# Frequency response with capacitors and inductors.

Discussion in 'Homework Help' started by Yakima, Jan 16, 2015.

1. ### Yakima Thread Starter Member

Jan 23, 2012
35
2
It would help any student to realize that capacitors and inductors can be seen as variable resistors that vary as a function of frequency. When you calculate the impedance of a capacitor or an inductor think of the result simply as a resistance. Reflect upon the behavior of resistors in parallel or series.

In parallel resistors create a resistance that is lower than either of the two resistors. In series the resistance is higher than either of the two resistors.

Next, consider that the "resistance" of a capacitor diminishes with increasing frequency and increases with decreasing frequency. An inductor's resistance increases with increasing frequency and diminishes with decreasing frequency.

Given this set of rules, how would you make a first order (just one frequency-dependent element; i.e. , a capacitor or inductor) low pass filter, given a resistor and a frequency dependent device? You have two choices:

1. The input to output consists of a resistor in series and a capacitor to ground, or
2. The input to output consists of resistor in series with an inductor in series.

In the first case an increase in frequency dumps the signal through the capacitor to ground, while in the second case an increase in frequency squeezes off the current through an increasing resistance due to the inductor.

Last edited: Jan 16, 2015
2. ### Yakima Thread Starter Member

Jan 23, 2012
35
2
The next thing to consider when designing with capacitors and inductors is how fast the roll off is. Experimentally, it has been determined that a single frequency element (a capacitor or an inductor) has been determined to roll off at a rate of 20 dB per decade of frequency. And that with two such elements involved the roll off is twice that rate -- 40 dB per decade of frequency. So when you're making a filter the use of two frequency dependent elements produces twice the roll off rate.

The relationships have been made predictable for certain multiple inductor/capacitor configurations and there have been tables compiled with respect to building filters. The two most famous are the Chebychev and Butterworth configurations for which the tables are readily available. Each configuration has characteristics with respect to the transfer function that are a bit different. The designer needs to determine which is more appropriate for his particular need.

Last edited: Jan 16, 2015
3. ### WBahn Moderator

Mar 31, 2012
20,214
5,743
There's no "experimental" about it -- it comes straight out of the theory.

Also, claiming that two frequency dependent elements produces twice the rolloff is far too simplistic. This implies that if I put a second capacitor in series (or a parallel) with the first that I get twice the rolloff when, in fact, all I have done is changed the equivalent capacitance of a single element. Then there is the case of a capacitive voltage divider which has two capacitors yet has a flat frequency response. Or what about an RC high-pass filter that is cascaded with an RC low-pass filter?

4. ### MrAl Distinguished Member

Jun 17, 2014
3,715
788
Hi,

It makes a little sense yes when the reactive elements will be used in filters. The behavior is roughly the same although the true impedance is imaginary so we end with with a slightly different response.

If we consider the capacitor to act as you are saying and for R=1 and C=1 we get:
Vout/Vin=1/(w+1)

and if we consider the more exact response we have:
Vout/Vin=1/sqrt(w^2+1)

So they are similar but not exactly the same. For w=0 then are the same but quickly deviate by up to about 70 percent, until around w=10 where they start to look similar again, and as w gets much larger they look the same again.

So they do act in a similar manner except for a certain range and then they deviate from each other by about 30 percent max.
It might help as long as they know the true response does differ somewhat.

The thing that is not the same at all is the phase shift. The first will not have a phase shift while the second could vary up to 90 degrees.

5. ### MrChips Moderator

Oct 2, 2009
14,494
4,268
Treating capacitance or inductance as a resistance is simplistic.

To solve for currents and voltages you must use complex impedance. This will allow you to determine phase shifts.

The impedance of an inductor of inductance L is

XL = jωL

The impedance of a capacitor of capacitance C is

Xc = 1/(jωC)

Take a look at this:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/impcom.html

6. ### Yakima Thread Starter Member

Jan 23, 2012
35
2
You're right, of course. This is really my seat-of-the-pants way of determining the behavior of circuit. I never memorized the RC, RL or RCL configurations with respect to their response. I just use the rules laid out above when I want to design a first or even a second order filter. I almost never worry about the phase shift with the AM radio circuits I design. I can explain the working of an RLC tank very easily with these rules, even including which frequency dependent device is causing the attenuation above the center frequency and which below the center frequency. I realize this is probably not a post proper to academia.

7. ### Yakima Thread Starter Member

Jan 23, 2012
35
2
You're right. As I explained in a post above, this really only a device of mine to avoid memorizing filter configurations. But it's very useful. I have calculated practical circuit designs that work quite well, and I've wound many a toroid core. I felt compelled to post the OP due to so many students having trouble with frequency response, but now I see my "short cut" method is not suitable for academia. They do need to "worry" about phase shift and I was sloppy in my presentation, as Baun pointed out, making it appear you could just line up a series of inductors and get a sharper roll off. Of course I didn't mean that but it was careless of me.

Last edited: Jan 17, 2015
8. ### Yakima Thread Starter Member

Jan 23, 2012
35
2
The design "rules" I gave above are for the case where I'm not worried about the phase shift. And that is the case more often than not. I've been working mostly with AM radio where phase shift is irrelevant. When working with FM there is the need, but not in the front end or the IF strip; the receiver employs demodulation technique involving phase shift, and then I have to be mindful of it.

9. ### MrChips Moderator

Oct 2, 2009
14,494
4,268
Fine, if you are not interested in phase shifts.
If you want to calculate the attenuation you cannot use the voltage divider formula. Use vector analysis instead.

That is because resistance and reactance are 90° out of phase.

10. ### MrAl Distinguished Member

Jun 17, 2014
3,715
788
Hi,

Yeah, it depends on how complicated the circuit is. For a simple circuit where you want a quick simple idea what is going on (like one resistor and one capacitor) then you are probably ok for the most part, but there are circuits where it wont work at all. For example, a circuit that uses an inductor and capacitor to generate phase shifts, then adds the responses together to produce a zero voltage output. Or to put it even simpler, two out of phase circuits with the same amplitude but when their outputs are added together the result is zero. The very reason they cancel is because of the phase alone so that would be a necessary thing to be able to determine.
So some circuits might be ok with this idea as a rough approximation and some wont be ok.

11. ### WBahn Moderator

Mar 31, 2012
20,214
5,743
Even in AM circuits, think of your LC tank. At the resonant frequency the impedances of both are very finite and are the same magnitude. The only difference is the phase shift. Yet what a difference that makes!