SIGNALS AND CIRCUITS 2

1. SIGNALS AND CIRCUITS 2 ELECTRONIC FILTERS

2. ELECTRONIC FILTERS One-Port Network • The basic idea of a one-port network is shown below. The one-port is a ``black box'' with a single pair of input/output terminals, referred to as a port. Network theory is normally described in terms of circuit theory elements, in which case a voltage is applied at the terminals and a current flows as shown. • A one-port network characterized by its driving point impedance Z(s) or admittance G(s). For any applied voltage V(s), the observed current I(s)= V(s)G(s).

3. ELECTRONIC FILTER A one-port network characterized by its driving point impedance Z(s) or admittance G(s). For any applied voltage V(s), the observed current I(s)= V(s)G(s).

4. ELECTRONIC FILTERS • Two-port network • A two-port network (a kind of four-terminal network or quadripole) is an electrical circuit or device with two pairs of terminals connected together internally by an electrical network.

5. ELECTRONIC FILTERS Alinear circuit • is an electronic circuit in which, for a sinusoidal input voltage of frequencyf, any output of the circuit (the current through any component, or the voltage between any two points) is also sinusoidal with frequency f. Note that the output need not be in phase with the input. • An equivalent definition of a linear circuit is that it obeys the superposition principle. This means that the output of the circuit F(x) when a linear combination of signals ax1(t) + bx2(t) is applied to it is equal to the linear combination of the outputs due to the signals x1(t) and x2(t) applied separately: • Informally, a linear circuit is one in which the values of the electronic components, the resistance, capacitance, inductance, gain, etc. don't change with the level of voltage or current in the circuit.

6. ELECTRONIC FILTERS The superposition theorem states: • The current or voltage associated with a branch in a linear network equals the sum of the current or voltage components set up in that branch due to each of the independent sources acting one at the time on the circuit

7. ELECTRONIC FILTERS Define nonlinear circuit?

8. ELECTRONIC FILTERS Define nonlinear circuit? If the system is non-linear, the current or voltage response will contain harmonics of the excitation frequency. A harmonic is a frequency equal to an integer multiplied by the fundamental frequency. For example, the “second harmonic” is a frequency equal to two times the fundamental frequency.Some researchers have made use of this phenomenon. Linear systems should not generate harmonics, so thepresence or absence of significant harmonic response allows one to determine the systems linearity. Other researchers have intentionally used larger amplitude excitation potentials. They use the harmonic response to estimate the curvature in the cell's current voltage curve.

9. ELECTRONIC FILTERS Transfer functions • are commonly used in the analysis of systems such as single-input single-outputfilters, typically within the fields of signal processing, communication theory, and control theory. The term is often used exclusively to refer to linear, time-invariant systems (LTI), as covered in this article. Most real systems have non-linear input/output characteristics, but many systems, when operated within nominal parameters (not "over-driven") have behavior that is close enough to linear that LTI system theory is an acceptable representation of the input/output behavior. • In its simplest form for continuous-time input signal x(t) and output y(t), the transfer function is the linear mapping of the Laplace transform of the input, X(s), to the output Y(s):

10. ELECTRONIC FILTERS or where H(s) is the transfer function of the LTI system. In discrete-time systems, the function is similarly written as (see Z transform) and is often referred to as the pulse-transfer function. We can express transfer function in the frequency domain also: TV(jω)=Vout(jω)/ Vin(jω) or TI(jω)=Iout(jω)/ Iin(jω)

11. ELECTRONIC FILTERS Frequency spectrum The frequency spectrum of a time-domain signal is a representation of that signal in the frequency domain. The frequency spectrum can be generated via a Fourier transform of the signal, and the resulting values are usually presented as amplitude and phase, both plotted versus frequency

12. ELECTRONIC FILTERS Spectrum analysis is the technical process of decomposing a complex signal into simpler parts. As described above, many physical processes are best described as a sum of many individual frequency components. Any process that quantifies the various amounts (e.g. amplitudes, powers, intensities, or phases), versus frequency can be called spectrum analysis. Spectrum analysis can be performed on the entire signal. Alternatively, a signal can be broken into short segments (sometimes called frames), and spectrum analysis may be applied to these individual segments. Periodic functions (such as sin(t)) are particularly well-suited for this sub-division. General mathematical techniques for analyzing non-periodic functions fall into the category of Fourier analysis.

