Frequency Response DR. GYURCSEK ISTVÁN

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DR. GYURCSEK ISTVÁN Frequency Response Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 Ch. Alexander, M. Sadiku: Fundamentals of Electric Circuits, 6th Ed., McGraw Hill NY 2016, ISBN: 978-0078028229 Dr. Selmeczi K. Schnöller A.: Villamosságtan 1. MK Budapest 2002, TK szám: 49203/I Dr. Selmeczi K. Schnöller A.: Villamosságtan 2. TK Budapest 2002, ISBN:9631026043 1 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Progress in Content The Decibel Scale (gain, level, db, neper) Transfer Function (freq. response, represent.) Bode Plots (magnitude, phase. building blocks) Use Cases (base circuits, Nyquist/Bode plots) Outlook (practical examples) 2 gyurcsek.istvan@mik.pte.hu 2018.03.23.

The Decibel Scale Before we start Gain in communication measured in bel. G db = 10 log 1 = 0 db log P 1 P 2 = log P 1 + log P 2 log P 1 P 2 = log P 1 log P 2 log P n = n log P log 1 = log P P log 1 = 0 G = log P 2 P 1 G db = 10 log P 2 P 1 2 G db = 10 log P V 2 2 = 10 log P 2 1 V 1 R 2 R 1 = 20 log G db = 10 log 2 3 db R 2 = R 1 G db = 20 log V 2 V 1 = 20 log I 2 I 1 G db = 10 log 0.5 3 db V 2 V 1 10 log R 2 R 1 3 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Normal Generator 1.55V 600 Ω0 Zero Level, Sound Level, Noise Level (Examples) Level (rel./abs.) dbm dbu dbi / dbd Gain (db) Ratio PWR standard reference in denominator (db, dbm, dbu) 1 mw reference 0.775 V reference (600ohm!) antenna gain (to isotropic / dipole) Ratio Intensity 0 1 1 3 2 1.41 6 4 2 10 10 3.16 20 100 10 30 1 000 31.6 40 10 000 100 4 gyurcsek.istvan@mik.pte.hu 2018.03.23.

History Np and/or db H = H e jφ ln H = ln H + jφ Re H = ln H, Im H = φ ratio db Np int H X = X 2 X 1 h X (db) = 20 log HX h X (Np) = ln HX pwr H P = P 2 P 1 h P (db) = 10 log HP h P (Np) = 1 2 ln H P 1 db = 0.115 Np, 1 Np = 8.686 db 5 gyurcsek.istvan@mik.pte.hu 2018.03.23.

