Filters PDF ECE102-1

ECE102-1 Filters Filter A filter is a circuit capable of passing (or amplifying) certain frequencies while attenuating other frequencies. The following are basic types of filters: Low-Pass filters High-Pass filters Band-Pass filters Band-Stop filters All-Pass filters Filter Frequ...

ECE102-1 Filters Filter A filter is a circuit capable of passing (or amplifying) certain frequencies while attenuating other frequencies. The following are basic types of filters: Low-Pass filters High-Pass filters Band-Pass filters Band-Stop filters All-Pass filters Filter Frequency Response Terminologies Passband: The range of frequencies where the output has a gain. Stopband: The range of frequencies where the output is zero or very small. Passband ripple: The variations or oscillations in the bandpass. Stopband ripple: It represents the variations in the stopband region. Critical frequency: The critical frequency, (also called the cutoff frequency) defines the end of the passband and is normally specified at the point where the response drops (70.7% of Av and 50% G in pass band) from the passband response. Filter Frequency Response Terminologies Stopband frequency: It is the frequency at which the first stopband ripple occurs. Transition band: This represents the range (fs - fc) of frequencies between the critical and stopband frequencies. The slope of the transition region is related to the number of poles in the transfer function of the response, also known as the filter's order. A pole is a root of the denominator of the transfer function. For a standard filter, every pole adds -20 dB/decade or -6 dB/octave to the slope of the response. The slope of the line is called the roll-off of the transition. Filter Frequency Response Terminologies Low-Pass Filter LPF attenuate or suppress signals with frequencies above a particular frequency called the cutoff or critical frequency. For example, a low- pass filter (LPF) with a cutoff frequency of 40 Hz can eliminate noise with a frequency of 60 Hz. High-Pass Filter HPF suppress or attenuate signals with frequencies lower than a particular frequency, also called the cutoff or critical frequency. For example, a high-pass filter (HPF) with a cutoff frequency of 100 Hz can be used to suppress the unwanted DC voltage in amplifier systems, if desired. Band-Pass Filter BPF attenuate or suppress signals with frequencies outside a band of frequencies. They are common in TV or radio tuning circuits. Band-Stop Filter BSF attenuate or suppress signals with a range of frequencies. For instance, a notch filter can reject signals with frequencies between 50 Hz and 150 Hz. They also called band-reject filters, notch filters. All-Pass Filter The figure shows the frequency response of an ideal all-pass filter. It has a passband and no stopband. Because of this, it passes all frequencies between zero and infinite frequency. All-Pass Filter It may seem rather unusual to call it a filter since it has zero attenuation for all frequencies. The reason it is called a filter is because of the effect it has on the phase of signals passing through it. The all- pass filter is useful when we want to produce a certain amount of phase shift for the signal being filtered without changing its amplitude. Order of a Filter The order of a passive filter (symbolized by n) equals the number of inductors and capacitors in the filter. If a passive filter has two inductors and two capacitors, n = 4. If a passive filter has five inductors and five capacitors, n = 10. Therefore, the order tells us how complicated the filter is. The higher the order, the more complicated the filter. The order of an active filter depends on the number capacitors it contains. If an active filter contains eight capacitors, n = 8. Filter Response Characteristics Butterworth Chebyshev Inverse Chebyshev Elliptic Bessel Butterworth Approximation The Butterworth approximation is sometimes called the maximally flat approximation because the passband attenuation is zero through most of the passband and decreases gradually at the edge of the passband. Roll-off Rate: 20n dB/decade 6n dB/octave Where: n is the order of the filter Butterworth Approximation The major disadvantage is the relatively slow roll-off rate compared with the other approximations. Chebyshev Approximation In some applications, a flat passband response is not important. In this case, a Chebyshev approximation may be preferred because it rolls off faster in the transition region than a Butterworth filter. The price paid for this faster roll-off is that ripples appear in the passband of the frequency response. Number of ripples in passband: n/2 Where: n is the order of the filter Chebyshev Approximation The ripples have the same peak-to-peak value. This is why the Chebyshev approximation is sometimes called the equal-ripple approximation. Typically, a designer will choose a ripple depth between 0.1 and 3 dB, depending on the needs of the application. Inverse Chebyshev Approximation In applications in which a flat passband