Radio Frequency Integrated Circuit Design - Webs

The Use and Design of Passive Circuit Elements in IC Technologies
frequency; however, as the self-resonance frequency is approached, the inductance
rises and then abruptly falls to zero. Beyond the self-resonant frequency,
the parasitic capacitance will dominate and the inductor will look capacitive.
Thus, the inductor has a finite bandwidth over which it can be used. For reliable
operation, it is necessary to stay well below the self-resonance frequency. Since
parasitic capacitance increases in proportion to the size of the inductor, the selfresonant
frequency decreases as the size of the inductor increases. Thus, the
size of on-chip inductors that can be built is severely limited.
5.14 The Quality Factor of an Inductor
The quality factor, or Q, of a passive circuit element can be defined as
Q = |Im(Z ind)|
|Re(Z ind)|
111
(5.13)
where Z ind is the impedance of the inductor. This is not necessarily the most
fundamental definition of Q, but it is a good way to characterize the structure.
A good way to think about this is that Q is a measure of the ratio of the desired
quantity (inductive reactance) to the undesired quantity (resistance). Obviously,
the higher-Q device is more ideal.
The Q of an on-chip inductor is affected by many things. At low frequencies,
the Q tends to increase with frequency, because the losses are relatively
constant (mostly due to metal resistance R s ), while the imaginary part of the
impedance is increasing linearly with frequency. However, as the frequency
increases, currents start to flow in the substrate through capacitive and, to a
lesser degree, magnetic coupling. This loss of energy into the substrate causes
an effective increase in the resistance. In addition, the skin effect starts to raise
the resistance of the metal traces at higher frequencies. Thus, most integrated
inductors have Qs that rise at low frequencies and then have some peak beyond
which the losses make the resistance rise faster than the imaginary part of the
impedance, and the Q starts to fall off again. Thus, it is easy to see the need
for proper optimization to ensure that the inductor has peak performance at
the frequency of interest.
Example 5.5 Calculating Model Values for the Inductor
Given a square inductor with the dimensions shown in Figure 5.14, determine
a model for the structure including all model values. The inductor is made out
of 3-�m-thick aluminum metal. The inductor is suspended over 5 �m of oxide
above a substrate. The underpass is 1-�m aluminum and is 3 �m above the
substrate. Assume the vias are lossless.