1. LIEC (todo)
2. What is a Filter?
It is sometimes desirable to have circuits capable of selectively filtering one frequency or range of frequencies out of a mix of different frequencies in a circuit. A circuit designed to perform this frequency selection is called a filter circuit, or simply a filter. A common need for filter circuits is in highperformance stereo systems, where certain ranges of audio frequencies need to be amplified or suppressed for best sound quality and power efficiency. You may be familiar with equalizers, which allow the amplitudes of several frequency ranges to be adjusted to suit the listener’s taste and acoustic properties of the listening area. You may also be familiar with crossover networks, which block certain ranges of frequencies from reaching speakers. A tweeter (highfrequency speaker) is inefficient at reproducing lowfrequency signals such as drum beats, so a crossover circuit is connected between the tweeter and the stereo’s output terminals to block lowfrequency signals, only passing highfrequency signals to the speaker’s connection terminals. This gives better audio system efficiency and thus better performance. Both equalizers and crossover networks are examples of filters, designed to accomplish filtering of certain frequencies.
Another practical application of filter circuits is in the "conditioning" of nonsinusoidal voltage waveforms in power circuits. Some electronic devices are sensitive to the presence of harmonics in the power supply voltage, and so require power conditioning for proper operation. If a distorted sinewave voltage behaves like a series of harmonic waveforms added to the fundamental frequency, then it should be possible to construct a filter circuit that only allows the fundamental waveform frequency to pass through, blocking all (higherfrequency) harmonics.
We will be studying the design of several elementary filter circuits in this lesson. To reduce the load of math on the reader, I will make extensive use of SPICE as an analysis tool, displaying Bode plots (amplitude versus frequency) for the various kinds of filters. Bear in mind, though, that these circuits can be analyzed over several points of frequency by repeated seriesparallel analysis, much like the previous example with two sources (60 and 90 Hz), if the student is willing to invest a lot of time working and reworking circuit calculations for each frequency.

REVIEW:

A filter is an AC circuit that separates some frequencies from others within mixedfrequency signals.

Audio equalizers and crossover networks are two wellknown applications of filter circuits.

A Bode plot is a graph plotting waveform amplitude or phase on one axis and frequency on the other.
2.1. LowPass Filters
By definition, a lowpass filter is a circuit offering easy passage to lowfrequency signals and difficult passage to highfrequency signals. There are two basic kinds of circuits capable of accomplishing this objective, and many variations of each one: The inductive lowpass filter in Figure below and the capacitive lowpass filter in Figure below
Inductive lowpass filter
The inductor’s impedance increases with increasing frequency. This high impedance in series tends to block highfrequency signals from getting to the load. This can be demonstrated with a SPICE analysis: (Figure below)
inductive lowpass filter v1 1 0 ac 1 sin l1 1 2 3 rload 2 0 1k .ac lin 20 1 200 .plot ac v(2) .end
The response of an inductive lowpass filter falls off with increasing frequency.
Capacitive lowpass filter.
The capacitor’s impedance decreases with increasing frequency. This low impedance in parallel with the load resistance tends to short out highfrequency signals, dropping most of the voltage across series resistor R_{1}. (Figure below)
capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u rload 2 0 1k .ac lin 20 30 150 .plot ac v(2) .end
The response of a capacitive lowpass filter falls off with increasing frequency.
The inductive lowpass filter is the pinnacle of simplicity, with only one component comprising the filter. The capacitive version of this filter is not that much more complex, with only a resistor and capacitor needed for operation. However, despite their increased complexity, capacitive filter designs are generally preferred over inductive because capacitors tend to be "purer" reactive components than inductors and therefore are more predictable in their behavior. By "pure" I mean that capacitors exhibit little resistive effects than inductors, making them almost 100% reactive. Inductors, on the other hand, typically exhibit significant dissipative (resistorlike) effects, both in the long lengths of wire used to make them, and in the magnetic losses of the core material. Capacitors also tend to participate less in "coupling" effects with other components (generate and/or receive interference from other components via mutual electric or magnetic fields) than inductors, and are less expensive.
However, the inductive lowpass filter is often preferred in ACDC power supplies to filter out the AC "ripple" waveform created when AC is converted (rectified) into DC, passing only the pure DC component. The primary reason for this is the requirement of low filter resistance for the output of such a power supply. A capacitive lowpass filter requires an extra resistance in series with the source, whereas the inductive lowpass filter does not. In the design of a highcurrent circuit like a DC power supply where additional series resistance is undesirable, the inductive lowpass filter is the better design choice. On the other hand, if low weight and compact size are higher priorities than low internal supply resistance in a power supply design, the capacitive lowpass filter might make more sense.
All lowpass filters are rated at a certain cutoff frequency. That is, the frequency above which the output voltage falls below 70.7% of the input voltage. This cutoff percentage of 70.7 is not really arbitrary, all though it may seem so at first glance. In a simple capacitive/resistive lowpass filter, it is the frequency at which capacitive reactance in ohms equals resistance in ohms. In a simple capacitive lowpass filter (one resistor, one capacitor), the cutoff frequency is given as:
Inserting the values of R and C from the last SPICE simulation into this formula, we arrive at a cutoff frequency of 45.473 Hz. However, when we look at the plot generated by the SPICE simulation, we see the load voltage well below 70.7% of the source voltage (1 volt) even at a frequency as low as 30 Hz, below the calculated cutoff point. What’s wrong? The problem here is that the load resistance of 1 kΩ affects the frequency response of the filter, skewing it down from what the formula told us it would be. Without that load resistance in place, SPICE produces a Bode plot whose numbers make more sense: (Figure below)
capacitive lowpass filter v1 1 0 ac 1 sin r1 1 2 500 c1 2 0 7u * note: no load resistor! .ac lin 20 40 50 .plot ac v(2) .end
For the capacitive lowpass filter with R = 500 Ω and C = 7 µF, the Output should be 70.7% at 45.473 Hz.
fcutoff = 1/(2πRC) = 1/(2π(500 Ω)(7 µF)) = 45.473 Hz
When dealing with filter circuits, it is always important to note that the response of the filter depends on the filter’s component values and the impedance of the load. If a cutoff frequency equation fails to give consideration to load impedance, it assumes no load and will fail to give accurate results for a reallife filter conducting power to a load.
One frequent application of the capacitive lowpass filter principle is in the design of circuits having components or sections sensitive to electrical "noise." As mentioned at the beginning of the last chapter, sometimes AC signals can "couple" from one circuit to another via capacitance (C_{stray}) and/or mutual inductance (M_{stray}) between the two sets of conductors. A prime example of this is unwanted AC signals ("noise") becoming impressed on DC power lines supplying sensitive circuits: (Figure below)
Noise is coupled by stray capacitance and mutual inductance into "clean" DC power.
The oscilloscopemeter on the left shows the "clean" power from the DC voltage source. After coupling with the AC noise source via stray mutual inductance and stray capacitance, though, the voltage as measured at the load terminals is now a mix of AC and DC, the AC being unwanted. Normally, one would expect E_{load} to be precisely identical to E_{source}, because the uninterrupted conductors connecting them should make the two sets of points electrically common. However, power conductor impedance allows the two voltages to differ, which means the noise magnitude can vary at different points in the DC system.
If we wish to prevent such "noise" from reaching the DC load, all we need to do is connect a lowpass filter near the load to block any coupled signals. In its simplest form, this is nothing more than a capacitor connected directly across the power terminals of the load, the capacitor behaving as a very low impedance to any AC noise, and shorting it out. Such a capacitor is called a decoupling capacitor: (Figure below)
Decoupling capacitor, applied to load, filters noise from DC power supply.
A cursory glance at a crowded printedcircuit board (PCB) will typically reveal decoupling capacitors scattered throughout, usually located as close as possible to the sensitive DC loads. Capacitor size is usually 0.1 µF or more, a minimum amount of capacitance needed to produce a low enough impedance to short out any noise. Greater capacitance will do a better job at filtering noise, but size and economics limit decoupling capacitors to meager values.

