Cut-off Frequency (Electrical Engineering)

In summary, the equations given in the book for wc for L/H pass filters are applicable but not always easy to find. They can be derived from generic examples but the cutoff frequency can be calculated using the cutoff frequency definition.
  • #1
jghlee
16
0
So this is a pretty general question regarding the cut-off frequency for any filter. In my study of filters, I've come across certain equations that the book gives for wc.

For example, Low/High Pass filters have wc = 1/(RC) or R/L, and Bandpass/stop filters have the center frequency wo^2 = 1/(LC). I'm able to derive these equations from some pretty generic L/H Pass Filter examples so I know where they come from.

Now my question is, do these equations apply to any filters of its kind? Meaning, can I always find the wc for a L/H pass filter by simply plugging in the 1/(RC) or R/L values? I'm terrible confused because all of my course homework seems to point this out but I'm not too sure in situations for when there's multiple L/C/R's or when I have RLC filters in which case a simple 1/(RC) doesn't seem to accurately depict the wc since it's missing the L value..

If somebody could clarify the meaning of those wc and wo equations, I'd appreciate it very much.
 
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  • #2
They don't apply to all filters but often apply.

The most general statement you can make, that always applies (by convention, not axiom) is that the cutoff wavelength is where the transmission is 3 dB down. The exact (to maybe 7 figures) cutoff wavelength is at 3.0103 dB down. This seems arcane but it's very simple.

If you have a simple voltage divider Z1/(Z1+Z2), the 3dB point is where |Z1|=|Z2|. Put in any element for Z1 and 2 to make any simple first order filter. The higher order filters also follow the 3dB convention even though it is less obvious that it would make sense.
 
  • #3
First off, thanks for reply!

So basically the sure way of knowing what the cut-off frequency is essentially using the cutoff frequency definition where H(jw) = Hmax / (root 2).

This leads me to another question. What are the purposes of these "general" equations like H(s) = (R/L) / [s + (R/L)] or H(s) = (1/RC) / [s + (1/RC)] in the case of LPF? Is there a special meaning in working/deriving our frequency response to look like this? Seems like for LPF/HPF we always want to get that s by itself in the denominator. My hunch is that, whenever you get a freq response equation that fits this mold, you're able to use those generic wc and wo values...
 
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  • #4
The magnitude of frequency response of the system with transfer function H(s) is equal to |H(jω)|.

if one takes

[tex]H(s) = \frac{1/RC}{s + 1/RC}[/tex]

and substitutes jω for s we have

[tex]H(jω) = \frac{1/RC}{jω + 1/RC}[/tex]

When ω = 1/RC, the denominator is a complex number with the real part equal in magnitude to the imaginary part.

[tex]H(jω) = \frac{1/RC}{ (1/RC)(j + 1)}[/tex]The magnitude of the denominator with ω = 1/RC is sqrt(2) times what it is when ω = 0.
 
  • #5
Not sure if this helps, but a quick rule you can apply to any circuit network (I think) is an LC in parallel has infinite impedance at w = 1/sqrt(lc) and one in series as zero impedance at that w. So for simple networks, you can figure out the passband/stopband instantly by replacing the LCs with shorts/opens in your mind and asking yourself if the signal fully passes or fully stops at w = 1/sqrt(LC).
 

1. What is the cut-off frequency?

The cut-off frequency in electrical engineering refers to the frequency at which a filter or circuit begins to attenuate a signal. It is the point at which the output signal drops to a certain percentage (usually 50%) of the input signal.

2. How is the cut-off frequency calculated?

The cut-off frequency can be calculated using the formula fc = 1/(2πRC), where fc is the cut-off frequency, R is the resistance in ohms, and C is the capacitance in farads. This formula applies to low-pass filters, while different formulas may be used for high-pass, band-pass, or band-stop filters.

3. What is the relationship between the cut-off frequency and the bandwidth?

The cut-off frequency and bandwidth are closely related. The bandwidth is the range of frequencies that a filter or circuit allows to pass through, while the cut-off frequency is the point at which the signal begins to be attenuated. In a filter with a sharp cut-off, the cut-off frequency and bandwidth will be close together, while in a filter with a gradual transition, they may be farther apart.

4. How does the cut-off frequency affect the frequency response of a system?

The cut-off frequency plays a significant role in the frequency response of a system. In a low-pass filter, frequencies below the cut-off frequency will pass through with little attenuation, while frequencies above the cut-off will be significantly attenuated. In a high-pass filter, the opposite is true. The cut-off frequency can also affect the shape and steepness of the filter's frequency response curve.

5. How does the choice of components affect the cut-off frequency of a filter?

The choice of components, specifically the values of resistance and capacitance, can greatly affect the cut-off frequency of a filter. Higher values of resistance and capacitance will result in a lower cut-off frequency, while lower values will result in a higher cut-off frequency. Additionally, the type and design of the filter will also impact the cut-off frequency and may require different component values for a desired cut-off frequency.

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