Chapter 19. Basic Filters - PDF Free Download

Chapter 19 Basic Filters

Objectives Analyze the operation of RC and RL lowpass filters Analyze the operation of RC and RL highpass filters Analyze the operation of band-pass filters Analyze the operation of band-stop filters Investigate a real-world application of filters

Low-Pass Filters The range of low frequencies passed by a low-pass filter within a specified limit is called the passband (or bandpass) of the filter. The point considered to be the upper end of the passband is at the critical frequency (f c ). Lowpass filters have an upper critical frequency usually denoted as f CH. Critical frequency is the frequency at which the filter s output voltage is 70.7% of the maximum. Critical frequency is also called the cutoff frequency, break frequency, or -3dB frequency.

Figure 19--11 Low-pass filter block diagram and response curve. Non-ideal: Frequencies from 0 Hz to -3dB are said to be in the passband. Frequencies greater than -3dB but less than or equal to -6dB are said to be in the transition band. Frequencies greater than -6dB are said to be in the stopband.

Low-Pass Filters For an RC low-pass filter, the output voltage is taken across the capacitor. For an RL low-pass filter, the output voltage is taken across the resistor.

RC Low-Pass Filter When the input is dc, the output voltage equals the input voltage because X C is infinitely large. As the input frequency is increased, X C decreases. As X C decreases, V out also decreases. The critical frequency (f c ) of the filter is reached when X C = R: f c = 1/(2 RC)

RC Low-Pass Filter At any frequency, the output voltage magnitude is: V out = (X C / R 2 + X 2 C)V in The ratio of output voltage to input voltage at the critical frequency can be expressed in decibels as: 20 log(v out /V in ) = -3 db

RC Low-Pass Filters (Inverted-L) IT = ES/ZT VR = IT x R VC = IT x XC This is a LAG network: VC (Vout) lags ES by (-90 + θ = φ) By Convention, φ is used to denote phase for filter circuits. The distinction between θ and φ is removed when dealing with filter circuits. We use only one symbol φ (phi) to denote phase on filter phasor diagrams.

The RC Circuit as a Low Pass Filter The RC Low Pass Filter is also an AC voltage divider, with R the output taken across the Capacitor. Es Vout V OUT = (X C / R 2 +X C2 ) x E S C Xc At low f, X C increases, so R is negligible and V OUT E S, so little attenuation; this is the pass band. At f c, V C = V R because R = X C ; V OUT = E S / 2 or (0.707)E S or 3dB. At f > f c, V OUT decreases, and becomes the stop band. Insertion loss or db ATTEN = 20 log V OUT /V IN = 20 log (X C / R 2 +X C2 )

RL Low-Pass Filter The output voltage is taken across the resistor. When the input is dc, the output voltage equals the input voltage because X L is a short. As the input frequency is increased, X L increases. As X L increases, V out decreases. The critical frequency (f c ) of the filter is reached when X L = R: f c = 1/(2 (L/R))

RL Low-Pass Filters (Inverted-L) ES = VR + jvl ZT = R + jxl This is a LAG network: VR lags ES by θ By Convention, φ is used to denote phase for filter circuits. The distinction between θ and φ is removed when dealing with filter circuits. We use only one symbol φ (phi) to denote phase on filter phasor diagrams.

The RL Circuit as a Low Pass Filter The RL Low Pass Filter is also an AC voltage divider, with the output taken across the Resistor. L V OUT = (R/ R 2 +X L2 ) x E S Es R At low f, X L decreases, so X L is negligible and V OUT E S, so little attenuation; this is the pass band. At f c, V L = V R because R = X L ; V OUT = E S / 2 or (0.707)E S or 3dB. At f > f c, V OUT decreases, and becomes the stop band. Insertion loss or db ATTEN = 20 log V OUT /V IN = 20 log (R/ R 2 +X L2 )

Response Curve Roll-Off The maximum output is defined to be 0 db as a reference (V out = V in ). The output drops from 0 db to -3 db at cutoff. Output drops at a fixed roll-off rate above cutoff.

