Analog Filters: Introduction

1. Analog Filters: Introduction Franco Maloberti

2. Historical Evolution Analog Filters: Introduction

3. Frequency and Size • Active filters will achieve ten of GHz in monolitic form Analog Filters: Introduction

4. 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. Input Output Analog Filter x(t) y(t) Analog Filters: Introduction

5. 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 Analog Filters: Introduction

6. 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 Analog Filters: Introduction

7. 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 Analog Filters: Introduction

8. 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. Analog Filters: Introduction

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

10. 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 Analog Filters: Introduction

11. 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 Analog Filters: Introduction

12. 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. Analog Filters: Introduction

13. 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 Analog Filters: Introduction

14. Minimum Phase Filters • When the zeros of N(s) lie on or to the left of the jw-axis H(s) is a minimum phase function. Analog Filters: Introduction

15. Type of Filters • Low-pass • High-pass • Band-pass • Band-Reject • All-Pass 1 f 0 1 fc f 1 0 fc f 1 0 fc1 fc2 f 0 1 fc fc2 f 0 Analog Filters: Introduction

16. Approximate Response • Pass-band ripple ap=20Log[Amax/Amin] • Stop-band attenuation, Asb • Transition-band ratio wp, ws Amax Amin Asb wp ws Analog Filters: Introduction

17. MATLAB • Works with matrices (real, complex or symbolic) • Multiply two polinomials f1(s)=5s3+4s2+2s +1; f2(s)=3s2+5 • clear all; • f1=[5 4 2 1]; • f2 = [3 0 5]; • f3 = conv(f1, f2) 15 12 31 23 10 5 f3(s)=15s5+12s4+ 31s3 + 23s2 + 10s+5 Analog Filters: Introduction

18. Frequency Scaling • If every inductance and every capacitance of a network is divided by the frequency scaling factor kf, then the network function H(s) becomes H(s/kf). • Xc=1/sC; X’c=1/[s(C/kf)]=1/[C(s/kf)] • XL=sL; X’L=s(L/kf)=L(s/kf) • What occurs at w’ in the original network now will occur at kf w’. Analog Filters: Introduction

19. Impedance Scaling • All elements with resistance dimension are multiplied by kz • R -> kz R; wL ->kzwL; (Vx=aIcont) a -> a kz • All elements with capacitance dimension are divided by kz • G -> G/kz; wC ->wC /kz; (Ix=bVcont) b -> b/kz • Impedences multiplied by kz • Admittances divided by kz • Dimensionless variables unchanged Analog Filters: Introduction

20. Normalization and Denormalization • Normalized filters use the key angular frequency of the filter (wp 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 Analog Filters: Introduction

21. 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, … Analog Filters: Introduction

22. 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 Analog Filters: Introduction

23. 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 Analog Filters: Introduction