Kienth Vincent Oracion BSEE-3 Allan R. Pangan October 8, 2023 Assignment PASSIVE FILTER CIRCUITS -----Low-pass filter circuit----A low-pass filter is an electronic circuit that allows all frequencies below a certain cutoff point to pass through without any reduction in amplitude while effectively blocking or reducing signals with frequencies above this cutoff point. These filters are also occasionally called high cut filters. Graph: Ideal response curve of a low-pass filter. Circuit: RC low-pass filter circuit and operation graph. Mathematical Equations: Cut off frequency derivation. The cut-off frequency of the low-pass filter circuit occurs when the resistance “R” and capacitive reactance “Xc” are equal. ππ = π 1 =π 2πππΆ ππππ−πππ = π ππ πΉπͺ Let’s say for example, ππ π = 3,000 πβππ , πΆ = 600ππΉ, π‘βπ ππ’π‘ πππ πππππ’ππππ¦ πππ ππ πππππ’πππ‘ππ ππ¦: πππ’π‘−πππ = 1 2π(3000)(600π₯10−12 ) πππ’π‘−πππ = 88,419.41 π»π§ ππ 88.4ππ»π§ Application: The low pass filter applications include the following. • • • • • • • Used to remove the noise of high-frequency signals Used in audio applications Used in biomedical applications Used in electronic applications like loudspeakers, subwoofers, etc Used in digital to analog converters Used as anti-analyzing filters Used in wave analyzers, audio amplifiers, and equalizers. RL low-pass filter: An uncommon low-pass filter An inductor can also be used to form a low-pass filter circuit. The response curve of this RL low-pass filter is the same as the RC one. However, these are uncommon because inductors are usually heavier, larger, and more expensive compared to capacitors. They also possess greater loss because of the nature of the windings. ππ = π and we know that, ππ = 2πππΏ, so 2πππΏ = π ππππ−πππ = πΉ ππ π³ Explanation: Operation behind the low-pass filter circuit. The simplest form of low-pass filter is the RC circuit shown above. The circuit forms a simple voltage divider with one frequency-sensitive component, in this case the capacitor. At very low frequencies, the capacitor has very high reactance com- pared to the resistance and therefore the attenuation is minimum. As the frequency increases, the capacitive reactance decreases. When the reactance becomes smaller than the resistance, the attenuation increases rapidly. -----High-pass filter circuit----A high-pass filter (HPF) is an electronic circuit or a digital signal processing algorithm designed to allow signals with frequencies higher than a certain cutoff frequency to pass through while attenuating or blocking signals with frequencies below the cutoff frequency. In essence, it lets high-frequency components of a signal through while reducing or eliminating low-frequency components. Graph: Frequency response curve of an ideal and practical high-pass filter. • IDEAL • PRACTICAL Circuit: RC and RL high-pass filter circuit. Mathematical Equation: Cut-off frequency derivation. The cut-off frequency of the RC high-pass filter circuit is the same as the low-pass filter which occurs when the resistance “R” and capacitive reactance “Xc” are equal. ππ = π 1 =π 2πππΆ ππππ−πππ = π ππ πΉπͺ Let’s say for example, ππ π = 4,000 πβππ , πΆ = 700ππΉ, π‘βπ ππ’π‘ πππ πππππ’ππππ¦ πππ ππ πππππ’πππ‘ππ ππ¦: πππ’π‘−πππ = 1 2π(4000)(700π₯10−12 ) πππ’π‘−πππ = 56,841.05 π»π§ ππ 56.8ππ»π§ RL high-pass filter circuit: cut-off frequency derivation. ππ = π and we know that, ππ = 2πππΏ, so 2πππΏ = π ππππ−πππ = πΉ ππ π³ Applications: Common applications of RC high-pass filter. • • • • • Audio: In audio processing, high-pass filters can be used to remove low-frequency noise, such as rumble and hum, while preserving the higher-frequency audio content. Image Processing: High-pass filters can be applied to images to enhance the edges and fine details, making them useful for tasks like edge detection and sharpening. Signal Processing: In signal analysis and communication systems, high-pass filters can help isolate and analyze the high-frequency components of a signal. RF (Radio Frequency) and Electronics: High-pass filters are used in RF circuits to allow radio frequency signals to pass while blocking lower-frequency interference. Control Systems: High-pass filters can be used in control systems to eliminate or reduce low-frequency disturbances that can affect the system’s performance. Theoretical Explanation: High-pass filter The basic RC high-pass filter is shown in the circuit above. 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, XC is very high. When XC 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 reactance is equal to or less than the resistance, the voltage divider gives very little attenuation. Therefore, high frequencies pass relatively unattenuated. -----Band-pass filter circuit----We can say that a Band pass filter is a combination of both low pass filter and high pass filter. A bandpass filter is one that allows a narrow range of frequencies around a center frequency fc to pass with minimum attenuation but rejects frequencies above and below this range. In audio applications, sometimes it is necessary to pass only a certain range of frequencies. This frequency range does not start at 0Hz or doesn’t end at very high frequency but these frequencies are within a certain range, either wide or narrow. These bands of frequencies are commonly termed Bandwidth. Graph: Ideal and practical response curve of a bandpass filter. Circuit: Bandpass circuit using R, L, and C components. Mathematical Equation: Derivation of bandpass working frequency. The frequency in which the bandpass filter circuit allow occurs at resonance frequency when XL=XC. ππΆ = ππΏ 1 = 2πππΏ 2πππΆ ππ2 = ππ = 1 4π 2 πΏπΆ π ππ √π³πͺ Lower range frequency (f1): π ππ ππ = ππ − π©πΎ π€βπππππ, π΅π = = π2 − π1 π π − ππππ‘ππ Upper range frequency (f2): π ππ ππ = ππ + π©πΎ π€βπππππ, π΅π = = π2 − π1 π π − ππππ‘ππ Alternatives: Cascading RLC bandpass filter circuit to improve selectivity Improved selectivity with steeper “skirts” on the curve can be obtained by cascading several bandpass sections. Several ways to do this are shown above. As sections are cascaded, the bandwidth