Self Resonance Frequency of an Inductor

The self resonance frequency (SRF) of an inductor is the frequency at which resonance between the inductance and the parasitic capacitance of the inductor occurs. At the SRF, the inductor looks like an open circuit. In this article, we’ll examine why this happens.

An ideal inductor has only inductance associated with it, but of course a real inductor has some resistance in the windings as well as capacitance between the layers of the windings and a little bit between the leads of the inductor. These non-ideal characteristics lead to an inductor model that looks like this:

Inductor Model

Figure 1. Inductor Model Showing Coil Resistance and Parasitic Capacitance

The part of this diagram to note when considering self resonance frequency is the fact that you have the inductance in series with the coil resistance and in parallel with the parasitic capacitance. To help in our understanding of the SRF, let’s start off ignoring the coil resistance which gives a circuit like this where the inductance is in parallel with the parasitic capacitance:

Another Inductor model

Figure 2. Inductor Model Showing Only Parasitic Capacitance

The impedance of this circuit is therefore:

\vec{Z_T} = \frac{(X_L \angle 90)(X_C \angle -90)}{jX_L + jX_C} = \frac{(X_L)(X_C)}{j(X_L-X_C)}

XL and XC are both frequency dependent (XL increases with frequency and XC decreases) and as they get closer and closer together, the denominator gets closer and closer to 0 which means the magnitude of the impedance gets bigger and bigger, approaching infinity.

The frequency at with this occurs can be easily calculated:

X_L = X_C

\omega L  = \frac{1}{\omega C}

\omega^2 = \frac{1}{LC}  \rightarrow \omega = \frac{1}{\sqrt{LC}}   or   f = \frac{1}{2\pi \sqrt{LC}}

This frequency is the resonant frequency and when using the inductance and the parasitic capacitance of an inductor in this equation, you get the self resonance frequency of the inductor. At this frequency, the inductor looks like an open circuit and can be used as a simple choke to block frequencies at the SRF.

Including the Coil Resistance

When including the coil resistance, we get a bit better modelling of the SRF. The calculation is a little more complicated, but not much.

Inductor Model

Figure 3. Inductor Model Showing Coil Resistance and Parasitic Capacitance

It is possible to convert the above circuit into an equivalent circuit like this:

Inductor Model C||R||L

Figure 4. Inductor Model Showing Parallel R and L Equivalents

It is beyond the scope of this article to explain why, but the calculations for determining RP; and XLP are:

R_P = \frac{(R_{coil})^2 + (X_L)^2}{R_{coil}}

X_{LP} = \frac{(R_{coil})^2 + (X_L)^2}{X_L}

In this equivalent circuit, resonance occurs when:

X_C = X_{LP}

X_C = \frac{(R_{coil})^2 + (X_L)^2}{X_L}

\frac{1}{\omega C} = \frac{(R_{coil})^2 + (\omega L)^2}{\omega L}

\frac{L}{C} = (R_{coil})^2 + (\omega L)^2

Solving for ω:

\omega = \sqrt{\frac{1}{LC} - \frac{(R_{coil})^2}{L^2}}     or

\omega = \frac{1}{\sqrt{LC}}\sqrt{1-\frac{(R_{coil})^2C}{L^2}}

This result is pretty similar to the first one when the coil resistance was ignored except for the \sqrt{1-\frac{(R_{coil})^2C}{L^2}} factor.

In practice though, a simulation tool (such as PSPICE) is typically used to determine the SRF (because there are some added complications like the inductance and the parasitic capacitance changing with frequency) but this article gives you an idea of the theory behind it.

Example

A 1μH inductor has a coil resistance of 0.5Ω and parasitic capacitance of 213fF.

Ignoring coil resistance gives an SRF of:

SRF = \frac{1}{2 \pi \sqrt{LC}} = \frac{1}{2 \pi \sqrt{1\mu H \times 213fF}} = 345MHz

Including coil resistance gives an SRF of:

SRF = 345MHz \times \sqrt{1-\frac{0.05^2 \times 213fF}{(1\mu H)^2}} =

345MHz \times 0.999 = 344.7MHz

The coil resistance and parasitic capacitance are so small there is hardly any difference between the two calculation methods. Bode plots of the change of impedance over frequency also show similar results (around the SRF) as you can see in the graphs below (click on the images for a better view):

Figure 5. Impedance over Frequency for LC Impedance Model

 

Figure 6. Impedance over Frequency For LRC Impedance Model