13. ELECTRONIC FILTERS Spectrum of the saw-toof type signal

14. ELECTRONIC FILTERS Electronic filters are electronic circuits which perform signal processing functions, specifically to remove unwanted frequency components from the signal, to enhance wanted ones, or both. Electronic filters can be: • passive or active • analog or digital • high-pass, low-pass, bandpass, band-reject (band reject; notch), or all-pass. • discrete-time (sampled) or continuous-time

15. ELECTRONIC FILTERS Other filter technologies • Quartz filters and piezoelectrics • In the late 1930s, engineers realized that small mechanical systems made of rigid materials such as quartz would acoustically resonate at radio frequencies, i.e. from audible frequencies (sound) up to several hundred megahertz. Some early resonators were made of steel, but quartz quickly became favored. The biggest advantage of quartz is that it is piezoelectric. This means that quartz resonators can directly convert their own mechanical motion into electrical signals. Quartz also has a very low coefficient of thermal expansion which means that quartz resonators can produce stable frequencies over a wide temperature range. Quartz crystal filters have much higher quality factors than LCR filters. When higher stabilities are required, the crystals and their driving circuits may be mounted in a "crystal oven" to control the temperature. For very narrow band filters, sometimes several crystals are operated in series. • Engineers realized that a large number of crystals could be collapsed into a single component, by mounting comb-shaped evaporations of metal on a quartz crystal. In this scheme, a "tapped delay line" reinforces the desired frequencies as the sound waves flow across the surface of the quartz crystal. The tapped delay line has become a general scheme of making high-Q filters in many different ways. • SAW filters • SAW (surface acoustic wave) filters are electromechanical devices commonly used in radio frequency applications. Electrical signals are converted to a mechanical wave in a device constructed of a piezoelectric crystal or ceramic; this wave is delayed as it propagates across the device, before being converted back to an electrical signal by further electrodes. The delayed outputs are recombined to produce a direct analog implementation of a finite impulse response filter. This hybrid filtering technique is also found in an analog sampled filter. SAW filters are limited to frequencies up to 3 GHz. • BAW filters • BAW (Bulk Acoustic Wave) filters are electromechanical devices. BAW filters can implement ladder or lattice filters. BAW filters typically operate at frequencies from around 2 to around 16 GHz, and may be smaller or thinner than equivalent SAW filters. Two main variants of BAW filters are making their way into devices, Thin film bulk acoustic resonator or FBAR and Solid Mounted Bulk Acoustic Resonators.

16. ELECTRONIC FILTERS Passive filters Passive implementations of linear filters are based on combinations of resistors (R), inductors (L) and capacitors (C). These types are collectively known as passive filters, because they do not depend upon an external power supply and/or they do not contain active components such as transistors.

17. ELECTRONIC FILTERS Single element types A low-pass electronic filter realised by an RC circuit The simplest passive filters, RC and RL filters, include only one reactive element, except hybrid LC filter which is characterized by inductance and capacitance integrated in one element.

18. ELECTRONIC FILTERS L filter An L filter consists of two reactive elements, one in series and one in parallel.

19. ELECTRONIC FILTERS T and π filters Low-pass π filter High-pass T filter

20. ELECTRONIC FILTERS • Three-element filters can have a 'T' or 'π' topology and in either geometries, a low-pass, high-pass, band-pass, or band-stop characteristic is possible. The components can be chosen symmetric or not, depending on the required frequency characteristics. The high-pass T filter in the illustration, has a very low impedance at high frequencies, and a very high impedance at low frequencies. That means that it can be inserted in a transmission line, resulting in the high frequencies being passed and low frequencies being reflected. Likewise, for the illustrated low-pass π filter, the circuit can be connected to a transmission line, transmitting low frequencies and reflecting high frequencies. Using m-derived filter sections with correct termination impedances, the input impedance can be increased.

21. ELECTRONIC FILTERS Digital filter • A digital filter system usually consists of an analog-to-digital converter to sample the input signal, followed by a microprocessor and some peripheral components such as memory to store data and filter coefficients etc. Finally a digital-to-analog converter to complete the output stage. • A general finite impulse response filter with n stages, each with an independent delay, di, and amplification gain, ai.

22. ELECTRONIC FILTERS

23. ELECTRONIC FILTERS FILTER SPECIFICATIONS

24. ELECTRONIC FILTERS FILTER SPECIFICATIONS

25. ELECTRONIC FILTERS FILTER SPECIFICATIONS

26. ELECTRONIC FILTERS FILTER SPECIFICATIONS

27. ELECTRONIC FILTERS

28. ELECTRONIC FILTERS

29. ELECTRONIC FILTERS Classification by topology • Electronic filters can be classified by the technology used to implement them. Filters using passive filter and active filter technology can be further classified by the particular electronic filter topology used to implement them. • Any given filter transfer function may be implemented in any electronic filter topology. • Some common circuit topologies are: • Cauer topology - Passive • Sallen Key topology - Active • Multiple Feedback topology - Active • State Variable Topology - Active • Biquadratic topologybiquad filter - Active

30. ELECTRONIC FILTERS Ladder topologies Ladder topology, often called Cauer topology after Wilhelm Cauer (inventor of the Elliptical filter)

31. ELECTRONIC FILTERS Unbalanced LHalf sectionT SectionΠ SectionLadder network Ladder network

32. ELECTRONIC FILTERS Modified ladder topologies Fig. m-derived topology Image filter design commonly uses modifications of the basic ladder topology. These topologies, invented by Otto Zobelhave the same passbands as the ladder on which they are based but their transfer functions are modified to improve some parameter such as impedance matching, stopband rejection or passband-to-stopband transition steepness.