becomes narrower and the response curve becomes steeper. Output graph of the cascading RLC filter circuit. Applications: Common applications of bandpass filter circuit. • • • • • • These are used in wireless communication medium at transmitter and receiver circuits. In transmitter section this filter will pass the only required signals and reduces the interfering of signals with other stations. In receiver section, it will help from unwanted signal penetration in to the channels. These are used to optimize the signal to noise ratio of the receiver. These are used in optical communication area like LIDARS. They are used in some of the techniques of colour filtering. These are also used in medical field instruments like EEG. In telephonic applications, at DSL to split phone and broad band signals. Explanation: • Series Bandpass RLC: A Series resonant circuit is connected in series with an output resistor, forming a voltage divider. At frequencies above and below the resonant frequency, either the inductive or the capacitive reactance will be high compared to the output resistance. Therefore, the output amplitude will be very low. However, at the resonant frequency, the inductive and capacitive reactances cancel, leaving only the small resistance of the inductor. Thus, most of the input voltage appears across the larger output resistance. • Parallel Bandpass RLC: A parallel resonant bandpass filter is shown in Fig. 2-36(b). Again, a voltage divider is formed with resistor R and the tuned circuit. This time the output is taken from across the parallel resonant circuit. At frequencies above and below the center resonant frequency, the impedance of the parallel tuned circuit is low compared to that of the resistance. Therefore, the output voltage is very low. Frequencies above and below the center frequency are greatly attenuated. At the resonant frequency, the reactances are equal and the impedance of the parallel tuned circuit is very high compared to that of the resistance. Therefore, most of the input voltage appears across the tuned circuit. -----Bandstop filter circuit----The Band Stop Filter, (BSF) is another type of frequency selective circuit that functions in exactly the opposite way to the Band Pass Filter we looked at before. The band stop filter, also known as a band reject filter, passes all frequencies with the exception of those within a specified stop band which are greatly attenuated. Graph: Bandstop filter circuit frequency response. Circuit: Simple RC bandstop filter circuit using R, L, and C components in (a.) series and (b.) parallel Mathematical Equation: RLC bandstop filter circuit ππΆ = ππΏ 1 = 2πππΏ 2πππΆ ππ2 = ππ = 1 4π 2 πΏπΆ π ππ √π³πͺ Bandstop filter design schematics: Sample Calculation: The upper and lower cut-off frequency points for a band stop filter can be found using the same 1 formula as that for both the low and high pass filters as shown. π = 2ππ πΆ • • Low-pass filter section ππ π πΏ = 8π πβππ πππ πΆ = 0.1ππΉ 1 ππΏ = 2π(8000)(0.1π₯10−6 ) • ππΏ = 200π»π§ High-pass filter section ππ π π» = 2π πβππ πππ πΆ = 0.1ππΉ 1 ππ» = 2π(2000)(0.1π₯10−6 ) • ππ» = 800π»π§ • ππΆ = √ππΏ ∗ ππ» = √200 ∗ 800 = 400π»π§ • ππ΅π = ππ» − ππΏ = 800 − 200 = 600π»π§ • ππΆ π=π π΅π 400 = 600 = 0.67 ππ − 3.5ππ΅ Applications: In different technologies, these filters are used at different varieties. • • • • • • • • In telephone technology, these filters are used as the telephone line noise reducers and DSL internet services. It will help to remove the interference on the line which will reduce the DSL performance. These are widely used in the electric guitar amplifiers. Actually,this electric guitar produces a ‘hum’ at 60 Hz frequency. This filter is used to reduce that hum in order to amplify the signal produced by the guitar amplifier and makes the best equipment. These are also used in some of the acoustic applications like mandolin, base instrument amplifiers. In communication electronics the signal is distorted due to some noise (harmonics) which makes the original signal to interfere with other signals which lead to errors in the output. Thus, these filters are used to eliminate these unwanted harmonics. These are used to reduce the static on radio, which are commonly used in our daily life. These are also used in Optical communication technologies, at the end of the optical fiber there may be some interfering (spurious) frequencies of light which makes the distortions in the light beam. These distortions are eliminated by band stop filters. The best example is in Raman spectroscopy. In image and signal processing these filters are highly preferred to reject noise. These are used in high quality audio applications like PA systems (Public address systems). These are also used in medical field applications,i.e., in biomedical instruments like EGC for removing line noise. Explanation: Operation behind bandstop filter • • Series bandstop RLC The series LC resonant circuit forms a voltage divider with input resistor R. At frequencies above and below the center rejection or notch frequency, the LC circuit impedance is high compared to that of the resistance. Therefore, signals at frequencies above and below center frequency are passed with minimum attenuation. At the center frequency, the tuned circuit resonates, leaving only the small resistance of the inductor. This forms a voltage divider with the input resistor. Since the impedance at resonance is very low compared to the resistor, the output signal is very low in amplitude. Parallel bandstop RLC A parallel version of this circuit is shown in Fig. 2-39(b), where the parallel resonant circuit is connected in series with a resistor from which the output is taken. At frequencies above and below the resonant frequency, the impedance of the parallel circuit is very low; there is, therefore, little signal attenuation, and most of the input voltage appears across the output resistor. At the resonant frequency, the parallel LC circuit has an extremely high resistive impedance compared to the output resistance, and so minimum voltage appears at the center frequency.
