Introduction to Filters: Understanding Filter Circuits and Their Frequency Response, Exams of Circuit Theory

Download Introduction to Filters: Understanding Filter Circuits and Their Frequency Response and more Exams Circuit Theory in PDF only on Docsity! EE 230 intro to filters – 1 Filter circuits From our work with Laplace transforms (and the AC analysis work from EE 201), it is clear that the behavior of a circuit depends on the frequency used. The impedance of the reactive elements varies with frequency. As s → 0, ZC → ∞. An open circuit. As s → ∞, ZC → 0. A short circuit. As s → 0, ZL → 0. A short circuit. As s → ∞, ZL → ∞. An open circuit. We can use this frequency dependence to build circuits that discriminate between frequencies. The circuit would allow signals at some frequencies to pass through while signals at other frequencies are attenuated – filtered out. =& =  V& =/ = V/ =& =  M֌& (=/ = M֌/) EE 230 intro to filters – 2 As s → 0, ZC → ∞ and ZL → 0. As s → ∞, ZC → 0 and ZL → ∞. Consider This would be a low-pass filter. Low frequency signals are passed through to the output. High-frequency signals are cut off — Vo = 0). + – – + Vo(s) = Vi(s)Vi(s) + – – + Vo(s) = 0Vi(s) + – L C – + Vo(s)Vi(s) ZL = sL =& =  V& EE 230 intro to filters – 5 High Pass: Frequencies above fc pass through, and those below are cut off. fc cut-off frequency EE 230 intro to filters – 6 Band pass: Only frequencies between fc1 and fc2 pass through. Everything else is cut off. fc1 fc2 bandwidth: BW = fc2 – fc1 = ∆f quality factor: QP = fpeak /∆f fpeak : frequency at which maximum occurs EE 230 intro to filters – 7 Band reject: Frequencies between fc1 and fc2 are blocked. Everything else is allowed to pass through. fc1 fc2 bandwidth: BW = fc2 – fc1 = ∆f fnull : frequency at which minimum occurs quality factor: QP = fnull /∆f EE 230 intro to filters – 10 Cut-off frequency For the ideal filters, it is fairly obvious where to locate the dividing line between the passband and the cut-off region. For realistic filters, the transition is gradual and it is less obvious where the dividing line between pass and cut-off should be. The usual convention is to choose the frequency where the magnitude of the transfer function is down by (= 0.707) from the peak value. 1. Find the peak of the magnitude of transfer function, |T(jω)|max — low frequencies for low pass, high frequencies for high-pass, in the middle of the pass band for bandpass. 2. Find the frequency where the magnitude is /  This is also the frequency at which the power being transmitted through a passive circuit is one-half of the power provided by the source. So it is also know as the half-power frequency. 7 PD[  7 (M֌F) = 7 PD[  EE 230 intro to filters – 11 7PD[7PD[  fc1 7PD[ fc2 f or ω ? 7PD[  To some degree, it doesn’t matter which you use, just make sure that you keep them straight. f is the real frequency and is the number of oscillations per second (Hz or hertz). This is used in the lab and is usually the number that is reported. The angular frequency ω is the number of radians per second. This is what goes into our equations. Important: ω = 2πf !!!! fc EE 230 intro to filters – 12 decibels As a ratio, the magnitude of the transfer function has no units. If such a ratio ranges over many orders of magnitude it is sometimes more convenient to talk about the logarithm of the ratio. magnitude: 1 → 10 → 100 → 1000 → 10,000 logarithm: 0 → 1 → 2 → 3 → 4 Using the logarithm, we would describe the ratio in bels (after Alexander Graham Bell). The bel is generally a bit coarse for practical use – instead we use decibels (dB). 10 decibels = 1 bel The idea of using logarithms like this comes from early work in quantifying human hearing. The dynamic range of the human ear is remarkable. The power of the weakest sound humans can hear is denoted as 0 dB (= 1). Normal conversation is about a million times more powerful (60 dB = 106.) The sound from a jet engine is about 1 million times more powerful than conversation (120 dB = 1012). decibel: 0 → 10 → 20 → 30 → 40