Attenuation for a Basic RC or RL Low-Pass Circuit The roll-off rate, or attenuation, as the frequency continues to increase above f C is at -20 db for each tenfold increase in frequency (or -6dB/octave). A tenfold change in frequency is called a decade. The roll-off is a constant -20 db/decade for a basic RC or RL filter (or -6dB/octave). Frequencies from 0Hz to f c are said to be in the passband of the filter. Frequencies above f c are said to be in the stopband of the filter. The frequency response is plotted on a semilog scale. The response curve is called a Bode plot.

RC or RL Low-Pass Filter Response Bode Plot

Phase Shift in a Low-Pass Filter The RC low-pass filter acts as a lag network. The phase shift between output and input is: = -90 + -tan -1 (X C /R) The RL low-pass filter also acts as a lag network. The phase shift between output and input is: θ = -tan -1 (X L /R) (which is denoted as on phase-shift diagrams) At the critical frequency, in both cases, phase-shift = -45. As f approaches 0 Hz, decreases and approaches 0.

Low-Pass Filters Example 1 Determine the output voltage of each filter below at the specified frequency when Vin = 10V. Determine f c for each filter.

Low-Pass Filters Example 1 (cont.)

Low-Pass Filters Example 2 For the filter below, calculate the value of C required for each of the following critical frequencies: (a) 60 Hz (b) 500 Hz (c) 1kHz (d) 5kHz Draw the idealized Bode plot for each critical frequency above.

Low-Pass Filters Example 2 (cont.)

The Order Of A Filter Filters are often expressed with respect to the rolloff rate designated as the number of poles or the filter order. Poles are defined as the number of capacitors or inductors used in a filter. There are some exceptions to this rule. More on this later!!! Each pole introduces a -20dB/decade roll-off rate. For example, a two-pole filter or second order filter will have a -40dB/decade roll-off rate.

Increasing Low-Pass RC Filter Roll-off Rates The number of poles or the order of RC low pass filters is defined as the number of capacitors used. The T and Inverted-L configurations are first-order filters (per section) while the PI configuration is a second-order filter.

Increasing Low-Pass RL Filter Roll-off Rates The number of poles or the order of RL low pass filters is defined as the number of inductors used. The PI and Inverted-L configurations are first-order filters (per section) while the T configuration is a second-order filter.

High-Pass Filters A high-pass filter allows signals with higher frequencies to pass from input to output while rejecting lower frequencies. The range of high frequencies passed by a highpass filter within a specified limit is called the passband (or bandpass) of the filter. The frequency considered to be the lower end of the passband is called the critical frequency f c or f CH. The critical frequency has an output which is 70.7% of the maximum.

Figure 19--11 High-pass filter block diagram and response curve. Non-ideal: Frequencies from 0 Hz to less than -6dB are said to be in the stopband. Frequencies greater than or equal to -6dB but less than -3dB are said to be in the transition band. Frequencies greater than or equal to -3dB are said to be in the passband.

High-Pass Filters For an RC high-pass filter, the output voltage is taken across the resistor. For an RL high-pass filter, the output voltage is taken across the inductor.

RC High-Pass Filter When the input frequency is at its critical value, X C =R, the output voltage is 0.707V (-3dB). As the input frequency increases above f C, X C decreases. As X C decreases, the output voltage increases, approaching the value of V in. The critical frequency is: f c = 1/2 RC

RC High-Pass Filters (Inverted-L) ES = VR jvc ZT = R jxc This is a LEAD network: VR leads ES by θ By Convention, φ is used to denote phase for filter circuits. The distinction between θ and φ is removed when dealing with filter circuits. We use only one symbol φ (phi) to denote phase on filter phasor diagrams.

The RC Circuit as a High Pass Filter The RC High Pass Filter is also an AC voltage divider, with the output taken across the Resistor. C V OUT = (R/ R 2 +X C2 ) x E S At high f, X C decreases, so X C is negligible and V OUT E S, so little attenuation; this is the pass band. At f c, V C = V R because R = X C ; V OUT = E S / 2 or (0.707)E S or 3dB. At f < f c, V OUT decreases, and becomes the stop band. Insertion loss or db ATTEN = 20 log V OUT /V IN = 20 log (R/ R 2 +X C2 ) Es Xc R Vout

RL High-Pass Filter When the input frequency is at its critical value, X L =R, the output voltage is 0.707V in (-3dB). As the input frequency increases above f c, X L increases. As X L increases, the output voltage increases, approaching the value of V in. The critical frequency is: f c = 1/(2 (L/R))

RL High-Pass Filters (Inverted-L) IT = ES/ZT VR = IT x R VL = IT x XL This is a LEAD network: VL leads ES by (90 - θ) = φ By Convention, φ is used to denote phase for filter circuits. The distinction between θ and φ is removed when dealing with filter circuits. We use only one symbol φ (phi) to denote phase on filter phasor diagrams.

