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I'm preparing to teach some people regarding filters, and I'm confused regarding the use of the LC-filter circuit, specifically the circuit below:

schematic

simulate this circuit – Schematic created using CircuitLab

Taking the output at the capacitor, it seems you get a strange looking low pass filter with the transfer function

\$H(\omega)=1/(1-(\omega^2LC)\$

It's similar to a RC circuit, but at the cutoff frequency, there's a spike in the magnitude. With the basic RC/RL filter circuits, the cutoff frequency occurs when the power is 3 dB below the max value, but with the LC circuit, calculating that leads to a very long equation to the 4th power. With that, why do we calculate the resonance frequencies when the denominator is equal to 0? Similarly, since there's no imaginary components, what does that say about the phase of the circuit with regards to frequency?

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  • \$\begingroup\$ With Zs(f)=0 and no load you get infinite Q at\$\omega_0\$ \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Feb 8 at 23:38
  • \$\begingroup\$ You better learn impedance ratios \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Feb 9 at 0:05
  • \$\begingroup\$ Use this to design your own filter falstad.com/afilter \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Feb 9 at 0:41
  • \$\begingroup\$ If \$d\$ is the damping value (not unitless and it should not be confused with the damping factor or damping ratio, \$\zeta\$, which is the unitless ratio between the damping value used below and the critical damping value), then: $$\frac{e_\text{out}}{e_\text{in}}=20\operatorname{log}_{10}\left(\omega^4+\left(d^2-2\right)\omega^2+1\right)$$ and, $$\phi=-\operatorname{tan}^{-1}\left(\frac{d\:\omega}{1-\omega^2}\right)$$To find the peak, you just take the derivative. So \$2\omega^2+d^2-2=0\$ (after reduction.) \$\endgroup\$ – jonk Feb 9 at 4:40
  • \$\begingroup\$ Therefore, \$\omega_\text{max}=\sqrt{1-\frac{d^2}{2}}\$ (if a peak exists, of course.) \$\endgroup\$ – jonk Feb 9 at 4:46
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Your filter is an ideal resonant circuit - practically non-existent! Components, especially inductors are lossy and signal sources also have something, typically a series resistance. Finally there's a load in the output. It also takes something. The resulted circuit no more has a pole (=zero denominator) at the resonant frequency. If there's low enough resistance load resistor, this circuit can be a 2nd order low pass filter.

This filter pulls the signal source output on the knees at the resonant frequency if there's some resistance in the source and the output is unloaded. It's extremely important to understand it qualitatively - the LC series impedance is lowest at the resonant frequency and the resistance of the signal source makes with it a frequency dependent voltage divider. You can calculate what's left at the output, but you can see it also by simulating.

You should do some simulations to see the effect of the various resistances. Even this site has a good enough simulator.

Calculations with equations really are complex, but doable. Advanced filter designers know some shortcuts how to use already calculated filter tables, how to transform them to other frequencies and how to change component values for different signal source and load resistances. The theory has been there already about 80 years. It's packed also in filter design software and printed to handbooks.

For teaching you maybe should show results and the methods qualitatively. Don't alienate people with heavy equation manipulations. More likely let them find something wanted with simulations.

If transfer function manipulation with complex numbers is the main subject, then simulation can be useful to reveal errors.

You can calculate and plot transfer functions with Excel, but simulator can show the frequency response of a circuit without forcing one to write complex equations which are error prone.

Time domain analysis with simulators, Excel and formally with Laplace transforms can be interesting, but have you time for it? You know. My opinion is that you should show with simulations that too high resonant peak in the frequency response causes too much unwanted ringing when the signal has pulses.

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  • \$\begingroup\$ Thank you for your reply. Your explanation really gives me an idea on how to start. Honestly, for the scope of my lecture, it's really the tip of the iceburg here (an introduction lecture), with the audience not really going into EE. In a sense, I'm trying to basically get them interested to show how it relates to real-world applications, such as working with audio signals, so that's roughly the scope of this particular lecture. \$\endgroup\$ – user101402 Feb 11 at 14:15
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Here is another view of LRC filters, with 4 placed in parallel, all fed from a shared50 ohm resistor. This use of multiple caps in parallel is often found in VDD bypassing on microcontrollers and on analog circuits (and on switch-regs).

Notice the several resonances: peaks and valleys.

enter image description here

Note that each capacitor has the same (small parasitic) inductance and resistance.

Note 0.01 ohm (10 milliOhms) does good job of dampening the larger capacitors, yet is wholly useless for the 2 smallest caps: their peaks at 15Mhz and 150MHz are 30dB stronger than the baseline increase (which comes from the 10nanoHenry inductor in each cap)

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The main concept of LC filters depends on the impedance equations for the source, load and Z of each reactance. Computing just as any other 2 element tap from a true 0 ohm voltage source.

Here , There is one frequency when the impedance of the Inductor is the exact opposite of the capacitor, both conducting voltage out of phase +90 deg , -90deg yet sharing the same current. When this happens with identical magnitude impedance in series (between 0 Ohms of the theoretical Voltage source and 0 Ohms of a theoretical ground,) both elements create a perfect 0 Ohm short circuit with no reactance ( as they cancel) with an infinite current loop and infinite voltage at the midpoint ( or output here).

In practice since there is always some wire and ground and component DC resistance or DCR in the magnet wire and ESR in the capacitor electrode interface, the current is never infinite, meaning the Q is never infinite as the gain of voltage ratio to source at this 1 very precise frequency. Practical Q of 100 is possible but difficult to control. Q of 1000 is extremely difficult if not impossible for most.

But it is useful to make oscillations grow for oscillators or a combination low pass + resonant bandpass filter. The only way to dampen this Q is to apply a load resistance and if that ratio equals 1 you get a damping factor which still has ringing and does not become critically damped with no overshoot until the resistance is equal to 0.707.

At this point the resonant frequency is shifted slightly due the impedance damping effect of R so that R and C share the same current from L so that when L and C impedances are equal, the voltage rise is damped by the R load and so the voltage overshoot can be minimized or eliminated depending on the impedance ratios of R to X at resonance for either L or C. This is important for passive crossover speakers.

However RF filters use LC components and often complex filters with many many LC components can be computed for a maximally flat frequency response or linear phase shift or even for data for ringing at the data period regardless of 1T, 2T, 3T/2 frequencies ( Raised Cosine Filters). for example a Chebychev maximally flat filter has much high Q’s than 1 but with shifted poles or resonant points make the bumps equally spaced and as small as possible (0.1 dB or 3dB) with the advantage of the higher Q skirt bandstop.

Not all inductors of the same value are identical since every component be it a cap or resistor or inductor or wire or air has some measurable capacitance, resistance and inductance even if beyond your measurement skills. The are called parasitics, because a wire lead or trace is an inductor 0.5~1 nH/mm roughly, and air or FR4 between close pads and tracks can be a capacitor at some pF/cm depending on geometric ratios of conductors. So when it comes down to high currents or wide bandwidth , these real parameters come into effect.

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