33. ELECTRONIC FILTERS Bridged-T topologies • Typical bridged-T Zobel network equaliser used to correct high end roll-off • The bridged-T topology is also used in sections intended to produce a signal delay but in this case no resistive components are used in the design.

34. ELECTRONIC FILTERS Lattice topology Lattice topology X-section phase

35. ELECTRONIC FILTERS ACTIVE FILTERS Sallen–Key topology The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity.

36. ELECTRONIC FILTERS The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure 1. The following analysis is based on the assumption that the operational amplifier is ideal. Figure 1: The generic Sallen–Key filter topology.

37. ELECTRONIC FILTERS • Because the operational amplifier (OA) is in a negative-feedback configuration, its v+ and v- inputs must match (i.e., v+ = v-). However, the inverting input v- is connected directly to the output vout, and so • By Kirchhoff's current law (KCL) applied at the vx node, • By combining Equations (1) and (2), • Applying Equation (1) and KCL at the OA's non-inverting input v+ gives • which means that • Combining Equations (2) and (3) gives • Rearranging Equation (4) gives the transfer function

38. ELECTRONIC FILTERS • Combining Equations (2) and (3) gives • Rearranging Equation (4) gives the transfer function

39. ELECTRONIC FILTERS Sallen-Key Lowpass filter example Sallen-Key Highpass filter example

40. ELECTRONIC FILTERS Multiple feedback topology Multiple feedback topology circuit

41. ELECTRONIC FILTERS The transfer function of the multiple feedback topology circuit, like all second-order linear filters, is: . In an MF filter, is the Q factor. is the DC voltage gain is the corner frequency

42. ELECTRONIC FILTERS Biquad filter A biquad filter is a type of linear filter that implements a transfer function that is the ratio of two quadratic functions. The name biquad is short for biquadratic. Biquad filters are typically active and implemented with a single-amplifier biquad (SAB)

43. ELECTRONIC FILTERS Tow-Thomas Biquad Example For example, the basic configuration in Figure 1 can be used as either a low-pass or bandpass filter depending on where the output signal is taken from. Figure 1: The common Tow-Thomas biquad filter topology.

44. ELECTRONIC FILTERS The second-order low-pass transfer function is given by where low-pass gain Glpf = R2 / R1. The second-order bandpass transfer function is given by • . • with bandpass gain Gbpf = − R4 / R2. In both cases, the • Natural frequency is • . • Quality factor is . The bandwidth is approximated by B = ω0 / Q, and Q is sometimes expressed as a damping constantζ = 1 / 2Q. If a noninverting low-pass filter is required, the output can be taken at the output of the second operational amplifier. If a noninverting bandpass filter is required, the order of the second integrator and the inverter can be switched, and the output taken at the output of the inverter's operational amplifier.

45. ELECTRONIC FILTERS Network synthesis filters Butterworth filter Butterworth filters are described as maximally flat, meaning that the response in the frequency domain is the smoothest possible curve of any class of filter of the equivalent orderThe Butterworth class of filter was first described in a 1930 paper by the British engineer Stephen Butterworth after whom it is named. The filter response is described by Butterworth polynomials

46. ELECTRONIC FILTERS Chebyshev filter A Chebyshev filter has a faster cut-off transition than a Butterworth, but at the expense of there being ripples in the frequency response of the passband. There is a compromise to be had between the maximum allowed attenuation in the passband and the steepness of the cut-off response. This is also sometimes called a type I Chebyshev, the type 2 being a filter with no ripple in the passband but ripples in the stopband. The filter is named after Pafnuty Chebyshev whose Chebyshev polynomials are used in the derivation of the transfer function

47. ELECTRONIC FILTERS Cauer (Eliptical) filter Cauer filters have equal maximum ripple in the passband and the stopband. The Cauer filter has a faster transition from the passband to the stopband than any other class of network synthesis filter. The term Cauer filter can be used interchangeably with elliptical filter, but the general case of elliptical filters can have unequal ripples in the passband and stopband. The filter is named after Wilhelm Cauer and the transfer function is based on elliptic rational functions.

48. ELECTRONIC FILTERS Bessel filter The Bessel filter has a maximally flat time-delay (group delay) over its passband. This gives the filter a linear phase response and results in it passing waveforms with minimal distortion. The Bessel filter has minimal distortion in the time domain due to the phase response with frequency as opposed to the Butterworth filter which has minimal distortion in the frequency domain due to the attenuation response with frequency. The Bessel filter is named after Friedrich Bessel and the transfer function is based on Bessel polynomials.

49. ELECTRONIC FILTERS As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth. In the particular implementation – analog or digital, passive or active – makes no difference; their output would be the same.