REVIEW:

A lowpass filter allows for easy passage of lowfrequency signals from source to load, and difficult passage of highfrequency signals.

Inductive lowpass filters insert an inductor in series with the load; capacitive lowpass filters insert a resistor in series and a capacitor in parallel with the load. The former filter design tries to "block" the unwanted frequency signal while the latter tries to short it out.

The cutoff frequency for a lowpass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is lower than 70.7% of the input, and vice versa.
2.2. HighPass Filters
A highpass filter’s task is just the opposite of a lowpass filter: to offer easy passage of a highfrequency signal and difficult passage to a lowfrequency signal. As one might expect, the inductive (Figure below) and capacitive (Figure below) versions of the highpass filter are just the opposite of their respective lowpass filter designs:
Capacitive highpass filter.
The capacitor’s impedance (Figure above) increases with decreasing frequency. (Figure below) This high impedance in series tends to block lowfrequency signals from getting to load.
capacitive highpass filter v1 1 0 ac 1 sin c1 1 2 0.5u rload 2 0 1k .ac lin 20 1 200 .plot ac v(2) .end
The response of the capacitive highpass filter increases with frequency.
Inductive highpass filter.
The inductor’s impedance (Figure above) decreases with decreasing frequency. (Figure below) This low impedance in parallel tends to short out lowfrequency signals from getting to the load resistor. As a consequence, most of the voltage gets dropped across series resistor R_{1}.
inductive highpass filter v1 1 0 ac 1 sin r1 1 2 200 l1 2 0 100m rload 2 0 1k .ac lin 20 1 200 .plot ac v(2) .end
The response of the inductive highpass filter increases with frequency.
This time, the capacitive design is the simplest, requiring only one component above and beyond the load. And, again, the reactive purity of capacitors over inductors tends to favor their use in filter design, especially with highpass filters where high frequencies commonly cause inductors to behave strangely due to the skin effect and electromagnetic core losses.
As with lowpass filters, highpass filters have a rated cutoff frequency, above which the output voltage increases above 70.7% of the input voltage. Just as in the case of the capacitive lowpass filter circuit, the capacitive highpass filter’s cutoff frequency can be found with the same formula:
In the example circuit, there is no resistance other than the load resistor, so that is the value for R in the formula.
Using a stereo system as a practical example, a capacitor connected in series with the tweeter (treble) speaker will serve as a highpass filter, imposing a high impedance to lowfrequency bass signals, thereby preventing that power from being wasted on a speaker inefficient for reproducing such sounds. In like fashion, an inductor connected in series with the woofer (bass) speaker will serve as a lowpass filter for the low frequencies that particular speaker is designed to reproduce. In this simple example circuit, the midrange speaker is subjected to the full spectrum of frequencies from the stereo’s output. More elaborate filter networks are sometimes used, but this should give you the general idea. Also bear in mind that I’m only showing you one channel (either left or right) on this stereo system. A real stereo would have six speakers: 2 woofers, 2 midranges, and 2 tweeters.
Highpass filter routes high frequencies to tweeter, while lowpass filter routes lows to woofer.
For better performance yet, we might like to have some kind of filter circuit capable of passing frequencies that are between low (bass) and high (treble) to the midrange speaker so that none of the low or highfrequency signal power is wasted on a speaker incapable of efficiently reproducing those sounds. What we would be looking for is called a bandpass filter, which is the topic of the next section.

REVIEW:

A highpass filter allows for easy passage of highfrequency signals from source to load, and difficult passage of lowfrequency signals.

Capacitive highpass filters insert a capacitor in series with the load; inductive highpass filters insert a resistor in series and an inductor in parallel with the load. The former filter design tries to "block" the unwanted frequency signal while the latter tries to short it out.