The RL Circuit as a High-Pass Filter The RL High Pass Filter is also an AC voltage divider, with the output taken across the Inductor. V OUT = (X L / R 2 +X L2 ) x E S At high f, X L increases, so R is negligible and V OUT E S, so little attenuation; this is the pass band. At f c, V L = V R because R = X L ; V OUT = E S / 2 or (0.707)E S or 3dB. At f < f c, V OUT decreases, and becomes the stop band. Insertion loss or db ATTEN = 20 log V OUT /V IN = 20 log (X L / R 2 +X L2 )

High-Pass Filter Response Curve Below f c, the output voltage decreases (rolls off) at a rate of - 20 db/decade. (Alternatively, we can also say that the output voltage increases at a rate of +20dB/decade up to f c.) The figure below shows actual and ideal response.

Attenuation for a Basic RC or RL High-Pass Circuit The roll-off rate, or attenuation, as the frequency continues to decrease below f C is at -20 db for each tenfold decrease in frequency (or -6dB/octave). A tenfold change in frequency is called a decade. The roll-off is a constant -20 db/decade for a basic RC or RL filter (or -6dB/octave). Frequencies from f c to infinity are said to be in the passband of the filter. Frequencies below f c are said to be in the stopband of the filter. The frequency response is plotted on a semilog scale. The response curve is called a Bode plot.

RC or RL High-Pass Filter Response Bode Plot

Phase Shift in a High-Pass Filter The RC high-pass filter acts as a lead network. The phase shift between output and input is: θ = tan -1 (X C /R) (denoted as on phase-shift diagrams) The RL high-pass filter acts as a lead network. The phase shift between output and input is: = 90 - tan -1 (X L /R) At the critical frequency, in both cases, = 45. As f increases, decreases and approaches 0.

High-Pass Filters Example 1 Determine the output voltage of each filter below at the specified frequency when Vin = 10V. Determine f c for each filter. Draw the idealized Bode plot for each filter.

High-Pass Filters Example 1 (cont.)

Increasing High-Pass RC Filter Roll-off Rates The number of poles or the order of RC high pass filters is defined as the number of capacitors used. The T and PI configurations are first-order filters (per section) while the T configuration is a second-order filter.

Increasing High-Pass RL Filter Roll-off Rates The number of poles or the order of RL high pass filters is defined as the number of inductors used. The T and Inverted-L configurations are first-order filters (per section) while the PI configuration is a second-order filter.

Band-Pass Filters A combination of low-pass and high-pass filters can be used to form a band-pass filter. If the critical frequencies of the low-pass (f c(l) ) and highpass (f c(h) ) filters overlap; responses overlap.

Band-Pass Filters (Non-Resonant) Vo Vo Exception to Rule: Bandpass filters with two capacitors or two inductors per section are one pole or first-order because the cutoff frequencies are chosen far enough apart so as to not introduce additional attenuation outside the passband.

Band-Pass Filter Bandwidth (Non-Resonant) Non-resonant passive filter circuits use f o to designate the center frequency since f r doesn t exist. The bandwidth (BW) of resonant or non-resonant bandpass filters is the range of frequencies for which the current, and therefore the output voltage, is equal to or greater than 70.7% of its value at the resonant or center frequency, respectively. Applies only to non-resonant passive bandpass filters: Since the cutoff frequencies are chosen far enough apart as to not introduce additional attenuation outside the passband, the order of the bandpass filter is defined differently from just a high-pass or low-pass filter individually. The order of the bandpass filter is the order of either the high-pass or the low-pass section per filter section.