response is required, as well as a fast roll-off, a designer may use the inverse Chebyshev approximation. It has a flat passband response and a rippled stopband response. The roll-off rate in the transition region is comparable to the roll-off rate of a Chebyshev filter. Monotonic means that the stopband has no ripples. With the approximations discussed so far, the Butterworth and Chebyshev filters have monotonic stopbands. The inverse Chebyshev has a rippled stopband. Inverse Chebyshev Approximation When specifying an inverse Chebyshev filter, the minimum acceptable attenuation throughout the stopband must be given because the stopband has ripples that may reach this value. For instance, in the previous figure, the inverse Chebyshev filter has a stopband attenuation of 60 dB. As you can see, the ripples do approach this level at different frequencies in the stopband. Elliptic Approximation Some applications need the fastest possible roll-off in the transition region. If a rippled passband and a rippled stopband are acceptable, a designer may choose the elliptic approximation. Also known as the Cauer filter, this filter optimizes the transition region at the expense of the passband and stopband. Given a set of specifications for any complicated filter, the elliptic approximation will always produce the most efficient design; that is, it will have the lowest order. Bessel Approximation The Bessel approximation has a flat passband and a monotonic stopband similar to those of the Butterworth approximation. For the same filter order, however, the roll-off in the transition region is much less with a Bessel filter than with a Butterworth filter. Given a set of specifications for a complicated filter, the Bessel approximation will always produce the least roll-off of all the approximations. Stated another way: It has the highest order or greatest circuit complexity of all approximations. Bessel Approximation The Butterworth, Chebyshev, inverse Chebyshev, and elliptic approximations are optimized for frequency response only. With these approximations, no attempt is made to control the phase of the output signal. On the other hand, the Bessel approximation is optimized to produce a linear phase shift with frequency. In other words, the Bessel filter trades off some of the roll-off rate to get a linear phase shift. Bessel Approximation A linear phase response implies a constant time delay, which means that all frequencies in the passband are delayed by the same amount of time as they pass through the filter. The amount of time it takes for a signal to pass through a filter depends on the order of the filter. With all filters except the Bessel filter, this amount of time changes with the frequency. With the Bessel filter, the time delay is constant at all frequencies in the passband. This is why the Bessel filter is sometimes referred to as a maximally flat delay filter. Bessel Approximation Elliptic Approximation Bessel Approximation Time Delay Time Delay Passive Filter Passive filter only uses passive components such as resistors, capacitors and inductors, etc. it has a very simple design and is very cheap. This filter does not require an external power source to operate and that is why they do not provide any power gain. However, they do use an inductor that makes them able to withstand high current. Passive Low Pass Filter A simple passive RC Low Pass Filter, can be easily made by connecting in series a single Resistor with a single Capacitor as shown below. In this type of filter arrangement, the input signal (VIN) is applied to the series combination (both the Resistor and Capacitor together) but the output signal (VOUT) is taken across the capacitor only. Output Voltage: 𝑋𝑐 𝑋𝑐 𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 = 𝑉𝑖𝑛 2 𝑅 + 𝑋𝑐 2 𝑍 Cut-off frequency: 1 𝑓𝑐 = Q: Can an LPF be designed using LR? 2𝜋𝑅𝐶 Example A Low Pass Filter circuit consisting of a resistor of 47kΩ in series with a capacitor of 47nF is connected across a 10v sinusoidal supply. Calculate the output voltage at a frequency of 100Hz and again at frequency of 10kHz. Also, calculate its cut-off frequency. 2nd-Order Low Pass Filter The circuit shown uses two passive first-order low pass filters connected or “cascaded” together to form a second-order or two-pole filter network. Cut-off frequency: 1 𝑓𝑐 = 2𝜋 𝑅1 𝐶1 𝑅2 𝐶2 nth-Order Low Pass Filter If there are n sections of RC low pass filter that are connected together, the roll-off rate and the cut-off frequency will change: Roll-off rate: 20n dB/decade 6n dB/octave Cut-off frequency: 𝑓𝑐 𝑛 = 𝑓𝑐 1 21/(𝑛−1) − 1 Passive High Pass Filter A High Pass Filter is the exact opposite to the low pass filter circuit as the two components have been interchanged with the filters output signal now being taken from across the resistor Output Voltage: 𝑅 𝑅 𝑉𝑜𝑢𝑡 = 𝑉𝑖𝑛 = 𝑉𝑖𝑛 𝑅2 + 𝑋𝑐2 𝑍 Cut-off frequency: 1 𝑓𝑐 = 2𝜋𝑅𝐶 Q: Can an HPF be designed using LR? Example Calculate the cut-off or “breakpoint” frequency (ƒc) for a simple passive high pass filter consisting of an 82pF capacitor connected in series with a 240kΩ resistor. 