the transmission line width. Figure 1.38 shows a model for a resistor mounted in the series and shunt configurations. Measurement examples to illustrate extracting these values will be shown in Chapter 11.

Figure 1.38 Models for a series resistor (left) and shunt resistor (right).

In many instances, one of the two parasitic elements will dominate the model for first‐order high‐frequency effects. In fact, one can use some simple calculations to estimate a rough order of magnitude for these parasitic elements. Take, for example, an 0603 resistor, which has dimensions of approximately 0.6 mm width, 0.4 mm height (considering some excess plating, and some edge effect), and 0.76 mm length. If one considers the contact of the resistor wrapped around the body, one might reasonably divide the effective length by 3, to about 0.25 mm. Remembering that SMT resistors are often constructed on ceramic substrates, with a relative dielectric constant of about 10, then the capacitance can be computed as

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The actual value may be substantially greater or less depending upon the exact attributes of the electrodes, but this gives a starting estimate. For inductance one can look to the formula of a transmission line, and assuming that the resistor is mounted on a narrow line, such that its impedance is high, the inductance is computed as

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Thus, from the model one can compute the values of resistance for which the inductance or capacitive term dominates, at some frequency. For example, at 3 GHz, the inductance has a value of about 15 Ω reactance in series with the resistive element; the capacitance has a value of about 1250 Ω reactive in shunt. At 50 Ω, the inductance value dominates, at 300 Ω, the capacitive value is the dominant parasitic effect. For low values of resistance and high frequency, the inductance becomes dominant, and the series impedance is larger than expected, causing the loss through the resistance to be larger than expected because of this effect. At high values of resistance, the parasitic capacitance reduces the series impedance, and the expected loss through the resistor is less than expected. The values change with the physical size of the component; thus, cross‐over points differ in resistance and frequency, but with similar effects. This can be used to advantage as there exists a crossover point where the inductive and capacitive effects cancel somewhat, and the resistor behaves in a more ideal way, to higher frequencies, than for values above or below the crossover value. Using this value of resistance can, in series or parallel arrangements, provide a range of resistances that avoid parasitic effects until higher frequencies. For these values, a 50 Ω resistor terminated to ground will have about −18 dB return loss at 3 GHz; however, two 100 Ω resistors terminated to ground will have about −36 dB return loss, thus providing a better RF resistance of 50 Ω than a single resistor. Thus, characterization of the parasitic effects, and proper compensation, can allow use of SMT parts to much higher than expected frequencies (Dunsmore 1988). Figure 1.39 shows the effective impedance of a single 50 Ω SMT resistor and two 100 Ω SMT resistors in parallel, when used as a 50 Ω load. This effect also occurs for SMT inductors and capacitors.

Figure 1.39 Input match of a single SMT resistor and two in parallel.