History Hendrik Wade Bode (eng., dutch, 1905 1982) Alexander Graham Bell (phisics, scottish,1847 1922) John Neper (of Merchiston) (maths, scottish,1550 1617) Harry Nyquist (eng., swedish,1889 1976) 6 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Progress in Content The Decibel Scale (gain, level, db, neper) Transfer Function (freq. response, represent.) Bode Plots (magnitude, phase. building blocks) Use Cases (base circuits, Nyquist/Bode plots) Outlook (practical examples) 7 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Frequency Response Frequency response Variation in circuits behavior with change in signal FRQ Variation of the GAIN and PHASE with FRQ Transfer function (network function H(ω) FRQ-dept. ratio of a phasor output Y(ω ) to a phasor input X(ω). Four possible transfer functions Voltage gain Current gain H ω = V o ω V i ω H ω = I o ω I i ω Transfer impedance Transfer admittance H ω = V o ω I i ω H ω = I o ω V i ω H ω = Y ω X ω = N ω D ω Roots of N ω = 0 zeros (jω = s = z 1, z 2, z 3, ) Roots of D ω = 0 poles (jω = s = p 1, p 2, p 3, ) 8 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Frequency Response Example Magnitude and phase of G(s) as a function of frequency. 9 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Graphical Representation 1 Nyquist Diagram Parametric plot frequency response (Harry Nyquist, engineer at Bell Labs, 1932) Frequency is swept as a parameter. Used in automatic control and signal processing (stability of a system with feedback). H ω = Y(ω) X(ω) = b m jω m + + b 1 jω 1 + b 0 a n jω n + + a 1 jω 1 + a 0 In Cartesian coordinates Re(H) X axis. Im (H) Y axis. In polar coordinates Gain of H radial coordinate Phase of H angular coordinate Work diagram param dependent analysis 10 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Graphical Representation 2 Bode Plot(s) Simple but accurate graphic method Gain plot of frequency response Phase-shift plot of frequency response Hendrik Wade Bode, eng. at Bell Labs, 1938, dutch) Used in electrical engineering and control theory Ideas Horizontal axis proportional to the log(ω) Gain is measured in db Use asymptote method H db = 20 log H(ω) φ = Arc H(ω = tan 1 Im H(ω) Re H(ω) 11 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Progress in Content The Decibel Scale (gain, level, db, neper) Transfer Function (freq. response, represent.) Bode Plots (magnitude, phase. building blocks) Use Cases (base circuits, Nyquist/Bode plots) Outlook (practical examples) 12 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Bode Plots Factorising the transfer function standard form H ω = N ω D ω = Seven types of factors A gain K K jω ±1 1 + jω z1 1 + j2ζ ω 1 ωk + jω 1 + p1 1 + j2ζ ω 2 ωn + K 2 jω ωn 2 jω ωk A pole or zero at the origin A simple pole or zero A quadratic pole or zero (ζ - damping factor) jω ±1 jω 1 + jω z1 or 1 + p1 1 + j2ζ 1 ω ωk + 2 jω ωl or 1 + j2ζ2 ω ωn + 7factors 2 jω ωn IDEA plot the building blocks (factors) separately and add them due to the log scale. 13 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Constant term (gain K) H ω = 20 log K = constant, φ ω = 0, if K < 0 φ ω = 180 14 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Pole / Zero at the Origin For pole H ω = 20 log jω, φ ω = 90, slope 20 db dec For zero H ω = 20 log jω, φ ω = 90, slope 20 db dec in general H ω = 20 log jω N, φ ω = N 90, slope N 20 db dec 15 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Simple Pole / Zero H ω = 20 log 1 + H ω = 20 log 1 + φ ω = tan 1 ω z1 jω z1 = 20 log 1 = 0 as ω 0 jω z1 = 20 log ω z1 as ω Asymptotes ω z 1 zero slope ω z 1 20 db dec slope Corner frequency (break frequency) ω = z 1 Line approx. in magnitude (error) 20 log 1 + j = 20 log 2 3 db Line approx in phase (see figure) 16 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Quadratic Pole / Zero H ω = 20 log 1 + j2ζ 2 ω ωn + 2 jω ωn = 20 log 1 = 0 as ω 0 H ω = 20 log 1 + j2ζ 2 ω ωn + 2 jω ωn φ ω = tan 2ζ ω 1 2 p1 1 ω 2 2 ω n = 40 log ω ω n as ω Asymptotes ω ω n zero slope ω ω n 40 db dec slope Damping factor dependent exact curves. 17 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Remark Damping Factor Dependent Exact Curves ω H ω n = 20 log 1 + j2ζ n 2 ωn + = 20 log j2ζ 2 = 20 log 2ζ 2 CASE A critically damped ζ 2 = 1 H ω 0 CASE B overdamped real roots 2 jω n ωn p 1,2 = repeated double roots = 20 log 2 1 = 3 db ζ 2 > 1 H ω 0 = 20 log 2ζ 2 3 db CASE C underdamped conjugate complex roots ζ 2 < 1 H ω 0 = 20 log 2ζ 2 < 3 db CASE D undamped conjugate imaginary roots ζ 2 = 0 H ω 0 = 20 log 0 ± Roots 2ζ 2 ωn ± 2 4ζ 2 2 ω 4 2 n ω n 2 2 ωn = ω n ζ 2 ± ζ 2 2 1 18 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Quick Procedure Idea Zeros cause an increase in slope Poles cause a decrease is slope Bode plot immediately from the transfer function without making individual plots and adding them Procedure Starting with the low-frequency asymptote of the Bode plot Increasing or decreasing the slope at each corner frequency moving along the FRQ axis Comment Software packages (MATLAB, Mathcad, Micro-Cap, Pspice) generate frequency response plots. 19 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Phase (deg); Magnitude (db) To: Y(1) Example Using Matlab For Frequency Response G s = 5.000 s + 10 s + 1 s + 500 = 5.000s + 50.000 s 2 + 501s + 500 40 30 Bode Diagrams num = [5000 50000]; den = [1 501 500]; Bode (num,den) 20 10 0-10 0-20 1 10 100 500 Blue - calculated magn. and phase plots (exact) Red Bode (approximate) plot for the magn. 3 db error at the corner frequencies! -40-60 -80 Bode for: G jω = 100 1 + jω 10 1 + jω 1 1 + jω 500-100 10-1 10 0 10 1 10 2 10 3 10 4 Frequency (rad/sec) 20 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Analytical Method for Determining the Phase For example H ω = 100 jω 10 + 1 jω + 1 jω 500 + 1 Arc H ω = φ = tan 1 ω 10 tan 1 ω 1 tan 1 ω 500 In general H ω = He jφ = A Beφ B Ce φ C De φ D Ee φ E φ = φ B +φ C φ D φ E ± = tan 1 im B re B + tan 1 im C re C tan 1 im D re D tan 1 im E re E ± 21 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Summary 1 Factor Magnitude Phase K 22 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Summary 2 Factor Magnitude Phase jω N 1 jω N 23 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Summary 3 Factor Magnitude Phase 1 + jω z N 1 + jω p N 24 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Summary 4 Factor Magnitude Phase 1 + j2ζ ω ωn + 2 jω N ωn 1 1 + j2ζ ω ωk + jω ωk 2 N 25 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Progress in Content The Decibel Scale (gain, level, db, neper) Transfer Function (freq. response, represent.) Bode Plots (magnitude, phase. building blocks) Use Cases (base circuits, Nyquist/Bode plots) Outlook (practical examples) 26 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 1 Proportional Transfer Function (P) TF in time domain v t = A u t TF in frequency domain Y jω = A Im Y A Nyquist / Bode Plots Re Y Y [db] 20lgA lg lg 27 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 2 Differential Transfer Function (D) TF in time domain: TF in frequency domain v t = T D du(t) dt Y jω = jωt D Y [db] 20 db/dek Nyquist / Bode Plots Im Y = 1 T D lg Re Y lg 28 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 3 Integral Transfer Function (I) TF in time domain: TF in frequency domain: Nyquist / Bode Plots T 1 dv(t) dt Im Y = A u t v t = A න u t dt + v 0, where A = T T 1 T I 1 Y jω = 1 jωt I Re Y 0 t Y [db] -20 db/dek lg = 1 T I lg 29 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 4 Proportional Diferential Transfer Function (PD) TF in time domain: TF in frequency domain Nyquist / Bode Plots Im Y v t = A 1 + T D du(t) dt Y jω = A 1 + jωt D A Re Y Y [db] 20lgA 20 db/dek = 1 lg T D lg 30 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 5 Proportional Integral Transfer Function (PI) t TF in time domain: TF in frequency domain: Nyquist / Bode Plots v t = A + A න u t dt + v(0) T I Y jω = A 1 + 1 jωt I 0 = A 1 + jωt I jωt I Y [db] 20lgA -20 db/dek = 1 T I lg Im Y A Re Y P tag I tag PD tag lg 31 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Use Case 6 Proportional Transfer Function with one Storage (PT1) TF in time domain: TF in frequency domain: Nyquist / Bode Plots T dv(t) dt Im Y + v(t) = A u(t) Y jω = A 1 T A 1 + jωt Re Y Y [db] 20lgA -20 db/dek lg = 1 T lg 32 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Progress in Content The Decibel Scale (gain, level, db, neper) Transfer Function (freq. response, represent.) Bode Plots (magnitude, phase. building blocks) Use Cases (base circuits, Nyquist/Bode plots) Outlook (practical examples) 33 gyurcsek.istvan@mik.pte.hu 2018.03.23.

