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PowerPoint Presentation Fundamentals of Electric Circuits AC Circuits Chapter 14 Frequency responses 14 1 Introduction 14 2 Transfer function 14 3 Decibel scale 14 4 Bode plots 14 5 Series/parallel re[.]

Fundamentals of Electric Circuits AC Circuits Chapter 14 Frequency responses 14.1 Introduction 14.2 Transfer function 14.3 Decibel scale 14.4 Bode plots 14.5 Series/parallel resonance 14.6 Passive/active filters 14.7 Scaling FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.1 Introduction + Previous chapters: learned how to find voltages and currents in a circuit with a constant frequency source + Let the amplitude of the sinusoidal source remain constant and vary the frequency  obtain the circuit’s frequency response The frequency response of a circuit is the variation in its behavior with change in signal frequency + The sinusoidal steady state frequency responses of circuits  significant in many applications (communications, control systems, filters) FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.2 Transfer function + Transfer function H(ɷ) (network function):  a useful analytical tool for finding the frequency response of a circuit + The frequency response of a circuit:  the plot of the circuit’s transfer function H(w) versus w with w varying from to ∞ Transfer fucntion H(ɷ) of a circuit:  ratio of a output phasor Y(ɷ) (voltage or current on an element) to an input phasor X(ɷ) (source voltage or current) Y   H    H     X   X() Input Linear network H() Y() Output FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.2 Transfer function + Since the input and output can be either voltage or current  possible transfer functions Vo   Voltage gain: H    Vi   Io   Current gain: H   Ii   Vo   Transfer impedance: H    Ii   Io   Transfer admittance: H   Vi   + To obtain the transfer functions: o Replace: R  R L  j L C j C o Apply any circuit analysis technique to find the defined ratio FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.2 Transfer function + Transfer function:  expressed in terms of its numerator polynomial N(ɷ), and denominator polynomial D(ɷ) N   H   D  Zero points z1, z2, …: The roots of N(ɷ) = Pole point p1, p2, …: The roots of D(ɷ) =  A zero, as a root of the numerator polynomial, is a value that results in a zero value of the function  A pole, as a root of the denominator polynomial, is a value for which the function is infinite FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.2 Transfer function + Example 1: Given a circuit as the next figure, find the transfer function V0/VS and its frequency response Solution + Replace the given circuit by the equivalent circuit in the frequency domain + The transfer function is: V ()  jC  H()  1 j RC VS() R jC H       1      0     RC    tan 1        0  + Frequency response is: FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.3 Decibel scale + It is not always easy to get a quick plot of the magnitude and phase of the transfer function + The frequency range required in frequency response is wide  it is inconvenient to use a linear scale + A more systematic way of obtaining the frequency response  use Bode plots which are based on logarithms Magnitude H 20log10H (dB) 0,001 -60 0,01 -40 0,1 -20 0,5 -6 1/sqrt(2) -3 sqrt(2) 10 20 20 26 100 40 1000 60 FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.3 Decibel scale P2 V22 / R2 V22 R GdB  10 log10  10 log10  10 log10  10 log10 P1 V1 / R1 V1 R2 For R1 = R2 : GdB  20 log10 GdB 10 log10 V2 V1 GdB  20 log10 I2 I1 I2 P2 V2 ; GdB 20 log10 ; GdB 20 log10 I1 P1 V1 + Note: o 10log is used for power, which 20log is used for voltage or current o dB value is dimensionless quantity FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.4 Bode plot + In Bode plots:  a logarithmic scale for the frequency axis, a linear scale in magnitude or phase Bode plots are semilog plots of the magnitude (in dB) and phase (in degrees) of a transfer function versus frequency   H   H      He j  lnH   ln H   ln e j  ln H  j Real part lnH is a function of the magnitude of the transfer function Imaginary part is the phase FUNDAMENTALS OF ELECTRIC CIRCUITS – AC Circuits 14.5 Series/parallel resonance Resonance is a condition in an R-L-C circuit in which the capacitive and inductive impedance are equal in magnitude, thereby resulting in a purely resistive impedance (reactance equals to zero) + Resonance occurs:  in any system that has a complex conjugate pair of poles  cause of oscillations of stored energy from one form to another + Resonance is a phenomenon:  allows frequency discrimination in communication networks, filter construction,… + The most prominent feature of the frequency response:  the sharp peak (resonant peak) exhibited in its amplitude characteristic