Impedance characteristics of power supply decoupling capacitors

Today we will look at the impedance characteristics of capacitors in power supply decoupling applications.

Schematic diagram of capacitors in PDN (power distribution network)

To understand how capacitors reduce PDN impedance, we must first understand the impedance characteristics of the capacitors themselves and the influence of their own parasitic parameters.

Without further ado, let's first think about what the impedance of a pure capacitor is like?

We might think of a formula like this:

Z=1/(2*pi*f*c)

Yes, but this does not help us understand the role of capacitors in PDN, and it is not intuitive enough;

Let's think about it again. What would it look like if we drew a curve of the impedance of a capacitor VS frequency ?

As shown below:

This is the impedance VS frequency curve of pure capacitance. In the curve, we can intuitively see the relationship between capacitance impedance and frequency;

Let's take a look at the impedance curves of capacitors with different capacitance values:

The larger the capacitance, the smaller the impedance, and the entire curve shows an overall downward shift;

Because capacitors have low impedance characteristics for high-frequency signals, many decoupling capacitors are added to the PDN system. However, as we have already introduced, capacitors also have parasitic inductance (ESL) and equivalent series resistance (ESR), so we still have to study the impact of parasitic parameters.

Next, let's take a look at the impedance characteristics of pure inductance. The formula is as follows;

Z=2*pi*L*f

Similarly, let's look at the impedance VS frequency curve of the inductor:

Through the curve, we can intuitively see the relationship between the impedance of the inductor and the frequency;

Let's take a look at the impedance curves of different inductance values. I believe that everyone has a direct impression at this point and can imagine the curves of different inductances:

Next is the resistor. The impedance VS frequency curve of a pure resistor is as follows:

There is nothing much to say about this, please read on.

Here comes the key point

Now that we know the impedance characteristics of a single device, what does it look like when they are connected in series? Let's look at the series connection of resistors and capacitors:

It can be seen that after RC is connected in series, the low-frequency impedance is capacitive, and its size is the impedance size of the series capacitor, and the high-frequency part is resistive, and its size is the impedance size of the resistor. Please see the series connection of resistor and inductor below:

It can be seen that after RL is connected in series, the impedance of the low-frequency part is resistive, and its size is the impedance of the resistor. The impedance of the high-frequency part is inductive, and its size is the impedance of the series inductor. Please see the series connection of capacitor and inductor below:

It can be seen that after LC is connected in series, the impedance of the low-frequency part is capacitive, and its size is the impedance of the capacitor. The impedance of the high-frequency part is inductive, and its size is the impedance of the series inductor. One point different from the connection with the resistor is that the LC series connection will form resonance. The frequency of the resonance point can be calculated as follows:

At the resonance point, the impedance of the capacitor is equal to the impedance of the inductor, that is,

Z = 1/(2*pi*f*c) = 2*pi*L*f

Derived f = 1/(2*pi*)

After knowing the capacitance and inductance values, we can find the resonant frequency. Before this frequency, it is capacitive, and after this frequency point, it is inductive. The phase in the figure below illustrates this point.

Well, the above are the basics, all of which are preparations for a better understanding of the impedance characteristics of non-ideal capacitors. Now let's look at the impedance characteristics of real capacitors, that is, RLC series (simple model)

The curve after RLC series connection is similar to that of LC. It also has a self-resonance frequency point and the calculation method is the same. It is capacitive before the self-resonance frequency point and inductive after the self-resonance frequency point. There is also resonance between LC. The difference is that the resonance size at this time is controlled and the size is equal to the size of the equivalent series resistance.

This is the impedance characteristic of non-ideal capacitors. You must remember it. It is precisely because of the low impedance characteristics of capacitors at high frequencies that we choose capacitors to reduce PDN impedance. However, due to the parasitic parameters of capacitors, they become inductive after a certain frequency band. Therefore, we need to select capacitors with appropriate self-resonance frequency. The following compares the impedance curves of several non-ideal capacitors. The curves come from the spice model on the official website of the capacitor manufacturer:

The above are several groups of capacitor impedance curves with different capacitance values, from 10nF to 100uF. It can be seen that within this range (all are chip ceramic capacitors), the parasitic inductance is basically the same. However, due to the different capacitance values, their self-resonant frequencies are different. Therefore, when decoupling in PDN, it is necessary to choose the appropriate capacitor.

In addition to the capacitor's own self-resonant frequency, different capacitors will also have anti-resonant frequencies when used in parallel.

From: Sig0 08