The cutoff frequency for a highpass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is greater than 70.7% of the input, and vice versa.
2.3. BandPass Filters
There are applications where a particular band, or spread, or frequencies need to be filtered from a wider range of mixed signals. Filter circuits can be designed to accomplish this task by combining the properties of lowpass and highpass into a single filter. The result is called a bandpass filter. Creating a bandpass filter from a lowpass and highpass filter can be illustrated using block diagrams: (Figure below)
System level block diagram of a bandpass filter.
What emerges from the series combination of these two filter circuits is a circuit that will only allow passage of those frequencies that are neither too high nor too low. Using real components, here is what a typical schematic might look like Figure below. The response of the bandpass filter is shown in (Figure below)
Capacitive bandpass filter.
capacitive bandpass filter v1 1 0 ac 1 sin r1 1 2 200 c1 2 0 2.5u c2 2 3 1u rload 3 0 1k .ac lin 20 100 500 .plot ac v(3) .end
The response of a capacitive bandpass filter peaks within a narrow frequency range.
Bandpass filters can also be constructed using inductors, but as mentioned before, the reactive "purity" of capacitors gives them a design advantage. If we were to design a bandpass filter using inductors, it might look something like Figure below.
Inductive bandpass filter.
The fact that the highpass section comes "first" in this design instead of the lowpass section makes no difference in its overall operation. It will still filter out all frequencies too high or too low.
While the general idea of combining lowpass and highpass filters together to make a bandpass filter is sound, it is not without certain limitations. Because this type of bandpass filter works by relying on either section to block unwanted frequencies, it can be difficult to design such a filter to allow unhindered passage within the desired frequency range. Both the lowpass and highpass sections will always be blocking signals to some extent, and their combined effort makes for an attenuated (reduced amplitude) signal at best, even at the peak of the "passband" frequency range. Notice the curve peak on the previous SPICE analysis: the load voltage of this filter never rises above 0.59 volts, although the source voltage is a full volt. This signal attenuation becomes more pronounced if the filter is designed to be more selective (steeper curve, narrower band of passable frequencies).
There are other methods to achieve bandpass operation without sacrificing signal strength within the passband. We will discuss those methods a little later in this chapter.

REVIEW:

A bandpass filter works to screen out frequencies that are too low or too high, giving easy passage only to frequencies within a certain range.

Bandpass filters can be made by stacking a lowpass filter on the end of a highpass filter, or vice versa.

"Attenuate" means to reduce or diminish in amplitude. When you turn down the volume control on your stereo, you are "attenuating" the signal being sent to the speakers.
2.4. BandStop Filters
Also called bandelimination, bandreject, or notch filters, this kind of filter passes all frequencies above and below a particular range set by the component values. Not surprisingly, it can be made out of a lowpass and a highpass filter, just like the bandpass design, except that this time we connect the two filter sections in parallel with each other instead of in series. (Figure below)
System level block diagram of a bandstop filter.
Constructed using two capacitive filter sections, it looks something like (Figure below).
"TwinT" bandstop filter.
The lowpass filter section is comprised of R_{1}, R_{2}, and C_{1} in a "T" configuration. The highpass filter section is comprised of C_{2}, C_{3}, and R_{3} in a "T" configuration as well. Together, this arrangement is commonly known as a "TwinT" filter, giving sharp response when the component values are chosen in the following ratios:
Given these component ratios, the frequency of maximum rejection (the "notch frequency") can be calculated as follows:
The impressive bandstopping ability of this filter is illustrated by the following SPICE analysis: (Figure below)
twint bandstop filter v1 1 0 ac 1 sin r1 1 2 200 c1 2 0 2u r2 2 3 200 c2 1 4 1u r3 4 0 100 c3 4 3 1u rload 3 0 1k .ac lin 20 200 1.5k .plot ac v(3) .end
+ (((22032.png)))
Response of "twinT" bandstop filter.
+

REVIEW:

A bandstop filter works to screen out frequencies that are within a certain range, giving easy passage only to frequencies outside of that range. Also known as bandelimination, bandreject, or notch filters.

Bandstop filters can be made by placing a lowpass filter in parallel with a highpass filter. Commonly, both the lowpass and highpass filter sections are of the "T" configuration, giving the name "TwinT" to the bandstop combination.

