# First and Second Order Filters

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1 First and Second Order Filters These functions are useful for the design of simple filters or they can be cascaded to form high-order filter functions First Order Filters General first order bilinear transfer function is given by: a1 s + ao T ( s) = s + ω o pole at s = - ω o and a zero at s = - a o / a 1 and a high frequency gain that approaches a 1 The numerator coefficients (a o, a 1 ) determine the type of filter (e.g. low-pass, high-pass, etc.)

2 Transfer Functions Low Pass High Pass Bandpass Notch All Pass T ( s) = T ( s) = T ( s) = T ( s) = s s s s 2 T ( s) = LC + s 1 CR + 1 LC s + s ω ο Q + ω s 1 CR + s 1 CR + 1 LC s + ω ο + s ω ο Q + ω s ο ο s ω ο Q + s ω ο Q + ω 2 2 ο

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4 Consider the Following Circuit

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6 t ( s) N ( s) = Loop Gain = L ( s) = At ( s) = D( s) A N ( s) D( s) For stability. 1+ L (s) = 0 which results in the poles s P of the closed - loop circuit as t ( sp ) = 1 A Assuming an ideal opamp with A = the poles are obtained from N ( s P ) = 0 That is, the poles are identical to the zeros of the RC network Since our objective is to realize a pair of complex conjugate poles we should select an RC network that has complex conjugate zeros The simplest such networks are Bridged - T networks

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9 The pole polynomial of the active filter circuit will be equal to the numerator polynomial of the Bridged - T network 2 ω ο s + s + ω ο = s Q C C R C C R R ω ο 1 C1 C2 R3 R4 1 1 = Q = C1 C2 R3 R + 4 R C C Common implementation C 1 = C 2 = C, R 3 = R, R 4 = R / m 2 m = 4Q CR = 2Q ω o

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11 A common implementation of the single amplifier biquad is the Salen - key filter A low - pass filter can be seen below Similar circuits are available for the other filter configurations ω c 1 = Q = R C C C 1 C 2

12 Sensitivity Because of telerances in component values and because of the finite opamp gain the response of the actual filter will deviate from the ideal response As a means of predicting such deviations, the filter designer employs the concept of sensitivity For second order filters one is normally interested in finding how sensitive their poles are relative to variations (Both initial tolerances and future changes) in RC component values and amplifier gain These sensitivities can be quantified using the classical sensitivity function S y x y y = lim = x 0 x x y x x y

13 Here x denotes the value of a component and y denotes a circuit parameter of interest (e.g. ωo, Q). For small changes S y x = y y x x Thus we use the value of S x y to determine the per unit change in y due to a given per - unit change in x If the sensitivity of Q relative to a particular resistance R 1 is 5, then a 1% increase in R 1 results in a 5% increase in the value of Q Switched - Capacitor Filters Active RC filters are difficult to implement totally on an IC due to the requirements of large - valued capacitors and accurate RC time constants The switched capacitor filter technique is based on the realization that a capacitor switched between two circuit nodes at a sufficiently high rate is equivalent to a resistor connecting these two nodes

14 Consider the Following Circuit

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17 From the above circuit we see that during each clock period T C an amount of charge q C1 = C 1 v 1 is subtracted from the input source and supplied to the integrator capacitor C 2 The average current flow between the input node and virtual ground (VG) is i AV C v T = 1 C i If T C is sufficiently short one can think of the process as continuous and define an equivalent resistance R EQ that is in effective resistance between nodes IN and VG R EQ v = i i = AV T C C 1 The time constant for the integrator is: Time constant = C R = 2 EQ T C C C 2 1

18 Thus the time constant that determines the frequency response of the filter is determined by the clock period T C and the capacitor ratio C 2 / C 1. Both of these parameters can be well controlled in an IC process Note the dependence on capacitor ratio rather than absolute value. The accuracy of capacitor ratios in MOS technology are on the order of 0.1% For reasonable clock frequencies (100 khz) and not too large capacitor ratios (10) one can obtain relatively large time constants (10-4 s) Switched capacitor filter ICs offer a low cost high order filter on a single IC The clock frequency must be higher than any frequency component of the signal (typically 100x) Can be easily programmed by changing the clock frequency Some of clock signal feeds through to the output, signals near the clock frequency can be aliased into the passband, overall increase in the noise floor

19 Oscillators and Waveform Shaping Circuits In the design of electronic systems the need frequently arises for signals having prescribed waveforms (e.g. sinusoidal, square, triangle, pulse, etc.) Commonly used in computers, control systems, communication systems and test and measurement systems Two common ways for generating sinusoids Positive feedback loop with non-linear gain limiting mechanism Appropriately shaping other waveforms such as triangle waves Circuits that directly generate square, triangle and pulse waveforms generally employ circuit blocks known as multivibrators. Three basic types are bistable, astable and monostable

20 Sinusoidal Oscillators Commonly referred to as linear sine-wave oscillators although some form of non-linearity has to be employed to limit the output amplitude Analysis of circuits is more difficult as s - plane analysis cannot be directly applied to the non-linear part of the circuit The basic structure of a sinusoidal oscillator consists of an amplifier and a frequency selective network connected in a positive feedback loop

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22 In actual oscillator no input will be present, included to help explain operation Note the feedback signal X F is summed with a positive sign Af ( s) A( s) = 1 A( s) β ( s) The loop gain is L( s) = A( s) β ( s) and the characteristic equation is 1 L( s) = 0 If at a specific frequency f o the loop gain Aβ is equal to unity it follows that A f will be infinite. Such a circuit is by definition an oscillator Thus for sinusoidal oscillation at ωo L( jω ) = A( jω ) β ( jω ) = 1 Barkhausen Criteria ο ο ο Unity gain, zero phase shift

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28 Active Filter Based on Two Loop Integrator (Biquad) Opamp - RC circuit that realizes second order filter functions based on the use of two integrators connected in cascade in an overall feedback loop Consider second order high pass filter Vhp T ( s) = = V i s 2 k s + s ω ο Q + ω 2 2 ο where k is the high frequency gain. Rearranging the equation gives

29 V 2 hp hp ο 2 Vhp = kvi 1 + Q ω s V + ο ω s ω ο The signal s V hp can be obtained by passing V hp through an integrator with a time constant equal to 1 ω 2 ο 2 V hp ο. Passing the resulting signal through another identical integrator ω generates s V 2 ο hp i hp V 2 hp 1 ω ο = kv Q s V ω s

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32 The signal at the output of the first integrator is ω ο V s i V hp which is a bandpass function ω ο V s i V hp = s k ω ο s + ω s ο Q + ω 2 2 ο = T bp ( s) Similarly the output of the second integrator is a low pass function ω 2 ο s V V 2 hp 2 i = s k ω ο ω + s ο Q ω 2 2 ο = T ( s) lp The two - integrator - loop biquad realizes three basic second order filter functions LP, BP and HP simultaneously. This circuit is commonly called the universal active filter

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35 If R f / R = 1 CR R = 1 3 R = 2Q 1 k = 2 1 Q ω ο 2 By summing the LP, BP, and HP outputs the overall transfer function of the KHN biquad and summer is V V o i k R F R s s R R F F H R ω + B ο R ω L = 2 s + s ω ο 2 Q + ω ο 2 2 ο An alternative two - loop integrator which uses three opamps in single ended mode is the Tow - Thomas biquad. Details can be found on pages 804 & 805 of the text Although Two - Integrator loop biquads are versatile and easy to design, their performance is adversly affected by the finite bandwidth of the opamps

36 Single Amplifier Biquad Filters Second order filter functions can also be implemented with a single amplifier. These minimal realizations are low power and low cost, however, they suffer from greater dependence on opamp gain and bandwidth and are generally more sensitive to tolerances in the resistors and capacitors The single amplifier biquads (SABs) are therefore generally limited to less stringent filter specifications (Q < 10)

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38 Active Filters In this section we study a family of opamp - RC circuits that realize the various second order filter functions by replacing the inductor L in the LCR resonator with an opamp - RC circuit that has an inductive input impedance Many opamp - RC circuits have been proposed for simulating the operation of an inductor One of the simplest is the negative impedance converter (NIC)

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40 Can be used to convert a capacitor to a backward inductor Z 1 = I = v jω C jω C C IN IN Z IN vin vin 1 j = = = = I v jω C jω C ω C IN IN Equivalent to an inductor of value 1 ω C One of the best circuits for simulating an inductor is the Antoniou inductance simulation circuit. This circuit is very tolerant to the non-ideal properties of opamps

41 The circuit is shown below

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43 The effective inductance of the circuit is L = C R R R R 2 Typically R = R = R = R = R C = C L = C R 2

44 Opamp - RC Resonator

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46 ω ο = 1 = 1 LC Q = ωο C6 R6 = R C 6 C C C R R R R 2 R R R R 2 If C = C = C R = R = R = R = R ω ο R 6 R = 1 CR Q =

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52 Second Order Functions The general second order (biquadratic) filter transfer function is given by T ( s) = 2 a2 s + a1 s+ a s + ω ο Q s + ω 2 2 ο where ωo and Q determine the poles according to o P1, P2 = ω ± j Q ω ο 4 Q o we are usually interested in the case of complex conjugate poles obtained for Q > 0.5

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54 The radial distance of the poles from the origin is the pole frequency The higher the value of Q the closer the poles are to the jω axis and the more selective (higher peak and initial rolloff) the filter response becomes An infinite value of Q locates the poles on the jω axis and can yield sustained oscillations Q is called the pole quality factor

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58 Second Order LCR Resonator V I o 1 1 s = = = C Y 1 sl + sc R s + s 1 CR + 1 LC Equating the denominator to standard form gives s + s o ω Q + ω 2 2 o ω 2 ο = 1 ω ο LC = 1 1 Q CR ω o = Q = LC ω ο C R

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