Series Resonant Band-Pass Filter A series resonant circuit has minimum impedance and maximum current at the resonant frequency, f r Most of the input voltage is dropped across the resistor at the resonant frequency, f o (center f ). Although the circuit Q is much greater than 1, there is no voltage magnification across R so the maximum filter voltage gain is 1. A series resonant bandpass filter is a first-order filter when the output voltage is taken across R. BW = f O /Q

Band-Pass Filter Example 1 Determine the center frequency for each filter. Neglect R W. If the coil resistance for each filter is 10 ohms, find the bandwidth.

Band-Pass Filter Example 1 (cont.)

Parallel Resonant Band-Pass Filter A parallel resonant circuit has maximum impedance at resonance. At resonance, the impedance of the tank is high, producing maximum output voltage at resonance. Although the circuit Q is much greater than 1, there is no voltage magnification across a tank circuit so the maximum filter voltage gain is 1. A parallel resonant bandpass filter is a first-order filter when the output voltage is taken across the tank.

Band-Pass Filter Example 2 Determine the center frequency for each filter. Neglect R W. If the coil resistance for each filter is 4 ohms, what is the output voltage at resonance when V in = 120V.

Band-Pass Filter Example 2 (cont.)

Band-Stop Or Notch Filters A band-stop filter can be formed from a low-pass and a high-pass filter connected in parallel. The low-pass critical frequency (f c(l) ) is set lower than the high-pass critical frequency (f c(h) ).

Band-Stop Filters (Non-Resonant) Exception to Rule: Bandstop filters with two capacitors or two inductors per section are one pole or first-order because the cutoff frequencies are chosen far enough apart as to not introduce additional attenuation (or gain) outside the stopband. Vo Vo

Band-Stop Filter Bandwidth (Non-Resonant) Non-resonant passive filter circuits use f o to designate the center frequency since f r doesn t exist. The bandwidth (BW) of resonant or non-resonant bandpass filters is the range of frequencies for which the current, and therefore the output voltage, is equal to or greater than 70.7% of its value at the resonant or center frequency, respectively. Applies only to non-resonant passive bandstop filters: Since the cutoff frequencies are chosen far enough apart as to not introduce additional attenuation (or gain) outside the stopband, the order of the bandstop filter is defined differently from just a highpass or low-pass filter individually. The order of the bandstop filter is the order of either the high-pass or the low-pass section per filter section.

Series Resonant Band-Stop Filter A series resonant circuit can be used in a band-stop configuration by taking the output across the series LC circuit. At resonant frequency, the impedance is minimum, and therefore the output voltage is minimum. As the LC impedance increases, above and below resonance, output voltage increases. Although the circuit Q is much greater than 1, there is maximum voltage attenuation across the L-C combination. A series resonant bandstop filter is a first-order filter when the output voltage is taken across the L-C combination. BW = f O /Q

Band-Stop Filter Example 1 Determine the center frequency for each filter. Neglect R W.

Parallel Resonant Band-Stop Filter A parallel resonant circuit can be used in a band-stop configuration by taking the output across the resistor. At resonant frequency, the tank impedance is maximum, and so most of the output voltage appears across it. As the tank impedance decreases, above and below resonance, output voltage increases. Although the circuit Q is much greater than 1, there is maximum voltage attenuation across R. A parallel resonant bandstop filter is a first-order filter when the output voltage is taken across R. BW = f O /Q

Band-Stop Filter Example 2 Determine the center frequency for each filter. Neglect R W. If the coil resistance for each filter is 8 ohms, what is the output voltage at resonance when V in = 50V.

Band-Stop Filter Example 2 (cont.)

Filters Application: Speaker Crossover Networks You should remember: 1. A passive crossover requires no external power source to operate. 2. A passive crossover uses caps, coils and resistors to attenuate the signal level above and/or below a certain frequency. 3. Passive crossover networks are designed to pass selected frequencies to speaker drivers that acoustically reproduce those selected frequencies the most efficiently and accurately. - Speaker systems are designed as one-way, two-way, three-way, and four-way systems. The way means how many speaker drivers are connected together (usually in parallel) per channel output. - For example, a three-way speaker system contains a woofer driver, a midrange driver, and a tweeter driver. The woofer reproduces the low frequencies the best, the midrange the middle frequencies the best, and the tweeter the high frequencies the best for a given frequency range. It is the crossover network s job to make sure each speaker driver receives the range of frequencies it can most accurately and efficiently reproduce acoustically.

