RC Filters

circuit is illustrated in Fig. 2-24(b). The cutoff frequency of this i lter is that point where R and X_C are equal. The cutoff frequency, also known as the critical frequency, is deter- mined by the expression

XC5R 1 2πfc

5R fco5 1

2πRC

For example, if R54.7 kV and C5560 pF, the cutoff frequency is

fco5 1

2π(4700) (560310²¹²) 560,469 Hz  or  60.5 kHz Figure 2-23 Ideal response curve of a low-pass fi lter.

Frequency

Cutoff frequency

Output

f_co Signals in the passband pass

unattenuated Signals above f_co are eliminated

Figure 2-24 RC low-pass fi lter. (a) Circuit. (b) Low-pass fi lter.

V_out V_in

X_C ⫽ R C

(a)

V_out(max) (0 dB) 0.707 V_out(max)

(⫺3 dB)

f_co

6 dB/octave or 20 dB/decade rate

(b) f_co ⫽ 1

2␲RC R

⫺9 dB 20 dB

6 dB

⫺23 dB

1200

6 kHz Hz

600

Example 2-23

What is the cutoff frequency of a single-section RC low-pass i lter with R58.2 kV and C50.0033 µF?

fco5 1

2πRC5 1

2π(8.2310³) (0.0033310²⁶) f_co55881.56 Hz  or  5.88 kHz

At the cutoff frequency, the output amplitude is 70.7 percent of the input amplitude at lower frequencies. This is the so-called 3-dB down point. In other words, this i lter has a voltage gain of 23 dB at the cutoff frequency. At frequencies above the cutoff frequency, the amplitude decreases at a linear rate of 6 dB per octave or 20 dB per decade. An octave is dei ned as a doubling or halving of frequency, and a decade represents a one-tenth or times-10 relationship. Assume that a i lter has a cutoff of 600 Hz. If the frequency doubles to 1200 Hz, the attenuation will increase by 6 dB, or from 3 dB at cutoff to 9 dB at 1200 Hz. If the frequency increased by a factor of 10 from 600 Hz to 6 kHz, the attenuation would increase by a factor of 20 dB from 3 dB at cutoff to 23 dB at 6 kHz.

If a faster rate of attenuation is required, two RC sections set to the same cutoff frequency can be used. Such a circuit is shown in Fig. 2-25(a). With this circuit, the rate of attenuation is 12 dB per octave or 40 dB per decade. Two identical RC circuits are used, but an isolation or buffer amplii er such as an emitter-follower (gain<1) is used between them to prevent the second section from loading the i rst. Cascading two RC sections without the isolation will give an attenuation rate less than the theoretically ideal 12-dB octave because of the loading effects.

If the cutoff frequency of each RC section is the same, the overall cutoff frequency for the complete i lter is somewhat less. This is caused by added attenuation of the second section.

With a steeper attenuation curve, the circuit is said to be more selective. The disad- vantage of cascading such sections is that higher attenuation makes the output signal considerably smaller. This signal attenuation in the passband of the i lter is called insertion loss.

A low-pass i lter can also be implemented with an inductor and a resistor, as shown in Fig. 2-26. The response curve for this RL i lter is the same as that shown in Fig. 2-24(b).

The cutoff frequency is determined by using the formula fco5 R

2πL Octave

Decade

Insertion loss

Figure 2-25

Two stages of RC fi lter improve the response but increase signal loss. (a) Circuit. (b) Response curve.

V_out V_in

R R

C C

Buffer amplifier to isolate RC sections

(a)

3 dB

12 dB/octave or 40 dB/decade roll-off rate

(b) V_out(max)

f_co 40 dB

12 dB

1200

6 kHz Hz

600

Figure 2-26 A low-pass fi lter implemented with an inductor.

f_co

X_L R

R 2L L

R

The RL low-pass i lters are not as widely used as RC i lters because inductors are usually larger, heavier, and more expensive than capacitors. Inductors also have greater loss than capacitors because of their inherent winding resistance.

