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
(1.93)
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.
1.13.2 SMT Capacitors
SMT capacitors have a different model from resistors. To a first order, their parasitic effects tend to be all in series, as shown in the model of Figure 1.40. The series inductance is due to primarily to the package size and is similar to that of a resistor. The series resistance is due to the manufacturing characteristics of the capacitor and thus cannot be easily estimated. If an SMT capacitor is used in a resonant structure, this resistance will have the principal effect on the Q of the resonator. However, its effect is typically small in most wideband applications, where the capacitor is used as a series DC blocking capacitor or a shunt RF bypass capacitor. This is because the series inductance will dominate the series resistance in these use cases and cause the impedance of the capacitor to rise with frequency (rather than go to zero). At high frequencies, there may also exist a parasitic shunt capacitance across the entire package, which may cause the impedance to fall again.
Figure 1.40 Model of an SMT capacitor.
The case where the series resistance is of consequence is when the capacitor is used in a turned circuit, where the package inductance may be subsumed in the resonating inductor and thus at resonance the series resistance adds to a degradation of the Q of the capacitor. With careful design, the capacitance value may be compensated for by the including the effects of the series inductance; this effect is to make the capacitor look larger than its prescribed value. In fact, where the reactance of the parasitic inductance equals the reactance of the capacitance, the effective value of capacitance goes to infinite and the series impedance becomes just the parasitic resistance. So, for characterizing capacitors for use in tuned circuits, one must really assess their value near the frequency on which they will be having the most effect on a circuit. Consider a one‐pole filter, where the cutoff of the filter starts to occur when the reactance of the capacitor reaches 50 Ω. In many cases, the inductance is quite significant and already altering the effective value of the capacitor. Thus, it is important to evaluate the effective capacitance near this point. A good rule of thumb is to evaluate a capacitor where the reactance is j50 Ω.
A further characteristic of capacitors that is significant is the internal assembly structure. Capacitors are typically formed by a set of interleaved parallel plates with alternate plates connected at each end to the terminals. The plates can be parallel to or vertical to the PC board. For some cases, the capacitor body itself can form a dielectric resonator at high frequency, but below that the capacitor can act as a single, large conductive block on a PC board trace, typically resulting in a model that might best be considered a transmission line of somewhat lower impedance than the mounting line.
Capacitors used as bypass capacitors have an additional parasitic effect from the series inductance of the ground via, and from the pad above the ground via.
1.13.3 SMT Inductors
Inductors are perhaps the most complicated of the simple passive components. Because they are constructed of coils of very fine wire, sometimes multiple layers of coils, their parasitic elements are greatly affected by the details of their construction. Some inductors have the axis of the coil parallel to the PC board, and some are wound with the axis perpendicular. In both cases, the model for the inductor is essentially the same as the resistor, as shown Figure 1.39, but with the value of the series inductance equal to the DC value of the inductor, and the series resistance equal to the DC resistance. Inductors, because of the nature of their construction, have very large relative parasitic capacitances. In cases where an inductor is used for a bias element (relying on its impedance to be high at high frequencies) one often finds that the parasitic capacitance will become the main effect over the band of interest. Thus, in many cases the value of inductance used is carefully selected based on the overall effective inductance and sometimes utilizes the shunt capacitance to provide a high impedance at a particular frequency of interest. It may quite difficult to make a single inductor provide good RF performance over a wide band.
When inductors are used as elements in filters, the parasitic capacitance can often have significant effects for use in band‐pass filters, and the inductance must be evaluated for each use to find the effective value considering the parasitic capacitance.
A common figure of merit for inductors is the self‐resonant frequency (SRF), above which they act more like a capacitor (impedance goes lower with increasing frequency) than an inductor. The value of the SRF can be estimated in one way by looking at the length of the wire used in making the inductor. The SRF will be less than the frequency for which the wire is one‐quarter wavelength.
1.13.4 PC Board Vias
The PC board via is perhaps the most common PC board component, and often the most overlooked. The effect of a via depends greatly upon how it is structured in the circuit. A single via to ground in the center of a transmission line appears as almost a pure inductance. However, a via between RF traces can have aspects of inductance and some parasitic capacitance (due to pads around the via) that can cancel, in part or all, the inductive
