Ideal Vs Practical Resonators
A resonant circuit is a circuit containing both an inductor and a capacitor — and therefore, both inductive and capacitive reactances. Its purpose is to produce a reflection zero that will ensure maximum transmission of the signal at a particular frequency. At this frequency, the impedance of the resonant circuit is at its lowest, allowing better signal throughput. A resonant circuit generally looks like below:
For any such circuit, there is a specific frequency at which the inductive and capacitive reactances are the same, that is
where f = frequency in Hertz (Hz), L = inductance in Henry (H), and C = capacitance in Farad (F), and π = 3.1416.
If we are to solve for f, taking only the positive root, we can find the resonant frequency of any combination of inductor and capacitor from the formula
The units of inductance (L) and capacitance © are micro- or nano-Henry ( μH or nH) and pico-Farad (pF) for most high-frequency radio or microwave which are in the megahertz ( MHz) or gigahertz ( GHz) range respectively.
Suppose, we want to create a resonant circuit at 50 MHz, if the inductor is 1 μ H, then we can calculate the capacitor in pF using the formula
Hence,
Using these values will yield the result as shown below. Notice the reflection zero at 50MHz resulting in almost no return loss.
Based on the formula, increasing the capacitor value or inductor value will move the resonant frequency to lower. We do this by making C = 20 pF.
In the beginning of the article, remember we mentioned that at resonant frequency, the impedance of the circuit is at its lowest. By adding a series resistance, notice how it causes the performance at the reflection zero to be worst due to the added loss which is generally true for all practical cases.
Now let’s move to the Z-parameters plot. For this plot we will make it a one port network by terminating another port because we are interested only in the input impedance.
Moving on to Z-plot. At 50 MHz, the imaginary part of the input impedance is zero (resonant), below resonant is positive i.e. inductor, and above resonant it is negative which is a capacitor. The resonant frequency at 50 MHz is the frequency at which the inductive reactance and capacitive reactance are equal and cancel on another.
This condition occurs in the inductor, it is a parallel resonant circuit which results in a very high real impedance as shown below. This should not be confused with the impedance of the two-port schematic. Where at resonant frequency, the impedance of the schematic is at its lowest, allowing maximum throughput, the Z-parameter response depicts the impedance of the reflection of the inductor. Since the impedance is high at Z11, this translates to a “high blockage” preventing reflection of signal back to its source, ensuring the signal is transmitted.
So how do we make use of this resonant concept in electrical component such as resistor, inductor, and capacitor?
Let’s take a look at an inductor
The plot show that the imaginary part is positive because inductor is
Therefore
at all frequency. Let’s try to extend the range and you could see that for the ideal inductor, the imaginary part is always positive.
It is worth taking note that the real inductor, one which you bought from the market is not an ideal component or a pure inductance. It is a network that includes parasitic capacitance and resistance which could be from the way the manufacturer package it or how we solder them in the circuit. The simple RLC network could be the representation of the inductor as shown below:
The resistor is representation of the impedance of the inductor (every real world element will have a small value of resistance). The parallel capacitor is representation of the parasitic capacitance that comes with the inductor element. One should note that based on the Z-parameter plot you can see that the real inductor has the resonant frequency called “Self-Resonant Frequency” or “SRF” above which it becomes negative which is a capacitor. The resonant frequency or the SRF is when the inductor and capacitor impedances cancel each other and remains as pure real as can see from the plot of Z-paramter is that the imaginary becomes zero. We can also observe using EDS software, the phase plot which shows that the positive phase is below the SRF and negative phase is above the SRF and zero phase is the SRF.
The S-parameter plot could show very clearly the SRF is at 50 MHz where the transmission coefficient is zero (S12 or S21 in dB is infinity) which is true from the Z-parameter plot which has a very high impedance Z11. With this EDS software plot, we could conclude that the inductor’s SRF is the parallel resonant frequency; not the series resonant frequency.
This concept is very important in designing filter or any microwave circuit especially when using lumped elements to realize them. Because the one we synthesis and their values are the ideal scenarios while the ones we can buy from the store or use to construct the device are actual components which have parasitic and its interference. Therefore, as the designer, we need to make sure that if that’s the inductor we design, it should really behave as inductor because after SRF, the inductor actually becomes a capacitor so you ended up having a capacitor rather than the inductor!
Thanks for all the manufacturers, they usually specify them in the datasheet of the component itself. That’s is why we need to include or at least consider all of these in our design before we come to constructing it as product or prototype or you will be surprised in what you are getting at the end. This could save time, money, and your reputation in delivering the right product specification.
Originally published at http://filpal.wordpress.com on April 4, 2021.
