Modeling the Q factor for accurate lumped element filter simulation

1. Introduction
When simulating a circuit structure it is crucial to be able to account for parasitics or non-ideal behavior of components such as non-linear quality factor (Q-value) and parasitic package capacitance and lead inductance. These parameters, especially the quality value, are usually approximated over a limited frequency bandwidth with a simple equation. 

Circuit simulators like Agilent's simulation suite ADS or Applied Wave Research's Microwave Office can implement frequency dependent variables or equations for the different components' values. Due to the fact that frequency dependent equations can be implemented to account for non-ideal behavior of the components makes the simulation results more accurate and reliable.

2. Basic Models

2a. The Capacitor Model

A generic capacitor model is shown in Figure 1.

Referring to Figure 1 the non-ideal behavior of the capacitor is shown as lead inductance (as well as inductance created by the layers of metallization in the capacitor) L, the lead resistance R_S due to imperfect metal (finite conductivity) and the conductance R_P due to the dielectric impurity. The series resistance R_S in Figure 1 can be determined from the given Q value and the parallel resistance R_P is determined from the dielectric loss tangent of the interior of the capacitor.

R_S in Figure 1 typically denotes the Equivalent Series Resistance (ESR) of a capacitor and L is used to denote the resonance frequency of the device. The resonance frequency of a surface mount (SMT) multilayer capacitor can vary for the same device under different mounting conditions depending on horizontal or vertical mounting [1].

The frequency-dependence of the quality factor Q determines the behavior of the capacitor. Typically, the Q factor can be simplified by using (1)

                                                     (1)

The quality factor of a capacitor can easily be determined when a sinusoidal voltage v(t)=V₀sin(ωt) is applied to the simplified equivalent circuit seen in Figure 2 of the capacitor

                                                                                              (2)

which becomes

                                                                                                      (3)

where the ideal capacitance C and the represented loss R_Q are as referred to in Figure 2. If a series resistor of a capacitor is used in Figure 2 the transformation of the resistance R_Q in (3) would be given by

                                                                                                     (4)

which is found by using the current i(t) instead of the voltage v(t) in (2).

The quality factor Q in (3) is a figure of merit and is furthermore a linear frequency dependent equation. The real-life capacitors typically have a Q that varies exponentially as seen in Figure 3 [2]. The frequency dependence of Q can typically be modeled as a one term decaying exponential term, which is a first order approximation.

The Q values in Figure 3 can be reconstructed by using a simple first order approximation together with (3)

                                                                                                (5)

where Q₀ is the Q value at frequency F_Q and f is the frequency of interest. The exponential factor α can be either a negative floating number (for increasing Q values with frequency) or a positive floating number for decreasing Q values. An example of (5) is shown in Figure 4 for three cases of ultra high Q ceramic capacitors.