OUTLOOK 1: PID Control Loop in Industrial Control System PID: control signal depends on the errors in past, present and the future. Effects of varying PID parameters (K p,k i,k d ) on the step response of a system. 34 gyurcsek.istvan@mik.pte.hu 2018.03.23.

OUTLOOK 2: Common Filter Transfer Function Families Butterworth filter (en.wikipedia.org/wiki/butterworth_filter) Maximally flat in passband and stopband for the given order Chebyshev filter (Type I) (en.wikipedia.org/wiki/chebyshev_filter) Max. flat in stopband, sharper cutoff than Butterworth Chebyshev filter (Type II) (en.wikipedia.org/wiki/chebyshev_filter) Max. flat in passband, sharper cutoff than Butterworth Bessel filter (en.wikipedia.org/wiki/bessel_filter) Best pulse response for a given order, no group delay ripple Elliptic filter (en.wikipedia.org/wiki/elliptic_filter) Sharpest cutoff (narrow transition bw. pass band and stop band) Optimum "L" filter (en.wikipedia.org/wiki/optimum_%22l%22_filter) Gaussian filter (en.wikipedia.org/wiki/gaussian_filter) Minimum group delay; gives no overshoot to a step function. Hourglass filter (en.wikipedia.org/wiki/hourglass_filter) Raised-cosine filter (en.wikipedia.org/wiki/raised-cosine_filter) Example: Fourth-order type I Chebyshev low-pass filter 35 gyurcsek.istvan@mik.pte.hu 2018.03.23.

OUTLOOK 3: Active Crossover Filter (Example) CIRCUIT EXAMPLE FOR AN ACTIVE FOURTH-ORDER CROSSOVER FILTER (LEFT CHANNEL) 36 gyurcsek.istvan@mik.pte.hu 2018.03.23.

Questions 37 gyurcsek.istvan@mik.pte.hu 2018.03.23.

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