6-5-2. Capacitor equivalent circuits
(1) Impedance at high frequencies tends to gather around the same
value
Looking closely at the graph in Fig. 4 shows that the right side
(high frequency side) of the V-shaped curves tend to gather at
around the same location, regardless of the capacitor.
Fig. 5 overlays the impedance for an inductance of 0.5nH (dotted
line) on the graph from Fig. 4 for comparison. Strangely enough, the
measured impedance of the capacitor (right side of the V-shaped
curve) is collected roughly on this line. In other words, the
capacitor measured here (MLCC) shows an inductance of around 0.5nH
at high frequencies.
(2) Equivalent circuits that take ESL into account
This inductance is referred to as the capacitor's ESL (Equivalent
Series Inductance) . To represent a capacitor with ESL in an
equivalent circuit, ESL is connected in series to the electrostatic
capacitance (Cap) , as shown in Fig. 6.
Note that, while an ESL of 0.5nH was used in Fig. 5, this value will
vary depending on the capacitor. MLCCs of the same size were used in
Fig. 5, so the ESL was around the same value. Using different
capacitors will result in significantly different values.
(3) Self-resonance
As mentioned previously, capacitor impedance often forms a
characteristic V-shaped curve, so the minimum point is in the
central portion of the curve. This property is referred to as the
capacitor's self-resonance; this is explained as the series
resonance occurring between Cap and ESL in the equivalent circuit
from Fig. 6. The frequency at the minimum point is called the SRF
(self-resonant frequency) .
Incidentally, calculating the impedance using the equivalent circuit
shown in Fig. 6 results in an impedance of zero at the self-resonant
frequency. In other words, an actual capacitor can express a smaller
impedance than the ideal capacitor at this frequency.
(4) ESR
Of course, an actual capacitor will exhibit a minor amount of loss,
so the impedance will not be exactly zero even at the self-resonant
frequency. In order to represent this loss, the ESR (Equivalent
Series Resistance) , which is the amount of resistance, is normally
included in capacitor equivalent circuits, as shown in Fig. 7.
The smaller the ESR is, the smaller the loss will be, and the better
the capacitor will function. Fig. 8 shows an example of the
impedance values of MLCCs with different ESRs. This shows that the
self-resonant frequency impedance of a temperature compensation
capacitor used in a resonant circuit is much smaller that of a
general-purpose high-dielectric constant capacitor. This is because
a capacitor with temperature compensation characteristics has a
lower ESR.
Note that when a capacitor is self-resonating, its impedance
demonstrates the ESR value of the capacitor. This is because the Cap
and ESL impedance cancel each other out to zero in Fig. 7.
(5) Impedance properties demonstrated by equivalent circuits
Fig. 9 (a) summarizes what has been covered thus far.
The impedance at the lower frequencies is roughly the same as that
of an ideal capacitor. This is because the electrostatic capacitance
impedance accounts for much of the total, so the effect of ESL and
ESR can be ignored. A capacitor in this state is said to be
"capacitive," and its impedance is inversely proportional to the
frequency and electrostatic capacitance.
The impedance at the higher frequencies is roughly the same as the
ESL impedance. This is because the ratio of this impedance is larger
at higher frequencies. The capacitor is said to be "inductive" in
this state, and the impedance is proportional to the frequency.
The self-resonant frequency is the region where the capacitor
switches from capacitive to inductive, and it is also the minimum
point of impedance. The impedance in this state is equivalent to
ESR.
As an example, Fig. 9 (b) shows the result of calculating the
impedance with representative values for a 0.1uF MLCC inserted in an
equivalent circuit. As shown in (a) , the overall impedance follows
the impedance of each element.
(6) How trustworthy are equivalent circuits?
Fig. 10 shows the actual measurements of a 1608 size MLCC overlapped
with the example calculations from Fig. 9 (b) . The figure shows
that, even when calculated using relatively approximate constants,
the LCR series equivalent circuit shown in Fig. 7 closely matches
the measured values and can reproduce the actual
characteristics.
Note that, in order to reproduce actual characteristics more
accurately, the ESR and ESL will need to be changed according to the
frequency. Also, an MLCC was used for explanatory purposes here,
this can also apply to other types of capacitors if the ESL and ESR
values are adjusted.