4Transfer Function – Rules Filter is characterized by its transfer functionThe poles should be at the left half of the s-plane.We only consider stable filter.Given a complex pole or zero, its complex conjugate is also pole or zero.
5Transfer Function – Rules Filter is characterized by its transfer functionAs the frequency increase, the output will become infinity.:improper filterRemember the two rules:proper filterWe only consider proper filer.The filters consider have more poles than zeros.
6Filter OrderOrder = nThe order of the denominator is the order of the filter.order=1order=4
9Firsr-order Filters Case 1: Case 2: zero or first order 0 or 1 zero 1 poleCase 1:1 pole, 0 zeroCase 2:1 pole, 1 zero
10Firsr-order Filters - Case 1 Lowpass filterAs ω increasesMagnitude decreasePhase decreasePole p is on the negative real axis
11Firsr-order Filters - Case 1 Amplitude of the transfer function of the first-order low pass filterIdeal Lowpass filterFirst-order Lowpass filter
12Firsr-order Filters - Case 1 Find cut-off frequency ωco of the first-order low pass filterLowpass filterAt DCFind cut-off frequency ωco such that
13Firsr-order Filters - Case 2 Case 2-1: Absolute value of zero is smaller than poleMagnitude is proportional to the length of green line divided by the length of the blue lineZero can be positive or negativeLow frequency ≈ |z|/|p|Because |z|<|p|The low frequency signal will be attenuatedIf z=0, the low frequency can be completely blockNot a low pass
14Firsr-order Filters - Case 2 Case 2-1: Absolute value of zero is smaller than poleMagnitude is proportional to the length of green line divided by the length of the blue lineHigh frequencyThe high frequency signal will passHigh passIf z=0 (completely block low frequency)
15First-order Filters - Case 2 Find cut-off frequency ωco of the first-order high pass filter(the same as low pass filter)
16First-order Filters - Case 2 Case 2-2: Absolute value of zero is larger than poleLow frequency ≈ |z|/|p|Because |z|>|p|The low frequency signal will be enhanced.High frequency: magnitude is 1The high frequency signal will pass.Neither high pass nor low pass
17First-order Filters Consider vin as input (pole) If vl is output Reasonable from intuitionIf vl is outputLowpass filterIf vh is outputHighpass filter(pole)
27Second-order Filter – Case 1 Real PolesThe magnitude isAs ω increasesThe magnitude monotonically decreases.Decrease faster than first order low pass
28Second-order Filter – Case 1 Complex PolesThe magnitude isAs ω increases,l1 decrease first and then increase.l2 always decreaseWhat will happen to magnitude?1. Increase2. Decrease3. Increase, then decrease4. Decrease, then increase
29Second-order Filter – Case 1 Complex PolesIf ω > ωdl1 and l2 both increase.The magnitude must decrease.What will happen to magnitude?1. Increase2. Decrease3. Increase, then decrease4. Decrease, then increase
30Second-order Filter – Case 1 Complex PolesWhen ω < ωdMaximize the magnitudeMinimize
31Second-order Filter – Case 1 MinimizeMinimize(maximize)
32Second-order Filter – Case 1 Lead to maximumThe maxima exists whenPeakingNo PeakingPeaking
33Second-order Filter – Case 1 Lead to maximumThe maxima exists whenPeakingAssume
60Suppose this band-stop filter were to suddenly start acting as a high-pass filter. Identify a single component failure that could cause this problem to occur: If resistor R3 failed open, it would cause this problem. However, this is not the only failure that could cause the same type of problem!
63High pass They are used as part of an audio crossover to direct high frequencies to a tweeter while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into a loudspeaker cabinet it is normally a passive filter that also includes a low-pass filter for the woofer and so often employs both a capacitor and inductor (although very simple high-pass filters for tweeters can consist of a series capacitor and nothing else).
64Any second-order bandpass filter may be described by Where quality factor:a: damping coefficientthe network is underdamped when a < w0 or Q > ½64
65The transfer function of a second-order notch filter is The notch effect comes from the quadratic numeratorThe notch width is B = wO / Q65
66Type Transfer Function Properties Table Simple FilterType Transfer Function PropertiesLowpassHighpassBandpassNotch66
67Example 11.6 Design of a Bandpass Filter bandpass filter: L = 1 mH, Rw = 1.2 W, C = ?, R = ?frequency: 20kHz ± 250Hz67
69From WikiButterworth filter – maximally flat in passband and stopband for the given orderChebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than Butterworth of same orderChebyshev filter (Type II) – maximally flat in passband, sharper cutoff than Butterworth of same orderBessel filter – best pulse response for a given order because it has no group delay rippleElliptic filter – sharpest cutoff (narrowest transition between pass band and stop band) for the given orderGaussian filter – minimum group delay; gives no overshoot to a step function.
70Only input signal at these frequencies can pass FilterA filter is a circuit that is designed to pass signals with desired frequencies and reject the others.Only input signal at these frequencies can passFilterMagnitude ratio
71Loudspeaker for home usage with three types of dynamic drivers 1 Loudspeaker for home usage with three types of dynamic drivers 1. Mid-range driver 2. Tweeter 3. Woofers
73Second Order Lowpass Filter As the frequency increases, the amplitude ratio drops faster than 1st order low pass filterCompare with 1st order low pass filter
74Firsr-order Filters - Case 1 Lowpass filter|p| is the cut-off frequencySmaller cut-off frequencyLarger cut-off frequency
75Firsr-order Transfer Function - Case 2 Highpass? Lowpass?Both possible?Absolute value of pole is equal to zeroMagnitude is the length of green line divided by the length of the blue linePositive zero can cause phase shiftPhase is the angle of the green minus the negative oneAll pass filter
76First Order Lowpass Filter Maximum:Cut-off Frequency ωco: