## Analogue Filters Tutorial |

**LOW FREQUENCIES RC FILTER**

Figure 1. Low pass RC filter.

A low pass filter allows the passage of low frequency signals and prevents the passage of high frequency signals. Figure 1 shows an RC filter low frequencies. The resistor and the capacitor form a voltage divider. At very low frequencies the capacitor has a very large capacitance (XC = 1 / 2nfc) to the R. Therefore almost all of the input voltage Vin is at the ends of the capacitor C. But as the signal frequency increases the capacitance Xc reduced and thus the output voltage of the filter, ie at the ends of the capacitor decreases. On very high frequencies the resistance of the capacitor is almost Oohm, and the capacitor appears as a short circuit, in this case all the input voltage is across the resistor R, and the output voltage of the filter is 0V.

Figure 2. Change in gain (Av = Vo/Vin) of RC filter.

Figure 2 shows the change in the gain of the RC low-pass filter to the frequency. Notice that the voltage gain (Av) is the ratio of the output voltage Vo to the input voltage Vin (Av = Vo/Vin) as with the amplifiers, although in the case of the filter aid, there is no amplification, but attenuation in the output voltage.

Usually we used unit Decibel (dB) for the measurement of gain filters. The voltage gain in decibels (dB) is given by the relationship:

So if the voltage gain is AV = 100 then the voltage gain in dB is:

If the output voltage is equal to the input voltage then the gain in dB is:

In RC low pass filter, the voltage gain equal to one (Av = 1) at very low frequencies because in that case the output voltage is approximately equal to the input voltage. If we use units of dB then the gain is OdB. As the frequency increases the gain of the filter voltage (Av = V0/Vin) begins to fall, since the output voltage decreases. At a certain frequency (fc) the value of the capacitive impedance Xc equals the value of the resistance R, so:

In this frequency, as will be demonstrated below, the gain is equal to 0.707 of the maximum value of gain, or if expressed in dB units corresponds to a gain -3dB. The frequency fc is called cutoff frequency or -3dB frequency and is given by the relationship:

Generally, the output voltage of the filter at any frequency is given by the relationship:

From the above equation we see that the cutoff frequency (fc) ie when Xc = R, we have:

or using units of dB we have:

Figure 3. Frequency response curve and Figure 4. Response curve of an ideal low pass filter.

since the logarithm of 0.707 relative to the base of 10 is -0.15. Figure 3 shows that varying the gain of one RC low-pass filter to the frequency, the curve is called frequency response curve. The gain is given in units of decibel. As seen from Figure 3, the curve is nearly flat up to a cutoff frequency fc, and the curve starts to "tumbling» (roll-off) as the frequency is increased the gain decreases. As Bandwidth filter is defined as the frequency range from OH2 to fcHz. Mathematically the gain of the filter for frequencies greater than Fc decreases by 6 dB per octave. With one octave we mean the frequency doubling, ie the gain of the filter reduced by 6dB with each doubling of frequency after the frequency Fc. if the cutoff frequency is 10Khz then the gain is reduced by 6dB when the frequency increase from 10Khz to 20Khz will decrease another 6dB when the frequency increased to 40 kHz, etc.

Figure 5.

Sometimes instead of saying that the gain of filter after Fc is reduced by 6dB/octave, we say that gain is reduced by 20dB/octave, which means that the gain is reduced after the Fc by 20dB every time the frequency tenfold. Ie if Fc = 10Khz then the gain is reduced by 20dB when the frequency increase from 10Khz to 10OKhz, the gain will be reduced by 20dB when the frequency increase from 100Khz to 1 MHz, etc. Observe that 6dB/octave and 20dB/octave express the same gain reduction (roll-off) of the curve after the frequency Fc. Figure 4 shows the curve response of an ideal low-pass filter frequencies. In practice it is impossible to construct a filter with such a response curve shape. But we can get close enough to the ideal response curve using more than one RC filters. Each additional filter increases the roll-off of the response curve by 6 or 20 dB/octave. Eg. With two filters (figure 5) the gain is reduced by 12dB/octave or 40dB/octave. With three RC filters the gain is reduced by 18dB/octave or 60dB/octave etc.

Figure 6. shows the response curves of 1, 2 and 3 RC sections.

**CHANGE OF PHASE ANGLE**

Figure 7. The voltage of the output signal following the voltage of the input signal.

An RC filter also creates a phase difference (f) between the voltage of the input signal and the voltage of the output signal or a time delay. In low-pass filter the signal voltage output follows the voltage of the input signal by an angle f as illustrated in Figure 7. From the basic theory of alternating circuits we know that the phase angle of the RF circuit is given by the relationship:

So, the phase angle dependent on the frequency f. The curve gives the variables of phase (f) to the frequency f, is shown in Figure 8 together with the response curve. When the signal frequency is (DC), there is no phase difference between input signal and the output signal. While frequency increases, so increases and the phase difference in very high frequencies the phase difference gets close to 90°. The phase difference in the cutoff frequency is 45°.