2nd-Order High Pass Filter The circuit shown uses two passive first-order high pass filters connected or “cascaded” together to form a second-order or two-pole filter network. Cut-off frequency: 1 𝑓𝑐 = 2𝜋 𝑅1 𝐶1 𝑅2 𝐶2 nth-Order Low Pass Filter If there are n sections of RC high pass filter that are connected together, the roll-off rate and the cut-off frequency will change: Roll-off rate: 20n dB/decade 6n dB/octave Cut-off frequency: 𝑓𝑐 𝑛 = 𝑓𝑐 1 21/(𝑛−1) − 1 Passive Band-Pass Filter Passive Band-Pass Filters can be made by connecting together a low pass filter with a high pass filter. The upper and lower cut-off frequency points for a band pass filter can be found using the same formula as that for both the low and high pass filters. Cut-off frequency (for both high-pass and low-pass components of BPF): 1 𝑓𝑐 = 2𝜋𝑅𝐶 It is important to note that the HPF part must have the lower cut-off frequency compared to the LPF part. Passive Band-Pass Filter 1 𝑓𝐻 = (set by LPF) 2𝜋𝑅1 𝐶1 1 𝑓𝐿 = (set by HPF) 2𝜋𝑅2 𝐶2 Bandwidth: 𝐵 = 𝑓𝐻 − 𝑓𝐿 Center frequency: 𝑓𝑐 = 𝑓𝐻 𝑓𝐿 Example A second-order band pass filter is to be constructed using RC components that will only allow a range of frequencies to pass above 1kHz (1,000Hz) and below 30kHz (30,000Hz). Assuming that both the resistors have values of 10kΩ, calculate the values of the two capacitors required. Passive Band-Pass Filter Passive Band-Pass Filters can also be made using LC components. We call these as resonant filters. Passive Band-Pass Filter Bandwidth: 𝑓𝑟 𝐵= 𝑄 Q-factor of the inductor: 𝑅 𝑅 𝑄= = 𝑋𝐿 2𝜋𝑓𝐿 A higher Q value of an inductor means lower losses and better suitability for use as a high frequency inductor. If Q > 10, the filter is considered as wideband. Otherwise, its narrowband. Active Filters Active filters are a type of electronic filter that uses active components, including op-amps and transistors. They work with passive components as well, like resistors and capacitors, but not inductors. Compared to passive filters, their design is complex, and they tend to be more expensive. However, they can produce have power gain. The op-amp active component acts as an amplifier. This allows for specific gain control and isolation of specific modules that can work independently of one another. Op-amps require a continuous power source. They also have high input impedance and very low output impedance. This means that the active filter will not have a loading effect problem at its source. Also, changing or adding variations to the load won’t impact the active filter’s performance because the load is isolated from the source. General Active Filter Configuration Active Low Pass Filters Single-Pole Filter Cut-off frequency 1 𝑓𝑐 = 2𝜋𝑅𝐶 Gain 𝑅1 𝐴𝑉 𝐶𝐿 =1+ 𝑅2 Active Low Pass Filters Two-Pole Filter The configuration shown is called the Sallen-Key LPF. Cut-off frequency 1 𝑓𝑐 = 2𝜋 𝑅𝐴 𝐶𝐴 𝑅𝐵 𝐶𝐵 Gain 𝑅1 𝐴𝑉 𝐶𝐿 =1+ 𝑅2 Active Low Pass Filters 3-Pole Filter 4-Pole Filter Active Low Pass Filters To design a Butterworth response, the following values are given Example For the four-pole filter in the previous slide, determine the capacitance values required to produce a critical frequency of 2680 Hz if all the resistors in the RC low-pass circuits are 1.8 kohm. Also select values for the feedback resistors to get a Butterworth response. Active High Pass Filters Single-Pole Filter Cut-off frequency 1 𝑓𝑐 = 2𝜋𝑅𝐶 Gain 𝑅1 𝐴𝑉 𝐶𝐿 =1+ 𝑅2 Active High Pass Filters Two-Pole Filter The configuration shown is called the Sallen-Key HPF. Cut-off frequency 1 𝑓𝑐 = 2𝜋 𝑅𝐴 𝐶𝐴 𝑅𝐵 𝐶𝐵 Gain 𝑅1 𝐴𝑉 𝐶𝐿 =1+ 𝑅2 Example Choose values for the Sallen-Key high-pass filter in the previous figure to implement an equal-value second-order Butterworth response with a critical frequency of approximately 10 kHz. Assume all resistors are 3.3kohm except for R1. Active Band-Pass Filters Cascaded LPF and HPF filters HPF Cut-off frequency 1 𝑓𝐿 = 2𝜋 𝑅1𝐴 𝐶1𝐴 𝑅1𝐵 𝐶1𝐵 LPF Cut-off frequency 1 𝑓𝐻 = 2𝜋 𝑅2𝐴 𝐶2𝐴 𝑅2𝐵 𝐶2𝐵 𝑓𝑟 = 𝑓𝐿 𝑓𝐻 Active Band-Pass Filters Multiple Feedback BPF Center frequency 1 𝑓𝑟 = 2𝜋 (𝑅1 | 𝑅3 𝑅2 𝐶1 𝐶2 Gain 𝑅2 𝐴𝑉 𝐶𝐿 = 2𝑅1 Q Factor 𝑄 = 𝜋𝑓𝑟 𝐶2 𝑅2 Example Determine the center frequency, maximum gain, and bandwidth for the filter shown. Active Band-Pass Filters State-Variable Filter Active Band-Pass Filters State-Variable Filter The state-variable or universal active filter is widely used for band- pass applications. As shown, it consists of a summing amplifier and two op-amp integrators (which act as single-pole low-pass filters) that are combined in a cascaded arrangement to form a second-order filter. Although used primarily as a band-pass (BP) filter, the state variable configuration also provides low-pass (LP) and high-pass (HP) outputs. The center frequency is set by the RC circuits in both integrators. When used as a band-pass filter, the critical frequencies of the integrators are usually made equal, thus setting the center frequency of the passband. Example Find the center frequency and bandwidth of the figure below: END