The frequency of maximum attenuation is called the notch frequency.
2.5. Resonant Filters
So far, the filter designs we’ve concentrated on have employed either capacitors or inductors, but never both at the same time. We should know by now that combinations of L and C will tend to resonate, and this property can be exploited in designing bandpass and bandstop filter circuits.
Series LC circuits give minimum impedance at resonance, while parallel LC ("tank") circuits give maximum impedance at their resonant frequency. Knowing this, we have two basic strategies for designing either bandpass or bandstop filters.
For bandpass filters, the two basic resonant strategies are this: series LC to pass a signal (Figure below), or parallel LC (Figure below) to short a signal. The two schemes will be contrasted and simulated here:
Series resonant LC bandpass filter.
Series LC components pass signal at resonance, and block signals of any other frequencies from getting to the load. (Figure below)
series resonant bandpass filter v1 1 0 ac 1 sin l1 1 2 1 c1 2 3 1u rload 3 0 1k .ac lin 20 50 250 .plot ac v(3) .end
Series resonant bandpass filter: voltage peaks at resonant frequency of 159.15 Hz.
A couple of points to note: see how there is virtually no signal attenuation within the "pass band" (the range of frequencies near the load voltage peak), unlike the bandpass filters made from capacitors or inductors alone. Also, since this filter works on the principle of series LC resonance, the resonant frequency of which is unaffected by circuit resistance, the value of the load resistor will not skew the peak frequency. However, different values for the load resistor will change the "steepness" of the Bode plot (the "selectivity" of the filter).
The other basic style of resonant bandpass filters employs a tank circuit (parallel LC combination) to short out signals too high or too low in frequency from getting to the load: (Figure below)
Parallel resonant bandpass filter.
The tank circuit will have a lot of impedance at resonance, allowing the signal to get to the load with minimal attenuation. Under or over resonant frequency, however, the tank circuit will have a low impedance, shorting out the signal and dropping most of it across series resistor R_{1}. (Figure below)
parallel resonant bandpass filter v1 1 0 ac 1 sin r1 1 2 500 l1 2 0 100m c1 2 0 10u rload 2 0 1k .ac lin 20 50 250 .plot ac v(2) .end
Parallel resonant filter: voltage peaks a resonant frequency of 159.15 Hz.
Just like the lowpass and highpass filter designs relying on a series resistance and a parallel "shorting" component to attenuate unwanted frequencies, this resonant circuit can never provide full input (source) voltage to the load. That series resistance will always be dropping some amount of voltage so long as there is a load resistance connected to the output of the filter.
It should be noted that this form of bandpass filter circuit is very popular in analog radio tuning circuitry, for selecting a particular radio frequency from the multitudes of frequencies available from the antenna. In most analog radio tuner circuits, the rotating dial for station selection moves a variable capacitor in a tank circuit.
Variable capacitor tunes radio receiver tank circuit to select one out of many broadcast stations.
The variable capacitor and aircore inductor shown in Figure above photograph of a simple radio comprise the main elements in the tank circuit filter used to discriminate one radio station’s signal from another.
Just as we can use series and parallel LC resonant circuits to pass only those frequencies within a certain range, we can also use them to block frequencies within a certain range, creating a bandstop filter. Again, we have two major strategies to follow in doing this, to use either series or parallel resonance. First, we’ll look at the series variety: (Figure below)
Series resonant bandstop filter.
When the series LC combination reaches resonance, its very low impedance shorts out the signal, dropping it across resistor R_{1} and preventing its passage on to the load. (Figure below)
series resonant bandstop filter v1 1 0 ac 1 sin r1 1 2 500 l1 2 3 100m c1 3 0 10u rload 2 0 1k .ac lin 20 70 230 .plot ac v(2) .end
Series resonant bandstop filter: Notch frequency = LC resonant frequency (159.15 Hz).
Next, we will examine the parallel resonant bandstop filter: (Figure below)
Parallel resonant bandstop filter.
The parallel LC components present a high impedance at resonant frequency, thereby blocking the signal from the load at that frequency. Conversely, it passes signals to the load at any other frequencies. (Figure below)
parallel resonant bandstop filter v1 1 0 ac 1 sin l1 1 2 100m c1 1 2 10u rload 2 0 1k .ac lin 20 100 200 .plot ac v(2) .end
Parallel resonant bandstop filter: Notch frequency = LC resonant frequency (159.15 Hz).
Once again, notice how the absence of a series resistor makes for minimum attenuation for all the desired (passed) signals. The amplitude at the notch frequency, on the other hand, is very low. In other words, this is a very "selective" filter.
In all these resonant filter designs, the selectivity depends greatly upon the "purity" of the inductance and capacitance used. If there is any stray resistance (especially likely in the inductor), this will diminish the filter’s ability to finely discriminate frequencies, as well as introduce antiresonant effects that will skew the peak/notch frequency.
A word of caution to those designing lowpass and highpass filters is in order at this point. After assessing the standard RC and LR lowpass and highpass filter designs, it might occur to a student that a better, more effective design of lowpass or highpass filter might be realized by combining capacitive and inductive elements together like Figure below.
Capacitive Inductive lowpass filter.
The inductors should block any high frequencies, while the capacitor should short out any high frequencies as well, both working together to allow only low frequency signals to reach the load.