High-Pass Filter.

A high-pass i lter passes frequencies above the cutoff frequency with little or no attenuation but greatly attenuates those signals below the cutoff. The ideal high-pass response curve is shown in Fig. 2-27(a). Approximations to the ideal response curve shown in Fig. 2-27(b) can be obtained with a variety of RC and LC i lters.

The basic RC high-pass i lter is shown in Fig. 2-28(a). Again, it is nothing more than a voltage divider with the capacitor serving as the frequency-sensitive component in a voltage divider. At low frequencies, X_C is very high. When X_C is much higher than R, the voltage divider effect provides high attenuation of the low-frequency signals. As the frequency increases, the capacitive reactance decreases. When the capacitive reac- tance is equal to or less than the resistance, the voltage divider gives very little attenu- ation. Therefore, high frequencies pass relatively unattenuated.

The cutoff frequency for this i lter is the same as that for the low-pass circuit and is derived from setting X_C equal to R and solving for frequency:

fco5 1 2πRC The roll-off rate is 6 dB per octave or 20 dB per decade.

A high-pass i lter can also be implemented with a coil and a resistor, as shown in Fig. 2-28(b). The cutoff frequency is

fco5 R 2πL

The response curve for this i lter is the same as that shown in Fig. 2-27(b). The rate of attenuation is 6 dB per octave or 20 dB per decade, as was the case with the low-pass i lter. Again, improved attenuation can be obtained by cascading i lter sections.

High-pass fi lter Figure 2-27

Frequency response curve of a high-pass filter. (a) Ideal. (b) Practical.

Output

Frequency

f_co f_co

(a)

Frequency (b) 0 dB

3 dB

6 dB/octave or 20 dB/decade

f_co⫽ 1 2␲RC Passband

Figure 2-28 (a) RC high-pass fi lter. (b) RL high-pass fi lter.

C

R

(a)

f_co ⫽ 1 2␲RC

R

L

(b)

f_co ⫽ R 2␲L

RC Notch Filter.

Notch i lters are also referred to as bandstop or band-reject i l- ters. Band-reject i lters are used to greatly attenuate a narrow range of frequencies around a center point. Notch i lters accomplish the same purpose, but for a single frequency.

A simple notch i lter that is implemented with resistors and capacitors as shown in Fig. 2-29(a) is called a parallel-T or twin-T notch i lter. This i lter is a variation of a bridge circuit. Recall that in a bridge circuit the output is zero if the bridge is balanced.

If the component values are precisely matched, the circuit will be in balance and produce an attenuation of an input signal at the design frequency as high as 30 to 40 dB. A typical response curve is shown in Fig. 2-29(b).

The center notch frequency is computed with the formula fnotch5 1

2πRC

For example, if the values of resistance and capacitance are 100 kV and 0.02 µF, the notch frequency is

fnotch5 1

6.28(10⁵) (0.02310²⁶) 579.6 Hz

Twin-T notch i lters are used primarily at low frequencies, audio and below. A com- mon use is to eliminate 60-Hz power line hum from audio circuits and low-frequency medical equipment amplii ers. The key to high attenuation at the notch frequency is Notch fi lter (bandstop or band-

reject fi lter)

Parallel-T (twin-T) notch fi lter

Example 2-24

What is the closest standard EIA resistor value that will produce a cutoff frequency of 3.4 kHz with a 0.047-µF capacitor in a high-pass RC i lter?

fco5 1 2πRC

R5 1

2πfcoC5 1

2π(3.4310³) (0.047310²⁶) 5996 V The closest standard values are 910 and 1000 V, with 1000 being the closest.

GOOD TO KNOW

Twin-T notch fi lters are used at low frequencies to eliminate power line hum from audio circuits and medical equipment amplifi ers.

Figure 2-29 RC notch fi lter.

f_notch ⫽ 1 2␲RC

R 2

f_notch

(a) (b)

R R

C C

2C

precise component values. The resistor and capacitor values must be matched to achieve high attenuation.