Figure 8. The voltage of the output signal following the voltage of the input signal.

Since the phase curve (f) is not a linear, a composite input signal, ie a signal which consists of several components will exit deformed. As shown in Figure 9, the signal consists of two sine waves of which one is twice the frequency of the other. Suppose now that the wave with less frequency will shift by an angle (f). Because the filter curve is not linear, the wave with the double frequency will not shift by the twice angle, but another angle. Thus, when added these two signals at the output, due to the non-linear curve (phase-frequency) output signal will be distorted. There are filters having nearly linear phase-frequency curve, eg the Bessel filters. Figure 10 shows that in a filter with linear phase-frequency curve all the waves of the complex input signal will have the same time delay in the output, so there there will be no distortion in the output signal.

Figure 9.

Figure 10.

**HIGH PASS RC FILTERS**

A high pass filter attenuates all frequencies which are smaller than a certain frequency fc called cutoff frequency, and allows the passage, without almost no attenuation, all the frequencies which are greater than the fc. Figure 11 shows the basic circuit of RC filter.

Figure 11.

At very low frequencies the capacitor presents large capacitance Xc and thus almost all the input voltage Vin is across the capacitor, and the voltage across R is almost 0V. As the frequency increases, the capacitance of the capacitor decreases at very high frequencies the voltage at the C is almost 0V, and the whole input voltage appears at the output of the filter. The response curve of the RC high pass filter shown in Figure 12 together with the phase shift curve.

Figure 12.

As in the RC low-pass filter so the RC high-pass filter when the Xc = R, the gain of the filter is 0.707 or -3dB and the cutoff frequency is fc = 1 / 2nRC. Also, the reduction in gain {roll-off) after frequency Fc is 6dB/octave or 20dB/decade. In case we have two RC filters the gain reduced by 12dB/octave or 40dB/decade etc. (Figure 13).

Figure 13.

**LC FILTERS**

It should also be mentioned that apart from the high-pass RC filters and low frequencies, we can construct filters using only capacitors and inductors (LC filters). These filters are more expensive and bulky than the corresponding RC filters because they use coils. The RC filters are suitable for applications at relatively low frequencies and for applications at high frequencies the LC filters are the most suitable. Figure 14 shows an LC high pass filter. The basic operation is as follows: The capacitance (Xc = 1/2nfC) and reactance (XL = 2nfl) form a voltage divider. At very low frequencies the Xc is much greater than the XL and thus almost all the input voltage is across the capacitor and consequently the output voltage of the filter, ie at the ends of the coil is almost 0V.

Figure 14.

As the frequency increases, the impedance inductance increases and therefore increases also the voltage drop across the coil that is at the output of the filter. At very high frequencies almost all the input voltage is at the ends of the coil, since at these frequencies the capacitor appears as a short circuit (0ohm).

**LOW PASS LC FILTERS**

Figure 15.

A simple LC low pass filter shown in Figure 15 with the response curve. The operation is similar to that of LC high pass filter. Notice that at very low frequencies the capacitor has a very large capacitance, and thus almost all of the input voltage is across the capacitor. On very high frequencies the capacitance of the capacitor is too small compared to the resistance and inductance, the voltage across the capacitor is almost 0V.

**TUNED FILTERS**

Parallel and series resonant circuits can be used to manufacture the band-pass filter (Band-pass) or cutoff frequency band (Band-stop). A tuned LC circuit has minimal resistance at the resonant frequency, and a parallel tuned LC circuit presents a high impedance, which is given by the relationship:

where Q is the quality factor of the coil, and Rl the resistance of the coil.

**BANDPASS FILTERS**

Figure 16.

The general response curve of a filter bandpass is shown in Figure 16. The filter allows the passage of ALL frequencies that lie within a zone between a lower cutoff frequency fci, fc2 and an upper cut-off frequency and attenuates all frequencies which are outside the bandpass. The bandpass (Bandwidth = BW) is defined as the difference fc2-fci. The cutoff frequencies, are the frequencies at which the gain is 0.707 of the maximum value. The frequency is called center frequency or resonance frequency. Figure 17 shows two circuits bandpass filters.

Figure 17.

**NOTCH FILTERS**

Figure 18.

This filter attenuates (rejects) all frequencies present in a given area and allow the passage of all other frequencies outside the cutting zone. Figure 18 shows a general response curve, and two circuits notch filter frequency.

**BUTTERWORTH FILTERS**

The general form of the response curve of a Butterworth filter is shown in Figure 21. The main characteristic of the Butterworth filter is that the gain is constant in the bandpass, ie until the cutoff frequency, and has a rate of 20dB/decade attenuation for frequencies above the cutoff frequency when the filter is first class.

Figure 21.

The class of a filter is determined by the number of capacitors and inductors. Eg a filter consisting of a capacitor and a resistor called a first order filter, a filter consisting of a capacitor and a coil is the second class etc. Note that as the order of the filter increases, also increases and the rate of attenuation of the gain. Eg A first class filter has a rate of attenuation 20dB/decade and a second order filter 40dB/decade, one third filter 60dB/decade etc. Ie, the rate of gain attenuation increases by 20dV/decade or 6dB/octave with each additional capacitor or inductor.

Figure 22.

The moving phase of the Butterworth filter as shown in Figure 21, is not linear. Figure 22 shows the response curves of Butterworth low pass filters of different classes. Notice that the response curves are designed for cutoff frequency Z = 1 represents 0,1592 Hz. To determine the response curve of a practical filter having a cutoff frequency, another frequency from Z = 1, we should multiply the values in the frequency axis, with the cutoff frequency value of the report filter.

**CHEBYSHEV FILTERS**

This filter is used in applications requiring a faster rate of gain attenuation (roll-off) than that of Butterworth and Bessel. The disadvantage of thAT filter is that the gain in the bandpass is not constant but shows a ripple as shown in Figure 23, two second class filters (n = 2) and fourth class (n = 4). The phase shift of the filter is not linear. The cutoff frequency (fc) is defined as the frequency at which the ripple channel ends.

Figure 23.

**BESSEL FILTERS**

The main feature of that filter is that the phase shift is linear and thus does not deform the signal at the output. Figure 25 shows the response curves Bessel low pass filters. All three types of filters use the same basic circuits, only the values of L and C are different.

Figure 25.

**FILTERS DESIGN USING TABLES**

The design of theSE filters is usually done by using tables which give us all the values of the components of the filter at a specified frequency. To design the filter to operate at another frequency than that used in the table, it is sufficient to multiply the values of the data delivered from the table with a certain constant which depends on the type of filter used. The most common frequency used for the preparation of the tables is the frequency 0.1592 Hz corresponding to angle frequency, omega = 1.

**DESIGNING BUTTERWORTH FILTERS**

We will now show how to design low band pass filters and high-band Butterworth types using Tables. We begin by designing lowpass filters. The Table shows the values of (L and C) needed to design filters range from 2 to 8. The cutoff frequency fc for which they calculated the values of the table are fc = 0,1592Hz. By the equations:

we can calculate the values of L and C which are required for the construction of the filter. Notice that f is the cutoff frequency of the filter you want to design, RL is the load resistance of filter, ln (table) and Cn (table) are the values given in the table for the cutoff frequency Fc = 0.1592 Hz, and Ln (filter) and Gn (filter) are the values for the filter you want to design at the new Cut-off frequency. There are two kind of filters that you can design using the tables, these are the T and W, as shown in Figure 26. The table gives us the freedom to design filters that have the same impedance source (Rs) and load (RL and salso filters whose source impedance or load is larger than the other.

Figure 26.

For designing a fifth class filter using the Table, we should be the first deside the class of the filter using the curves. Then you must decide which type of filter you are going to use. In cases where the input impedance is equal to the output we can use either the F or the T configuration. However, the F configuration is preferred because it requires less inductance (coils). If the output impedance is substantially greater than the impedance you can use the device T, and if the output impedance is much smaller than the input impedance use the device F.

DESIGN EXAMPLE 1

Draw a fifth class low pass filter. The impedance of the source and the load is 75W, and the cutoff frequency (-3dB) is 500KHz. We will use the F device to limit the number of coils. The circuit of the filter shown in Figure 27.

Figure 27. Butterworth filter type II, low-pass 5th-order.

From the table we have:

C1 (table) = C5 (table) = 0.6180

L2 (table) = L4 (table) = 1,618

and C3 (table) = 2

Using the relations 1 and 2 we have:

DESIGN EXAMPLE 2

Design a low pass 3rd class filter. The impedance is 600 ohms and the load impedance is 50Kohm. The cutoff frequency (-3dB) is 70Khz. Because RL is sufficiently greater than the Rs we would use the device T.

From Table 1 we have:

Li (table) = 1.5

L3 (table) = 0.5

C2 (table) = 1.3333

Substituting the above values to relations 1 and 2 we have:

The complete circuit of the filter shown in Figure 28.

Figure 28.

DESIGN EXAMPLE 3

Design a low-pass 2nd class filter. The source impedance is 1Mohm and the load resistance is 100ohm. The cutoff frequency is 10Khz. will use P device, because Rs is greater than the RL.

From Table 1 we have:

C, (table) = 1.4142

L2 (table) = 0.7071

By substituting the relations 1 and 2 we have: