1. Analog Filters: Introduction Franco Maloberti

2. Analog Filters: Introduction2 Historical Evolution

3. Franco MalobertiAnalog Filters: Introduction3 Frequency and Size Active filters will achieve ten of GHz in monolitic form

4. Franco MalobertiAnalog Filters: Introduction4 Introduction An analog filter is the interconnection of components (resistors, capacitors, inductors, active devices) It has one input (excitation) and one input (response) It determines a frequency selective transmission. Analog Filter InputOutput x(t) y(t)

5. Franco MalobertiAnalog Filters: Introduction5 Classification of Systems Time-Invariant and Time-Varying The shape of the response does not depends on the time of application of the input Casual System The response cannot precede the excitation

6. Franco MalobertiAnalog Filters: Introduction6 Classification of Systems Linear and Non-linear A system is linear if it satisfies the principle of superposition Continuous and Discrete-time In a continuous-time or continuous analog system the variables change continuously with time In discrete-time or sampled-data systems the variables change at only discrete instants of time

7. Franco MalobertiAnalog Filters: Introduction7 Linear Continuous Time-Invariant If a system is composed by lumped elements (and active devices) Linear differential equations, constant coefficients x(t), input, and y(t), output,are current and/or voltages For a given input and initial conditions the output is completely determined

8. Franco MalobertiAnalog Filters: Introduction8 Responses of a linear system Zero-input response Is the response obtained when all the inputs are zero. Depends on the initial charges of capacitors and initial flux of inductors Zero-state response Is the response obtained with zero initial conditions The complete response will be a combination of zero-input and zero-state.

9. Franco MalobertiAnalog Filters: Introduction9 Frequency-domain Study Remember that the Laplace transform of The equation Becomes ICy(s) and ICx(s) accounts for initial conditions

10. Franco MalobertiAnalog Filters: Introduction10 Transfer Function If X(s) is the input and Y(s) the zero-state output Input voltage, output voltage: voltage TF Inpur current, output current: Current TF Input votage output current: Transfer impedance Input current, ourput voltage: Trasnsfer admittance

11. Franco MalobertiAnalog Filters: Introduction11 Transfer Function Input and output ar normally either voltage or current Where Y(s) and X(s) are the Laplace transforms of y(t) and x(t) respectively. In the frequency domain the focus is directed toward Magnitude and/or Phase on the j axis of s

12. Franco MalobertiAnalog Filters: Introduction12 Magnitude and Phase Magnitude is often expressed in dB Important is also the group delay When both magnitude and phase are important the magnitude response is realized first. Then, an additional circuit, the delay equalizer, improves the delay function.

13. Franco MalobertiAnalog Filters: Introduction13 Real Transfer Function The coefficients of the TF are real for a linear, time-invariant lumped network. Only real or conjugate pairs of complex poles For stability the zeros of D(s) in the half left plane D(s) is a Hurwitz polynomial

14. Franco MalobertiAnalog Filters: Introduction14 Minimum Phase Filters When the zeros of N(s) lie on or to the left of the j  -axis H(s) is a minimum phase function.

15. Franco MalobertiAnalog Filters: Introduction15 Type of Filters Low-pass High-pass Band-pass Band-Reject All-Pass 1 0 f fcfc 1 0 f fcfc 1 0 f f c1 1 0 f fcfc f c2 1 0 f

16. Franco MalobertiAnalog Filters: Introduction16 Approximate Response Pass-band ripple  p =20Log[A max /A min ] Stop-band attenuation, A sb Transition-band ratio  p,  s A max A min A sb pp ss

17. Franco MalobertiAnalog Filters: Introduction17 MATLAB Works with matrices (real, complex or symbolic) Multiply two polinomials f 1 (s)=5s 3 +4s 2 +2s +1; f 2 (s)=3s 2 +5 clear all; f1=[5 4 2 1]; f2 = [3 0 5]; f3 = conv(f1, f2) 15 12 31 23 10 5 f 3 (s)=15s 5 +12s 4 + 31s 3 + 23s 2 + 10s +5

18. Franco MalobertiAnalog Filters: Introduction18 Frequency Scaling If every inductance and every capacitance of a network is divided by the frequency scaling factor k f, then the network function H(s) becomes H(s/k f ). X c =1/sC; X’ c =1/[s(C/k f )]=1/[C(s/k f )] X L =sL; X’ L =s(L/k f )=L(s/k f ) What occurs at  ’ in the original network now will occur at k f  ’.

19. Franco MalobertiAnalog Filters: Introduction19 Impedance Scaling All elements with resistance dimension are multiplied by k z R -> k z R;  L ->k z  L; (V x =  I cont )  ->  k z All elements with capacitance dimension are divided by k z G -> G/k z ;  C ->  C /k z ; (I x =  V cont )  ->  /k z Impedences multiplied by k z Admittances divided by k z Dimensionless variables unchanged

20. Franco MalobertiAnalog Filters: Introduction20 Normalization and Denormalization Normalized filters use the key angular frequency of the filter (  p in a low-pass, …) equal to 1. One of the resistance of the filter is set to 1 or One capacitor of the filter is set to 1 Frequency scaling and impedance scaling are eventually performed at the end of the design process

21. Franco MalobertiAnalog Filters: Introduction21 Design of Filters Procedure Specifications Kind of network Input network Infinite, zero load Single terminated/Double terminated Mask of the filter Magnitude response Delay response Other features Cost, volume, power consumption, temperature drift, aging, …

22. Franco MalobertiAnalog Filters: Introduction22 Design of Filters Procedure (ii) Normalization Set the value of one key component to 1 Set the value of one key frequency to 1 Approximation To find the transfer function that satisfy the (normalized) amplitude specifications (and, when required, the delay specification. Many transfer functions achieve the goal. The key task is to select the “cheapest” one

23. Franco MalobertiAnalog Filters: Introduction23 Design of Filters Procedure (iii) Network Synthesis (Realization) To find a network that realizes the transfer function Many networks achieve the same transfer function Active or passive implementation The behavior of networks implementing the same transfer function can be different (sensitivity, cost, … Denormalization Impedance scaling Frequency scaling Frequency transformation