At first, this seems to be a good strategy, and eliminates the need for a series resistance. However, the more insightful student will recognize that any combination of capacitors and inductors together in a circuit is likely to cause resonant effects to happen at a certain frequency. Resonance, as we have seen before, can cause strange things to happen. Let’s plot a SPICE analysis and see what happens over a wide frequency range: (Figure below)
lc lowpass filter v1 1 0 ac 1 sin l1 1 2 100m c1 2 0 1u l2 2 3 100m rload 3 0 1k .ac lin 20 100 1k .plot ac v(3) .end
+ (((22037.png)))
Unexpected response of LC lowpass filter.
What was supposed to be a lowpass filter turns out to be a bandpass filter with a peak somewhere around 526 Hz! The capacitance and inductance in this filter circuit are attaining resonance at that point, creating a large voltage drop around C_{1}, which is seen at the load, regardless of L_{2}'s attenuating influence. The output voltage to the load at this point actually exceeds the input (source) voltage! A little more reflection reveals that if L_{1} and C_{2} are at resonance, they will impose a very heavy (very low impedance) load on the AC source, which might not be good either. We’ll run the same analysis again, only this time plotting C_{1}'s voltage, vm(2) in Figure below, and the source current, I(v1), along with load voltage, vm(3):
Current inceases at the unwanted resonance of the LC lowpass filter.
Sure enough, we see the voltage across C_{1} and the source current spiking to a high point at the same frequency where the load voltage is maximum. If we were expecting this filter to provide a simple lowpass function, we might be disappointed by the results.
The problem is that an LC filter has an input impedance and an output impedance which must be matched. The voltage source impedance must match the input impedance of the filter, and the filter output impedance must be matched by "rload" for a flat response. The input and output impedance is given by the square root of (L/C).
Z = (L/C)1/2
Taking the component values from (Figure below), we can find the impedance of the filter, and the required , R_{g} and R_{load} to match it.
For L= 100 mH, C= 1µF Z = (L/C)1/2=((100 mH)/(1 µF))1/2 = 316 Ω
In Figure below we have added R_{g} = 316 Ω to the generator, and changed the load R_{load} from 1000 Ω to 316 Ω. Note that if we needed to drive a 1000 Ω load, the L/C ratio could have been adjusted to match that resistance.
Circuit of source and load matched LC lowpass filter.
LC matched lowpass filter V1 1 0 ac 1 SIN Rg 1 4 316 L1 4 2 100m C1 2 0 1.0u L2 2 3 100m Rload 3 0 316 .ac lin 20 100 1k .plot ac v(3) .end
Figure below shows the "flat" response of the LC low pass filter when the source and load impedance match the filter input and output impedances.
The response of impedance matched LC lowpass filter is nearly flat up to the cutoff frequency.
The point to make in comparing the response of the unmatched filter (Figure above) to the matched filter (Figure above) is that variable load on the filter produces a considerable change in voltage. This property is directly applicable to LC filtered power supplies– the regulation is poor. The power supply voltage changes with a change in load. This is undesirable.
This poor load regulation can be mitigated by a swinging choke. This is a choke, inductor, designed to saturate when a large DC current passes through it. By saturate, we mean that the DC current creates a "too" high level of flux in the magnetic core, so that the AC component of current cannot vary the flux. Since induction is proportional to dΦ/dt, the inductance is decreased by the heavy DC current. The decrease in inductance decreases reactance X_{L}. Decreasing reactance, reduces the voltage drop across the inductor; thus, increasing the voltage at the filter output. This improves the voltage regulation with respect to variable loads.
Despite the unintended resonance, lowpass filters made up of capacitors and inductors are frequently used as final stages in AC/DC power supplies to filter the unwanted AC "ripple" voltage out of the DC converted from AC. Why is this, if this particular filter design possesses a potentially troublesome resonant point?
The answer lies in the selection of filter component sizes and the frequencies encountered from an AC/DC converter (rectifier). What we’re trying to do in an AC/DC power supply filter is separate DC voltage from a small amount of relatively highfrequency AC voltage. The filter inductors and capacitors are generally quite large (several Henrys for the inductors and thousands of µF for the capacitors is typical), making the filter’s resonant frequency very, very low. DC of course, has a "frequency" of zero, so there’s no way it can make an LC circuit resonate. The ripple voltage, on the other hand, is a nonsinusoidal AC voltage consisting of a fundamental frequency at least twice the frequency of the converted AC voltage, with harmonics many times that in addition. For pluginthewall power supplies running on 60 Hz AC power (60 Hz United States; 50 Hz in Europe), the lowest frequency the filter will ever see is 120 Hz (100 Hz in Europe), which is well above its resonant point. Therefore, the potentially troublesome resonant point in a such a filter is completely avoided.
The following SPICE analysis calculates the voltage output (AC and DC) for such a filter, with series DC and AC (120 Hz) voltage sources providing a rough approximation of the mixedfrequency output of an AC/DC converter.
+ (((02129.png)))
AC/DC power suppply filter provides "ripple free" DC power.
ac/dc power supply filter v1 1 0 ac 1 sin v2 2 1 dc l1 2 3 3 c1 3 0 9500u l2 3 4 2 rload 4 0 1k .dc v2 12 12 1 .ac lin 1 120 120 .print dc v(4) .print ac v(4) .end
v2 v(4) 1.200E+01 1.200E+01 DC voltage at load = 12 volts freq v(4) 1.200E+02 3.412E05 AC voltage at load = 34.12 microvolts
With a full 12 volts DC at the load and only 34.12 µV of AC left from the 1 volt AC source imposed across the load, this circuit design proves itself to be a very effective power supply filter.
The lesson learned here about resonant effects also applies to the design of highpass filters using both capacitors and inductors. So long as the desired and undesired frequencies are well to either side of the resonant point, the filter will work OK. But if any signal of significant magnitude close to the resonant frequency is applied to the input of the filter, strange things will happen!

REVIEW:

Resonant combinations of capacitance and inductance can be employed to create very effective bandpass and bandstop filters without the need for added resistance in a circuit that would diminish the passage of desired frequencies.

2.6. Summary
As lengthy as this chapter has been up to this point, it only begins to scratch the surface of filter design. A quick perusal of any advanced filter design textbook is sufficient to prove my point. The mathematics involved with component selection and frequency response prediction is daunting to say the least — well beyond the scope of the beginning electronics student. It has been my intent here to present the basic principles of filter design with as little math as possible, leaning on the power of the SPICE circuit analysis program to explore filter performance. The benefit of such computer simulation software cannot be understated, for the beginning student or for the working engineer.
Circuit simulation software empowers the student to explore circuit designs far beyond the reach of their math skills. With the ability to generate Bode plots and precise figures, an intuitive understanding of circuit concepts can be attained, which is something often lost when a student is burdened with the task of solving lengthy equations by hand. If you are not familiar with the use of SPICE or other circuit simulation programs, take the time to become so! It will be of great benefit to your study. To see SPICE analyses presented in this book is an aid to understanding circuits, but to actually set up and analyze your own circuit simulations is a much more engaging and worthwhile endeavor as a student.
3. Navy (todo)
3.1. Resonant Circuits as Filter Circuits
The principle of series or parallelresonant circuits have many applications in radio, television, communications, and the various other electronic fields throughout the Navy. As you have seen, by making the capacitance or inductance variable, the frequency at which a circuit will resonate can be controlled.
In addition to station selecting or tuning, resonant circuits can separate currents of certain frequencies from those of other frequencies.
Circuits in which resonant circuits are used to do this are called FILTER CIRCUITS.
If we can select the proper values of resistors, inductors, or capacitors, a FILTER NETWORK, or "frequency selector," can be produced which offers little opposition to one frequency, while BLOCKING or ATTENUATING other frequencies. A filter network can also be designed that will "pass" a band of frequencies and "reject" all other frequencies.
Most electronic circuits require the use of filters in one form or another. You have already studied several in modules 6, 7, and 8 of the NEETS.
One example of a filter being applied is in a rectifier circuit. As you know, an alternating voltage is changed by the rectifier to a direct current. However, the d.c. voltage is not pure; it is still pulsating and fluctuating. In other words, the signal still has an a.c. component in addition to the d.c. voltage. By feeding the signal through simple filter networks, the a.c. component is reduced. The remaining d.c. is as pure as the designers require.
Bypass capacitors, which you have already studied, are part of filter networks that, in effect, bypass, or shunt, unwanted a.c. components to ground.
3.1.1. The Idea of "Q"
Several times in this chapter, we have discussed "ideal" or theoretically perfect circuits. In each case, you found that resistance kept our circuits from being perfect. You also found that low resistance in tuners was better than high resistance. Now you will learn about a factor that, in effect, measures just how close to perfect a tuner or tuner component can be. This same factor affects BANDWIDTH and SELECTIVITY. It can be used in figuring voltage across a coil or capacitor in a seriesresonant circuit and the amount of circulating (tank) current in a parallelresonant circuit. This factor is very important and useful to designers. Technicians should have some knowledge of the factor because it affects so many things. The factor is known as Q. Some say it stands for quality (or merit). The higher the Q, the better the circuit; the lower the losses (I_{2}R), the closer the circuit is to being perfect.
Having studied the first part of this chapter, you should not be surprised to learn that resistance (R) has a great effect on this figure of merit or quality.
3.1.2. Q Is a Ratio
Q is really very simple to understand if you think back to the tunedcircuit principles just covered. Inductance and capacitance are in all tuners. Resistance is an impurity that causes losses. Therefore, components that provide the reactance with a minimum of resistance are "purer" (more perfect) than those with higher resistance. The actual measure of this purity, merit, or quality must include the two basic quantities, X and R.
The ratio
does the job for us. Let’s take a look at it and see just why it measures quality.
First, if a perfect circuit has zero resistance, then our ratio should give a very high value of Q to reflect the high quality of the circuit. Does it?
Assume any value for X and a zero value for R.
Then:
Remember, any value divided by zero equals infinity. Thus, our ratio is infinitely high for a theoretically perfect circuit.
With components of higher resistance, the Q is reduced. Dividing by a larger number always yields a smaller quantity. Thus, lower quality components produce a lower Q. Q, then, is a direct and accurate measure of the quality of an LC circuit.
Q is just a ratio. It is always just a number — no units. The higher the number, the "better" the circuit. Later as you get into more practical circuits, you may find that low Q may be desirable to provide certain characteristics. For now, consider that higher is better.
Because capacitors have much, much less resistance in them than inductors, the Q of a circuit is very often expressed as the Q of the coil or:
The answer you get from using this formula is very near correct for most purposes. Basically, the Q of a capacitor is so high that it does not limit the Q of the circuit in any practical way. For that reason, the technician may ignore it.
3.1.3. The Q of a Coil
Q is a feature that is designed into a coil. When the coil is used within the frequency range for which it is designed, Q is relatively constant. In this sense, it is a physical characteristic.
Inductance is a result of the physical makeup of a coil  number of turns, core, type of winding, etc. Inductance governs reactance at a given frequency. Resistance is inherent in the length, size, and material of the wire. Therefore, the Q of a coil is mostly dependent on physical characteristics.
Values of Q that are in the hundreds are very practical and often found in typical equipment.
3.1.4. Application of Q
For the most part, Q is the concern of designers, not technicians. Therefore, the chances of you having to figure the Q of a coil are remote. However, it is important for you to know some circuit relationships that are affected by Q.
3.1.5. Q Relationships in Series Circuits
Q can be used to determine the "gain" of seriesresonant circuits. Gain refers to the fact that at resonance, the voltage drop across the reactances are greater than the applied voltage. Remember, when we applied Ohm’s law in a seriesresonant circuit, it gave us the following characteristics:

Low impedance, high current.