Filters Application: Speaker Crossover Networks You should remember: 4. Passive crossover networks are designed to pass selected frequencies to speaker drivers that acoustically reproduce those selected frequencies the most efficiently and accurately. Speaker systems are designed as one-way, two-way, three-way, and four-way systems. The way means how many speaker drivers are connected together (in parallel or series) per channel output. For example, a three-way speaker system contains a woofer driver, a midrange driver, and a tweeter driver. The woofer reproduces the low frequencies the best, the midrange the middle frequencies the best, and the tweeter the high frequencies the best for a given frequency range. It is the crossover network s job to make sure each speaker driver receives the range of frequencies it can most accurately and efficiently reproduce acoustically. Cut-off frequency guidelines; note that they are based on the use of 12-dB-per-octave slopes. SUBWOOFERS: Below 100 Hz (low-pass). WOOFERS/MIDBASSES: Between 100 and 500 Hz (low-pass or bandpass). MIDRANGES: Between 300 and 500 Hz (high-pass) and 3,500 and 8,000 Hz (low-pass) bandpass. TWEETERS: Above 5,000 Hz (high-pass). 5. In general, a high-pass filter with a slope of at least 12 db per octave should be used with midranges and tweeters, since this will protect them from potentially damaging low frequencies.

Filters Application: Speaker Crossover Networks Simple, one-way speaker systems. Most of the crossover networks shown are LC filters in Inverted-L and T configurations. For low and high pass LC filters, an L-C pair is two-poles or second order when the output voltage is taken across L or C. For bandpass or bandstop, an L-C filter pair is one-pole.

Filters Application: Speaker Crossover Networks

Passive Crossover Design Tables

Filters Application: Speaker Crossover Networks Passive High Pass Crossovers In the next four slides, there are 4 different crossover configurations. The graph shown after the 4 systems shows the slopes for -6dB (first order), -12dB (second order), -18dB (third order) and -24dB (fourth order) per octave crossovers. The crossover components' colors match its corresponding curve on the graph.

Passive High Pass Crossovers In this crossover, the capacitor blocks the lower frequencies while allowing the higher frequencies to pass. Roll-off Rate is -6dB/octave or -20dB/decade

Passive High Pass Crossovers In this crossover, the capacitor does the same thing as in the previous diagram. The inductor shunts, to ground, some of the low frequencies that are allowed to pass through the capacitor. This causes a higher roll off rate. Roll-off Rate is -12dB/octave or -40dB/decade

Passive High Pass Crossovers Roll-off Rate is -18dB/octave or -60dB/decade

Passive High Pass Crossovers Roll-off Rate is -24dB/octave or -80dB/decade

Passive High Pass Crossovers

Passive Low Pass Crossovers In the next four slides, there are 4 different crossover configurations. The graph shown after the 4 systems shows the slopes for -6dB (first order), -12dB (second order), -18dB (third order) and -24dB (fourth order) per octave crossovers. The crossover components' colors match its corresponding curve on the graph.

Passive Low Pass Crossovers In this crossover, the inductor blocks the higher frequencies while allowing the lower frequencies to pass. Roll-off Rate is -6dB/octave or -20dB/decade

Passive Low Pass Crossovers In this crossover, the inductor does the same thing as in the previous diagram. The capacitor shunts, to ground, some of the higher frequencies that are allowed to pass through the inductor. This causes a higher roll off rate. Roll-off Rate is -12dB/octave or -40dB/decade

Passive Low Pass Crossovers Roll-off Rate is -18dB/octave or -60dB/decade

Passive Low Pass Crossovers Roll-off Rate is -24dB/octave or -80dB/decade

Passive Low Pass Crossovers

Other Types of Crossover Networks Basics: When you double the cone area (add a second speaker with equal properties) while keeping the power constant, you gain 3dB of output. If you reduce the number of speakers by 1/2, you lose 3dB. If you double the power to a driver, you gain 3dB. If you cut the power by 1/2 you lose 3dB. If you double the cone area and the power (by paralleling the second speaker on the amplifier), you gain 6dB. The 'Q' of a filter (crossover) indicates the shape of the curve. For a second order crossover, it can be calculated with the formula: Q=[(R 2 C)/L] 1/2 Where R is the speaker's impedance. C is the capacitor used in the filter. And L is the inductor used in the filter.

