Dissipation factor (DF) and Tan δ — that is the Q. This month, we'll look at DF and lay the groundwork for next month's blog about Q.
When discussing power losses through a capacitor, DF and Tan δ are often used interchangeably, which in most cases is not an issue. In fact, the Q of a capacitor is often just referenced (correctly but unhelpfully) as the reciprocal of DF. Just how are these parameters related, and when is it best to reference them?
As we already discussed, capacitors are not ideal. There will always be a difference between the amount of power going into a capacitor and the amount of power available at the other side due to the internal resistive and inductive processes in the dielectric and electrode systems.
DF and Tan δ are parameters used when a capacitor is employed in an AC circuit. Such a capacitor may be charged to a certain bias voltage or, if there is no applied voltage, have an AC voltage superimposed. Let's first consider the simplest, unbiased option and envision the voltage signal as a sine wave. In this case, a current will be generated as the voltage varies from its maximum to minimum value and back again. The current generated is a function of dV/dt (the rate of voltage change over time). There is zero current when voltage reaches its peak and maximum current as the voltage passes through 0 V, which directly corresponds to the greatest rate of change for voltage. These variances in voltage and current continue in 360° full-circle cycles; however, the current is 90° out of phase with the voltage.
The phase relationship between current and voltage in an ideal capacitor circuit is analogous to the relationship between velocity and acceleration in a swinging pendulum. At its extremes (i.e., the points at which it changes direction), a pendulum is at maximum acceleration and zero velocity; that all its energy is potential energy. Alternatively, at the bottom of the swing — the midpoint between the two extremes — a pendulum is at maximum velocity and zero acceleration. All its potential energy has been converted into kinetic energy.
In an electrical system, the kinetic energy is the current of the electron flow, and the potential energy is a function of the charge held by the capacitor, a relationship that can be expressed as: i = C (dV/dt). Analogous to the mass of an ideal pendulum determining its period independently of the height where it starts, an electrical system will have a period determined solely by the size of the capacitor and independent of the voltage to which it is initially charged.
If you refer back to the phase diagram in Figure 1, you'll see that the phase shift between voltage and current is 90° for a perfect capacitor. In the real world, however, the resistive and inductive parasitics in the system create a divergence that causes the phase shift to be a few degrees less than 90°. This power loss, or dissipation, is typically referred to as the loss angle and is defined as the ratio of the resistive power loss (proportional to ESR) to the capacitive reactance (Xc). Reactance, like impedance, is a resistance with an associated phase (in the same way that velocity is speed with an associated direction) and can be plotted with the ESR in an impedance plane.
The ratio of ESR to Xc is determined by the ratio of the opposite side to the hypotenuse in the right triangle formed by the loss angle, δ. Geometrically, this ratio is Tan δ or, for small angle approximations (e.g., systems with low losses), Tan δ ~ δ, and values are typically in the range of 0.001 to 0.5, depending on the capacitor technology. When these values are expressed as a percentage (DF = Tan δ x 100%), they are referred to as dissipation factor or, for low-DF values and low frequencies where inductance is unimportant, as power factor.