* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download pasive filters - Portal UniMAP
Analog-to-digital converter wikipedia , lookup
Oscilloscope history wikipedia , lookup
Spark-gap transmitter wikipedia , lookup
Crystal radio wikipedia , lookup
Loudspeaker wikipedia , lookup
Power electronics wikipedia , lookup
Amateur radio repeater wikipedia , lookup
Spectrum analyzer wikipedia , lookup
405-line television system wikipedia , lookup
Atomic clock wikipedia , lookup
Standing wave ratio wikipedia , lookup
Switched-mode power supply wikipedia , lookup
Mechanical filter wikipedia , lookup
Opto-isolator wikipedia , lookup
Resistive opto-isolator wikipedia , lookup
Distributed element filter wikipedia , lookup
Analogue filter wikipedia , lookup
Audio crossover wikipedia , lookup
Wien bridge oscillator wikipedia , lookup
Rectiverter wikipedia , lookup
Regenerative circuit wikipedia , lookup
Phase-locked loop wikipedia , lookup
Superheterodyne receiver wikipedia , lookup
Valve RF amplifier wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Zobel network wikipedia , lookup
Index of electronics articles wikipedia , lookup
Radio transmitter design wikipedia , lookup
Equalization (audio) wikipedia , lookup
TOPIC 4: FREQUENCY SELECTIVE CIRCUITS 1 INTRODUCTION •Transfer Function •Frequency Selective Circuits 2 TRANSFER FUNCTION • The s-domain ratio of the Laplace transform of the output (response) to the Laplace transform of the input (source) when all initial conditions are zero. • The transfer function depends on what is defined as the output signal. 3 DEFINITION Y (s) H (s) X ( s) all ICs 0 H ( j ) H ( j ) 4 POLES AND ZEROS • The roots of the denominator polynomial are called the poles of H(s): the values of s at which H(s) becomes infinitely large. • The roots of the numerator polynomial are called the zeros of H(s): the values of s at which H(s) becomes zero. 5 FREQUENCY RESPONSE • The transfer function is a useful tool to compute the frequency response of a circuit (i.e. the steady state response to a varying-frequency sinusoidal source). • The magnitude and phase of the output signal depend only on the magnitude and phase of the transfer function, H(j). 6 FREQUENCY RESPONSE • Frequency response analysis is used to analyze the effect of varying source frequency on circuit voltages and currents. • The circuit’s response depends on: – the types of elements – the way the elements are connected – the impedance of the elements 7 FREQUENCY SELECTIVE CIRCUITS • Frequency selective circuits is a circuits that pass to the output only those input signals that reside in a desired range of frequencies. • Can be constructed with the careful choice of circuit elements, their values, and their connections. 8 PASSIVE FILTERS • Passband & Stopband • Cutoff Frequency • Bode plot 9 PASSIVE FILTERS • Frequency selective circuits are also called filters. • Filters attenuate, that is weaken or lessen the effect of any input signals with frequencies outside a particular frequency. • Called passive filters because their filtering capabilities depend only on the passive elements (i.e. R,L,C). 10 PASSBAND & STOPBAND • The signal passed from the input to the output fall within a band of frequencies called Passband. • Frequencies not in a circuit’s passband are in its Stopband. • Filters are categorized by the location of the passband. 11 FREQUENCY RESPONSE PLOT • One way of identifying the type of filter circuit is to examine a frequency response plot. • Two parts: one is a graph of H(j) vs frequency. Called magnitude plot. • The other part is a graph of (j) vs frequency. Called phase plot. 12 TYPE OF FILTERS •Low Pass Filter •High Pass Filter •Band Pass Filter Band Reject Filter 13 FILTER’S FREQUENCY RESPONSE lowpass highpass c c bandpass c1 c2 bandreject c1 c2 14 MAGNITUDE TYPE H(0) H(∞) H(ωC)@H(ωo) LOWPASS 1 0 1/√2 HIGHPASS 0 1 1/√2 BANDPASS 0 0 1 BANDREJECT 1 1 0 15 CUT OFF FREQUENCY • LPF and HPF have one passband and one stopband, which are defined by the cut off frequency that separates them. • BPF passes a input signal to the output when the input frequency is within the band defined by the two cut off frequencies. • BRF passes a input signal to the output when the input frequency is outside the band defined by the two cut off frequencies. 