High current; high voltage across the comparatively high reactances.
This high voltage is usable where little power is required, such as in driving the grid of a vacuum tube or the gate of a field effect transistor (F.E.T.). The gain of a properly designed seriesresonant circuit may be as great or greater than the amplification within the amplifier itself. The gain is a function of Q, as shown in the following example:
If the Q of the coil were 100, then the gain would be 100; that is, the voltage of the coil would be 100 times that of the input voltage to the series circuit.
Resistance affects the resonance curve of a series circuit in two ways — the lower the resistance, the higher the current; also, the lower the resistance, the sharper the curve. Because low resistance causes high Q, these two facts are usually expressed as functions of Q. That is, the higher the Q, the higher and sharper the curve and the more selective the circuit.
The lower the Q (because of higher resistance), the lower the current curve; therefore, the broader the curve, the less selective the circuit. A summary of the major characteristics of series RLCcircuits at resonance is given in Figure 1.
3.1.6. Q Relationships in a ParallelResonant Circuit
There is no voltage gain in a parallelresonant circuit because voltage is the same across all parts of a parallel circuit. However, Q helps give us a measure of the current that circulates in the tank.
Given:
Again, if the Q were 100, the circulating current would be 100 times the value of the line current. This may help explain why some of the wire sizes are very large in highpower amplifying circuits.
The impedance curve of a parallelresonant circuit is also affected by the Q of the circuit in a manner similar to the current curve of a series circuit. The Q of the circuit determines how much the impedance is increased across the parallelLC circuit. (Z = Q × X_{L})
The higher the Q, the greater the impedance at resonance and the sharper the curve. The lower the Q, the lower impedance at resonance; therefore, the broader the curve, the less selective the circuit. The major characteristics of parallelRLC circuits at resonance are given in Figure 2.
3.1.7. Summary of Q
The ratio that is called Q is a measure of the quality of resonant circuits and circuit components. Basically, the value of Q is an inverse function of electrical power dissipated through circuit resistance. Q is the ratio of the power stored in the reactive components to the power dissipated in the resistance. That is, high power loss is low Q; low power loss is high Q.
Circuit designers provide the proper Q. As a technician, you should know what can change Q and what quantities in a circuit are affected by such a change.
3.2. Filters
In many practical applications of complex circuits, various combinations of direct, lowfrequency, audiofrequency, and radiofrequency currents may exist. It is frequently necessary to have a means for separating these component currents at any desired point. An electrical device for accomplishing this separation is called a FILTER.
A filter circuit consists of inductance, capacitance, and resistance used singularly or in combination, depending upon the purpose. It may be designed so that it will separate alternating current from direct current, or so that it will separate alternating current of one frequency (or a band of frequencies) from other alternating currents of different frequencies.
The use of resistance by itself in filter circuits does not provide any filtering action, because it opposes the flow of any current regardless of its frequency. What it does, when connected in series or parallel with an inductor or capacitor, is to decrease the "sharpness," or selectivity, of the filter. Hence, in some particular application, resistance might be used in conjunction with inductance or capacitance to provide filtering action over a wider band of frequencies.
Filter circuits may be divided into four general types: LOWPASS, HIGHPASS, BANDPASS, AND BANDREJECT filters.
Electronic circuits often have currents of different frequencies. The reason is that a source produces current with the same frequency as the applied voltage. As an example, the a.c. signal input to an audio amplifier can have high and lowaudio frequencies; the input to an rf amplifier can have a wide range of radio frequencies.
In such applications where the current has different frequency components, it is usually necessary for the filter either to accept or reject one frequency or a group of frequencies. The electronic filter that can pass on the higherfrequency components to a load or to the next circuit is known as a HIGHPASS filter. A LOWPASS filter can be used to pass on lowerfrequency components.
Before discussing filters further, we will review and apply some basic principles of the frequency response characteristics of the capacitor and the inductor. Recall the basic formula for capacitive reactance and inductive reactance:
Assume any given value of L and C. If we increase the applied frequency, X_{C} decreases and X_{L} increases. If we increase the frequency enough, the capacitor acts as a short and the inductor acts as an open. Of course, the opposite is also true. Decreasing frequency causes X_{C} to increase and X_{L} to decrease. Here again, if we make a large enough change, X_{C} acts as an open and X_{L} acts as a short. Figure 3 gives a pictorial representation of these two basic components and how they respond to low and high frequencies.
If we apply these same principles to simple circuits, such as the ones in Figure 4, they affect input signals as shown. For example, in view (A) of the figure, a low frequency is blocked by the capacitor which acts as an open and at a high frequency the capacitor acts as a short. By studying the figure, it is easy to see how the various components will react in different configurations with a change in frequency.
As mentioned before, highpass and lowpass filters pass the specific frequencies for which circuits are designed.
There can be a great deal of confusion when talking about highpass, lowpass, discrimination, attenuation, and frequency cutoff, unless the terms are clearly understood. Since these terms are used widely throughout electronics texts and references, you should have a clear understanding before proceeding further.

HIGHPASS FILTER. A highpass filter passes on a majority of the high frequencies to the next circuit and rejects or attenuates the lower frequencies. Sometimes it is called a lowfrequency discriminator or lowfrequency attenuator.

LOWPASS FILTER. A lowpass filter passes on a majority of the low frequencies to the next circuit and rejects or attenuates the higher frequencies. Sometimes it is called a highfrequency discriminator or highfrequency attenuator.

DISCRIMINATION. The ability of the filter circuit to distinguish between high and low frequencies and to eliminate or reject the unwanted frequencies.

ATTENUATION. The ability of the filter circuit to reduce the amplitude of the unwanted frequencies below the level of the desired output frequency.

