Resonators with high quality factors (Qs) are used throughout high-frequency circuits. For example, matching networks are most efficient when constructed with components having high unloaded Qs. By increasing the amount of stored energy to the loss of a circuit, it is possible to improve its unloaded Q. Fortunately, a low-cost commercial computer-aided-engineering (CAE) program can help design high Q transmission-line resonators that are electrically short (less than one-quarter wavelength).

In a multisection lowpass filter, the insertion loss is inversely proportional to the average unloaded component Q (Q_U).¹ For a single resonator, the insertion loss in dB approaches zero as the ratio of unloaded to loaded Q becomes infinitely large. For a parallel LC resonator, this implies that the individual component Q will need to be large compared to the loaded Q (terminated circuit Q) for low insertion loss.

A variety of oscillator performance parameters can benefit from a high-Q resonator, including phase noise and frequency drift (long-term frequency stability). It has been shown that the single-sideband (SSB) phase noise of a feedback oscillator is in part inversely related to the square of the loaded circuit Q.² Other factors, such as amplifier noise factor and signal power, affect overall phase noise in direct proportion only to their magnitudes. The exponential (second power) influence of the Q factor gives it a prominent impact on oscillator phase noise.

A circuit's Q can basically be defined as equal to two times the product of π and the ratio of the maximum energy stored to the energy dissipated per cycle. In an electrical circuit, energy is stored in the electric or magnetic fields associated with reactive circuit components and electrical energy is lost (to heat) whenever current flows through a resistance. From this basic definition, the way to improve Q is to increase the ratio of stored energy to losses in a circuit. Although how to achieve this may not always be apparent, a commercial software program that combines synthesis with simulation, the LINC2 program from Applied Computational Sciences (Escondido, CA), can simplify the task.

A section of transmission line shorted at one end will form a resonant circuit at the frequency represented by its quarter wavelength (90-deg. electrical length). Ports coupled into and out of the top side of the resonator form a bandpass circuit similar to a parallel-resonant resistive-inductive-capacitive (RLC) circuit. The Q of this full-length (quarter-wave) resonator can be found from:

where:

α = the line loss (dB/in.), and

λ = the wavelength (in.).

As an example, consider a resonator constructed on FR-4 printed-circuit-board (PCB) material with the following material properties at 1 GHz: relative dielectric constant, ε_r = 4.6, dielectric height = 30 mil, 50-Ω trace width of 54.4 mil on 0.5-oz. copper, α = 0.094 dB/in., and λ = 6.35 in. Using these values in Eq. 1 gives Q = 27.3/0.094/6.35 = 45.7. Because of the lossy PCB material this is not a very high-Q resonator, and higher Q values could be achieved by using lumped elements. The Q of a parallel LC resonator is largely determined by the Q of the inductor, which can be as high as 60 or more at this frequency for a surface-mount 0603 size part. However, by shortening the microstrip resonator and restoring resonance with a low-loss shunt capacitor, the Q can be increased by 50 to 100 percent. The Q for the printed resonator can exceed that of the lumped LC version without the expense of the discrete inductor.

The unloaded Q of a nonresonant shorted section of transmission line can be determined as a function of its length by the relation³:

where:

ω = 2πf

and the real and imaginary components of the line impedance are:

R = Z₀sinh(2αl)/[cosh(2αl) + cos(2βl)]

X = Z₀sin(2βl)/[(cosh(2αl) + cos(2β l)]

where:

α is the attenuation factor, and

β = 2π/λ, the propagation constant of the line.

Figure 1 shows the Q as a function of length (to one-quarter wavelength) for transmission-line sections with varying amounts of loss. The bottom curve plots the Q for a 50-Ω trace on 30-mil-thick FR-4 PCB with dielectric loss tangent of 0.02. The top trace shows what can be achieved with low-loss PCB material having one-tenth of this loss tangent (or 0.002). Constructing a resonator from the sub-quarter-wavelength line requires a shunt capacitor to restore resonance as shown in Fig. 2. The capacitor has loss that will add to the total losses of the resonator and thus lower the Q. Figure 1 represents the unloaded Q of the line by itself, which would be equal to the Q of a resonator in the limiting case of an ideal (lossless) capacitor and transmission-line combination.

Figure 3 plots the unloaded Q of the resonator in Fig. 2 for various values of effective series resistance (ESR) for a capacitor resonating with the section of transmission line shown in Fig. 1 (with loss tangent of 0.008). For an ideal capacitor with ESR = 0 (top trace of Fig. 3), the resonator Q is the same as the transmission line Q in Fig. 1. Note that as the line length decreases the value of the capacitor increases to maintain resonance at the desired frequency (1 GHz in this case). As the resonator length is shortened (C increasing), the Q rises above the quarter-wavelength value of Q = 8.686π/(αλ). However, for all resonating capacitors with ESR values greater than zero, the Q eventually falls again.