16 CUT OFF FREQUENCY • The cutoff frequency (fc) is the frequency either above which or below which the power output of a circuit, such as a line, amplifier, or filter, is reduced to 1/2 of the passband power; the half-power point. • This is equivalent to a voltage (or amplitude) reduction to 70.7% of the passband, because voltage, V2 is proportional to power, P. 17 CUT OFF FREQUENCY • This happens to be close to −3 decibels, and the cutoff frequency is frequently referred to as the −3 dB point. • Also called the knee frequency, due to a frequency response curve's physical appearance. 18 CUT OFF FREQUENCY VO ( jc ) H ( jc ) Vi 1 H max 2 6 1 H max 2 1 Vo max 2 19 BODE PLOT • The most common way to describe the frequency response is by so called Bode plot. • Bode lot is a log-log plot for amplitude vs frequency and a linear-log plot for phase vs frequency. • Many circuits (e.g. amplifiers, filters, resonators, etc.) uses Bode plot to specify their performance and characteristics. 20 FILTER’S RESPONSE BODE PLOT 21 LOW PASS FILTER 22 LOW PASS FILTER (LPF) • The filter preserves low frequencies while attenuating the frequencies above the cut off frequencies. • There are two basic kinds of circuits that behave as LPFs: a) Series RL b) Series RC. 23 (a) LPF RL CIRCUIT sL Vi (s) Vo (s ) R OUTPUT 24 Transfer Function Vo R H (s) Vi R sL 25 The voltage transfer function R L H ( s) sR L • To study the frequency response, substitute s=j: R Vo L H ( j ) Vi j ( R ) L 26 Magnitude and Phase R L H ( j ) 2 2 R ( L) 1 L ( j ) tan R 27 When =0 and = Vi R Vi R 28 Qualitative Analysis • At low frequencies (L<< R): – jL is very small compared to R, and inductor functions as a short circuit. V0 Vi V0 Vi 29 Qualitative Analysis • At high frequencies (L>> R): – jL is very large compared to R, and inductor functions as a open circuit. V0 0 V0 Vi 90 30 LPF Frequency Response H(j) 1.0 0 c -90 31 32 Cutoff Frequency • At the cutoff frequency, voltage magnitude is equal to (1/2)Hmax : R 1 L H (C ) 2 2 R 2 ( L) R C L 33 Ex. • Electrocardiograph is an instrument that is used to measure the heart’s rhythmic beat. This instrument must be capable of detecting periodic signals whose frequency is about 1 Hz (the normal heart rate is 72 beats per minute). • The instrument must operate in the presence of sinusoidal noise consisting of signals from the surrounding electrical environment, whose fundamental frequency is 50 Hz- the frequency at which electric power is supplied. 34 Ex. i. Choose values for R and L in the series RL circuit such that the resulting circuit could be used in an electrocardiograph to filter out any noise above 10 Hz and pass the electric signals from the heart at or near 1 Hz. (choose L=100 mH) ii. Then compute the magnitude of Vo at 1 Hz, 10 Hz, and 50 Hz to see how well the filter performs. 35 Known quantities • Inductor, L = 100 mH • Cut off frequency, fc = 10 Hz – therefore, c = 2fc = 20 rad/s 36 Find R R c L (20 )(100 10 ) 3 6.28 37 Find the magnitude of Vo • Using the transfer function, the output voltage can be computed: Vo H ( j ) Vi Vo ( ) R L Vi 2 2 R ( L) 20 Vi 2 2 400 38 Vo ( ) 20 400 2 2 Vi f(Hz) Vi Vo 1 1.0 0.995 10 1.0 0.707 50 1.0 0.196 39 (b) LPF RC CIRCUIT R Vi vo C OUTPUT 40 Transfer Function 1 Vo jC H ( j ) Vi R 1 jC 1 RC j 1 RC 41 Magnitude and Phase of H(j) Vo H ( j ) Vi 1 RC 2 1 RC 1 ( j ) tan RC 2 42 • Zero frequency (=0): – the impedance of the capacitor is infinite, and the capacitor acts as an open circuit. – Vo and Vi are the same. • Infinite frequency (=): – the impedance of the capacitor is infinite, and the capacitor acts as an open circuit. – Vo is zero. • Frequency increasing from zero: – the impedance of the capacitor decreases relative to the impedance of the resistor – the source voltage