FREQUENCY CUTOFF (f_{co}). The frequency at which the filter circuit changes from the point of rejecting the unwanted frequencies to the point of passing the desired frequency; OR the point at which the filter circuit changes from the point of passing the desired frequency to the point of rejecting the undesired frequencies.
3.2.1. LowPass Filter
A lowpass filter passes all currents having a frequency below a specified frequency, while opposing all currents having a frequency above this specified frequency. This action is illustrated in its ideal form in Figure 5. At frequency cutoff, known as f_{c} the current decreases from maximum to zero. At all frequencies above f_{c} the filter presents infinite opposition and there is no current. However, this sharp division between no opposition and full opposition is impossible to attain. A more practical graph of the current is shown in Figure 6, where the filter gradually builds up opposition as the cutoff frequency (f) is approached. Notice that the filter cannot completely block current above the cutoff frequency.
Figure 7 shows the electrical construction of a lowpass filter with an inductor inserted in series with one side of a line carrying both low and high frequencies. The opposition offered by the reactance will be small at the lower frequencies and great at the higher frequencies. In order to divert the undesired high frequencies back to the source, a capacitor must be added across the line to bypass the higher frequencies around the load, as shown in Figure 8.
The capacitance of the capacitor must be such that its reactance will offer little opposition to frequencies above a definite value, and great opposition to frequencies below this value. By combining the series inductance and bypass capacitance, as shown in Figure 9, the simplest type of lowpass filter is obtained. At point P, a much higher opposition is offered to the low frequencies by the capacitor than by the inductor, and most of the lowfrequency current takes the path of least opposition. On the other hand, the least amount of opposition is offered to the high frequencies by the capacitor, and most of the highfrequency energy returns to the source through the capacitor.
3.2.2. HighPass Filter
A highpass filter circuit passes all currents having a frequency higher than a specified frequency, while opposing all currents having a frequency lower than its specified frequency. This is illustrated in Figure 10. A capacitor that is used in series with the source of both high and low frequencies, as shown in Figure 11, will respond differently to highfrequency, lowfrequency, and direct currents. It will offer little opposition to the passage of highfrequency currents, great opposition to the passage of lowfrequency currents, and completely block direct currents. The value of the capacitor must be chosen so that it allows the passage of all currents having frequencies above the desired value, and opposes those having frequencies below the desired value. Then, in order to shunt the undesired lowfrequency currents back to the source, an inductor is used, as shown in Figure 12. This inductor must have a value that will allow it to pass currents having frequencies below the frequency cutoff point, and reject currents having frequencies above the frequency cutoff point, thus forcing them to pass through the capacitor. By combining inductance and capacitance, as shown in Figure 13, you obtain the simplest type of highpass filter. At point P most of the highfrequency energy is passed on to the load by the capacitor, and most of the lowfrequency energy is shunted back to the source through the inductor.
3.2.3. Resonant Circuits as Filters
Resonant circuits can be made to serve as filters in a manner similar to the action of individual capacitors and inductors. As you know, the seriesLC circuit offers minimum opposition to currents that have frequencies at or near the resonant frequency, and maximum opposition to currents of all other frequencies.
You also know that a parallelLC circuit offers a very high impedance to currents that have frequencies at or near the resonant frequency, and a relatively low impedance to currents of all other frequencies.
If you use these two basic concepts, the BANDPASS and BANDREJECT filters can be constructed. The bandpass filter and the bandreject filter are two common types of filters that use resonant circuits.
3.2.3.1. Bandpass Filter
A bandpass filter passes a narrow band of frequencies through a circuit and attenuates all other frequencies that are higher or lower than the desired band of frequencies. This is shown in Figure 14 where the greatest current exists at the center frequency (f_{r}). Frequencies below resonance (f_{1}) and frequencies above resonance (f_{2}) drop off rapidly and are rejected.
In the circuit of Figure 15, the seriesLC circuit replaces the inductor of Figure 7, and acts as a BANDPASS filter. It passes currents having frequencies at or near its resonant frequency, and opposes the passage of all currents having frequencies outside this band.
Thus, in the circuit of Figure 16, the parallelLC circuit replaces the capacitor of Figure 8. If this circuit is tuned to the same frequency as the seriesLC circuit, it will provide a path for all currents having frequencies outside the limits of the frequency band passed by the seriesresonant circuit. The simplest type of bandpass filter is formed by connecting the two LC circuits as shown in Figure 17. The upper and lower frequency limits of the filter action are filter cutoff points.
3.2.3.2. BandReject Filter
A bandreject filter circuit is used to block the passage of current for a narrow band of frequencies, while allowing current to flow at all frequencies above or below this band. This type of filter is also known as a BANDSUPPRESSION or BANDSTOP filter. The way it responds is shown by the response curve of Figure 18. Since the purpose of the bandreject filter is directly opposite to that of a bandpass filter, the relative positions of the resonant circuits in the filter are interchanged. The parallelLC circuit shown in Figure 19 replaces the capacitor of Figure 11. It acts as a bandreject filter, blocking the passage of currents having frequencies at or near resonant frequency and passing all currents having frequencies outside this band. The seriesLC circuit shown in Figure 20 replaces the inductor of Figure 12. If this series circuit is tuned, to the same frequency as the parallel circuit, it acts as a bypass for the band of rejected frequencies. Then, the simplest type of bandreject filter is obtained by connecting the two circuits as shown in Figure 21.
Q14. What is the device called that will separate alternating current from direct current, or that will separate alternating current of one frequency from other alternating currents of different frequencies? Q15. What are the four general types of filters? Q16. What is the filter called in which the low frequencies do not produce a useful voltage? Q17. What is the filter called that passes low frequencies but rejects or attenuates high frequencies? Q18. How does a capacitor and an inductor react to (a) low frequency and (b) high frequency? Q19. What term is used to describe the frequency at which the filter circuit changes from the point of rejecting the unwanted frequencies to the point of passing the desired frequencies? Q20. What type filter is used to allow a narrow band of frequencies to pass through a circuit and attenuate all other frequencies above or below the desired band? Q21. What type filter is used to block the passage of current for a narrow band of frequencies, while allowing current to flow at all frequencies above or below this band?
3.2.4. Multisection Filters
All of the various types of filters we have discussed so far have had only one section. In many cases, the use of such simple filter circuits does not provide sufficiently sharp cutoff points. But by adding a capacitor, an inductor, or a resonant circuit in series or in parallel (depending upon the type of filter action required), the ideal effect is more nearly approached. When such additional units are added to a filter circuit, the form of the resulting circuit will resemble the letter T, or the Greek letter π (pi). They are, therefore, called T or πtype filters, depending upon which symbol they resemble. Two or more T or π−type filters may be connected together to produce a still sharper cutoff point.
Figure 22, Figure 23, and Figure 24; and Figure 25, Figure 26, and Figure 27 depict some of the common configurations of the T and πtype filters. Further discussion about the theory of operation of these circuits is beyond the intended scope of this module. If you are interested in learning more about filters, a good source of information to study is the Electronics Installation and Maintenance Handbook (EIMB), section 4 (Electronics Circuits), NAVSEA 0967LP0000120.
3.3. Safety Precautions
When working with resonant circuits, or electrical circuits, you must be aware of the potentially high voltages. Look at Figure 28. With the series circuit at resonance, the total impedance of the circuit is 5 ohms.
Remember, the impedance of a seriesRLC circuit at resonance depends on the resistive element. At resonance, the impedance (Z) equals the resistance (R). Resistance is minimum and current is maximum. Therefore, the current at resonance is:
The voltage drops around the circuit with 2 amperes of current flow are:
E_{C} = I_{T} × X_{C} E_{C} = 2 × 20 E_{C} = 40 volts a.c. E_{L} = I_{T} × X~L E_{L} = 2 × 20 E_{L} = 40 volts a.c. E_{R} = I_{T} × R E_{R} = 2 × 5 E_{R} = 10 volts a.c.
You can see that there is a voltage gain across the reactive components at resonance.
If the frequency was such that X_{L} and X_{C} were equal to 1000 ohms at the resonant frequency, the reactance voltage across the inductor or capacitor would increase to 2000 volts a.c. with 10 volts a.c. applied. Be aware that potentially high voltage can exist in seriesresonant circuits.