Other Types of Crossover Networks Basics (continued): Acoustical Output & Power Whenever two loudspeakers are playing the same frequencies in phase, in the same location, there will be up to a 3 db rise in the acoustical output. When crossovers are used, both the low pass filtered loudspeaker and the high pass filtered loudspeaker are down by 3 db at the crossover frequency. Their combined output will be up to 3 db higher at the crossover point. The net result is neither a rise or fall in resistance to the amplifier or acoustical output at the crossover frequency as indicated by the yellow line between the low pass and high pass curves. This means if both loudspeakers are 4 ohm, the amplifier will see a 4 ohm load below, at, and above the crossover frequency (*excluding the natural impedance curves of the loudspeakers). The crossover not only separates the frequency ranges for the different loudspeakers in a speaker system, but also separates these frequency ranges and impedance (resistance) ranges for the amplifier.

2nd order Linkwitz-Riley In the following graph, you can see the response for both the high and low frequency drivers. You can also see that the crossover point is 150hz. As previously noted, the on-axis acoustic output of a L-R crossover has a flat response at the crossover frequency. To do this, the crossover point has to be 6 db down (2 nd -order). Since there are 2 drivers (a midrange and a woofer) operating at the crossover point and they presumably have a comparable output and they are receiving the same power (both 6dB down from full power), the output (their summed on-axis acoustic output) will be as if a single driver were playing at the crossover point. This will provide a flat overall frequency response at the crossover point.

2nd order Bessel On the next graph, you'll see the response of a 2nd order Bessel crossover. You can see that the crossover point is 5dB down from the pass band. The summed response will give you a slight peak at the crossover point. The high pass and low pass curves have a Q of 0.58.

2nd order Butterworth On the next graph, you'll see the response of a 2nd order Butterworth crossover. The crossover point is 3dB down from the pass band. The summed response will give you a 3dB peak at the crossover point. The high pass and low pass curves have a Q of 0.707.

2nd order Chebychev On the next graph, you'll see the response of a 2nd order Chebychev crossover. The crossover point is at the same level as the pass band. The summed response will give you a 6dB peak at the crossover point. The high pass and low pass curves have a Q of 1.0.

Open Crossover Output Warning You may damage your amplifier if you drive a second (or higher) order crossover when the speaker's voice coil is open (the speaker is blown) or if no speaker is connected to the crossover's output. When the speaker is removed (or the voice coil opens), the circuit becomes a resonant circuit. This circuit will, at the crossover frequency (or some multiple of the crossover frequency), present a 0 ohm load to the amplifier. The actual resistance will be only the resistance in the speaker wire and the inductor. Any time that there is audio at the resonant frequency, the amplifier will be stressed the same as if the speaker wires were shorted together. This will drive some amplifiers into protection. Others will blow a fuse or die a horrible painful death. The following graph shows how the impedance of the normal circuit (violet line) never drops below 4 ohms (the speaker's impedance). It also shows how the impedance of the circuit without a speaker (yellow line) drops to 0 ohms at the crossover frequency (for a 2nd order crossover, the resonant frequency is the same as the crossover frequency).

Summary In an RC low-pass filter, the output voltage is taken across the capacitor and the output lags the input. In an RL low-pass filter, the output voltage is taken across the resistor and the output lags the input. In an RC high-pass filter, the output is taken across the resistor and the output leads the input.

Summary In an RL high-pass filter, the output is taken across the inductor and the output leads the input. The roll-off rate of a basic RC or RL filter is -20 db per decade. A band-pass filter passes frequencies between the lower and upper critical frequencies and rejects all others.

Summary The bandwidth of a resonant filter is determined by the quality factor (Q) of the circuit and the resonant frequency. Critical frequencies are also called -3 db frequencies. The output voltage is 70.7% of its maximum at the critical frequencies.