divides between the resistive impedance and the capacitive impedance. – Vo is smaller than Vi. 43 Cutoff Frequency • The voltage magnitude is equal to (1/2) Hmax at the cutoff frequency: 1 RC H (C ) 2 C ( 1 RC 1 C RC 1 2 2 ) 44 GENERAL LPF CIRCUITS sL OUTPUT Vi (s) Vo (s ) R c H (s) s c R Vi vo C OUTPUT 45 Ex. • For the series RC circuit of LPF: a) Find the transfer function between the source voltage and the output voltage b) Choose values for R and C that will yield a LPF with cutoff frequency of 3 kHz. 46 a) Find the transfer function • The magnitude of H(j): 1 Vo RC H ( j ) Vi j 1 RC 1 Vo RC H ( j ) 2 Vi ( ) ( 1 RC ) 2 47 b) Find R & C • R and C cannot be computed independently, so let’s choose C=1F. • Convert the specified cutoff frequency from 3 kHz to c=2(3x10-3) rad/s. 48 Calculate R 1 R c C 1 3 6 (2 )(3 10 )(110 ) 53.05 49 HIGH PASS FILTER 50 HIGH PASS FILTER • HPF offer easy passage of a high frequency signal and difficult passage to a low frequency signal. • Two types of HPF: – RC circuit – RL circuit 51 a) CAPACITIVE HPF C Vi vo R OUTPUT 52 s-Domain Circuit 1 Vi (s) sC Vo (s ) R 53 When =0 and = Vi vo R Vi R 54 • Zero frequency (=0): – the capacitor acts as an open circuit, so there is no current flowing in R. – Vo is zero. • Infinite frequency (=): – the capacitor acts as an short circuit and thus there is no voltage across the capacitor. – Vo is equal to Vi. • Frequency increasing from zero: – the impedance of the capacitor decreases relative to the impedance of the resistor – the source voltage divides between the resistive impedance and the capacitive impedance. – Vo begins to increase. 55 HPF Frequency Response 56 Transfer Function s H (s) s 1 RC s j ; j H ( j ) j 1 RC 57 Magnitude & Phase H ( s) (1 2 RC ) 2 ( j ) 90 tan RC 1 58 Ex: INDUCTIVE HPF • Show that the series RL circuit below also acts as a HPF. R Vi vo L 59 Ex. a) Derive an expression for the circuit’s transfer function b) Use the result from (a) to determine an equation for the cutoff frequency c) Choose values for R and L that will yield HPF with fc = 15 kHz. 60 s-Domain circuit R Vi (s) Vo (s ) sL 61 Transfer Function s H (s) sR s j ; L j H ( j ) j R L 62 Magnitude & Phase H ( j ) 2 R ( L) 2 • H()=1 andH(0)=0 HPF 63 Cutoff Frequency H ( j ) 2 R ( L) 2 c 1 H (c ) 2 2 R 2 c ( L ) c R L 64 R and L • Choose R=500 , and convert fc to c: LR c 500 5 . 31 mH 3 (2 )(15 10 ) 65 GENERAL HPF CIRCUITS 1 sC OUTPUT Vi (s) Vo (s ) R s H (s) s c R Vi (s) Vo (s ) sL OUTPUT 66 REMARKS • The components and connections for LPF and HPF are identical but, the choice of output is different. • The filtering characteristics of a circuit depend on the definition of the output as well as circuit components, values, and connections. • The cutoff frequency is similar whether the circuit is configured as LPF or HPF. 67 BANDPASS FILTER 68 Bandpass Filter • BPF is essential for applications where a particular band or frequencies need to be filtered from a wider range of mixed signals. • There are 3 important parameters that characterize a BPF, only two of them can be specified independently: – Center frequency (and two cutoff frequencies) – Bandwidth – Quality factor 69 Cutoff freq. & Center freq. • Ideal bandpass filters have two cutoff frequencies, c1 and c2, which identify the passband. • c1 and c2 are the frequencies for which the magnitude of H(j) equal (1/2). • The center frequency, o is defined as the frequency for which a circuit’s transfer function is purely real. • Also called as the resonant frequency. 70 Bandwidth, and Quality Factor, Q • The bandwidth, tells the width of the passband. • The quality factor, Q is the ratio of the center frequency to the bandwidth. • The quality factor describes the shape of the magnitude plot, independent of frequency. 71 a) BPF: Series RLC L vi C vo R 72 At =0 and = L C vi vo R L vi C vo R 73 At =0 and = • Zero frequency (=0): – the capacitor acts as an open circuit and the inductor behaves like a short circuit, so there is no current flowing in R. – Vo is zero. • Infinite frequency (=): – the capacitor acts as an short circuit and the inductor behaves like an open circuit, so again there is no current flowing in R. – Vo is zero 74 Between =0 and = • Both capacitor and inductor have finite impedances. • Voltage supplied by the source will drop across both L and C, but some voltage will reach R. • Note that the impedance of C is negative, whereas the impedance of L is positive. 75 • At some frequency, the impedance of C and the impedance of L have equal magnitudes and opposite sign cancel out! • Causing Vo to equal Vi • This is happen at a special frequency, called the center frequency, o. 76 BPF Frequency Response 77 Center Frequency H max H ( jo ) 78 s-Domain Circuit sL Vi (s) 1 sC Vo (s ) R 79 Transfer Function ( R )s L H (s) 2 s ( R )s ( 1 ) L LC s j; j ( R ) L H ( j ) 2 ( R ) j ( 1 ) L LC 80 Magnitude & Phase H ( j ) ( R L) 2 ( 1 ) ( R ) j LC L j ( R ) L H ( j ) 2 2 2 1 R [( ) ] [( ) ] LC L (R ) 1 L ( j ) 90 tan ( 1 ) 2 LC 81 Center Frequency • For circuit’s transfer function is purely real: j o L 1 o j o C 0 1 LC 82 Cutoff Frequencies c1 R 2L c 2 R 2L R 2L R 2L 2 2 1 LC 1 LC 83 Relationship Between Center Frequency and Cutoff Frequencies o c1 c 2 84 Bandwidth, c1 c 2 R L 85 Quality Factor, Q o Q o L 1 Q R oCR L 2 RC 86 Cutoff Frequencies in terms of 2 c1 o 2 2 2 2 c 2 o 2 2 2 87 b) BPF: Parallel RLC R vi C vo L 88 GENERAL BPF CIRCUITS • BPF series RLC: ( R )s L H ( s) 2 s ( R )s ( 1 ) L LC o 1 , R LC L • BPF parallel RLC: (s ) RC H (s) 2 s s (1 ) RC LC o 1 , 1 LC RC 89 Remarks • The general circuit transfer functions for both series and parallel BPF: s H ( s) 2 2 s s o 90 BANDREJECT FILTER 91 Bandreject Filter • A bandreject attenuates voltages at frequencies within the stopband, which is between c1 and c2. It passes frequencies outside the stopband. • BRF are characterized by the same parameters as BPF: – Center freq. (and two cutoff frequencies) – Bandwidth – Quality Factor 92 BRF: Series RLC R vi L vo OUTPUT C 93 At =0 and = R vi L vo C R vi L vo C 94 At =0 and = • Zero frequency (=0): – the capacitor acts as an open circuit and the inductor behaves like a short circuit. – the output voltage is defined over an effective open circuit. – Magnitude of Vo and Vi are similar • Infinite frequency (=): – the capacitor acts as an short circuit and the inductor behaves like an open circuit,. – the output voltage is defined over an effective open circuit. – Magnitude of Vo and Vi are similar 95 Between =0 and = • Both capacitor and inductor have finite impedances of opposite signs. • As the frequency is increased from zero, the impedance of the inductor increases and that for the capacitor decreases. • At some frequency between the two passbands, the impedance of C and L are equal but opposite sign. • The series combination of L and C is that short circuit so the magnitude of Vo must be zero. This is happen at a special frequency, called the center frequency, o. 96 BRF Frequency Response 97 Center Frequency • The center freq. is still defined as the frequency for which the sum of the impedances of L and C is zero. • Only, the magnitude at the center freq. is minimum. H min H ( jo ) 98 s-Domain Circuit R Vi (s) sL Vo (s ) 1 sC 99 Transfer Function sL 1 s 1 2 sC LC H ( s) R 2 R sL 1 1 s s sC LC L s j ; H ( j ) 1 LC 2 R 1 j LC L 2 100 Magnitude & Phase H ( j ) 1 1 LC LC 2 2 2 R L 2 R 1 L ( j ) tan 2 1 ( ) LC 101 Center Frequency • For circuit’s transfer function is purely real: j o L 1 o j o C 0 1 LC 102 Cutoff Frequencies 2 R 1 R c1 2L 2 L LC 2 R 1 R c 2 2L 2 L LC 103 Relationship Between Center Frequency and Cutoff Frequencies o c1 c 2 104 Bandwidth, c1 c 2 R L 105 Quality Factor, Q o Q o L 1 Q R oCR L 2 RC 106 Cutoff Frequencies in terms of 2 c1 o 2 2 2 2 c 2 o 2 2 2 107 Remarks H (s) R sL Vi (s) Vo (s ) 1 sC Vi (s) 1 o R Vo (s ) 1 LC L s2 1 LC s 2 s 1 LC RC o 1 LC 1 RC H ( s) sC LC R 2 s s 1 LC L R sL s2 1 108 GENERAL BRF CIRCUIT • The general circuit transfer functions for both series and parallel BRF: s H ( s) 2 2 s s o 2 2 o 109