Frequency Response Instructor: Chia-Ming Tsai Electronics Engineering National Chiao Tung University Hsinchu, Taiwan, R.O.C.

Contents Introduction Transfer Function The Decibel Scale Bode Plots Series Resonance Parallel Rosonance Passive Filters Active Filters Applications

Introduction Analysis with a constant frequency is already learned. To obtain the frequency response –Keep the amplitude and the phase of the sinusoidal source constant (amplitude=1, phase=0) –Sweep the frequency from a starting frequency to a stop frequency –Plot the amplitude and the phase of the desired voltage or current versus frequency

Transfer Function The transfer function H(  ) can be expressed as Zeros: the roots of N(  )=0 Poles: the roots of D(  )=0

Example 1 Phasor domain Time domain

Example 2

Example 3

The Thought of Bode Plots It is quite difficult to handle the plotting of the transfer function in a linear scale. If the transfer function is transformed to a logarithmic scale, then the plotting becomes much more easy.

The Decibel (dB) Scale

Bode Plots

Steps to construct a Bode plot: –Plot each factor separately –Additively combine all of them graphically because of the logarithms involved The mathematical convenience of the logarithm makes the Bode plots a powerful tool Straight-line plots used instead of actual plots

Bode Plots: A Gain K

Bode Plots: Zero/Pole at Origin

Bode Plots: Simple Zero 20 dB/decade

Bode Plots: Simple Pole -20 dB/decade -45  /decade

Bode Plots: Quadratic Pole

Bode Plots: Quadratic Zero

Summary

Example 1

Example 1 (Cont’d)

Example 2

Example 2 (Cont’d)

Example 3

Example 3 (Cont’d)

Series Resonance

When Resonance Occurs 1.The impedance is purely resistive. The LC series combination acts like a short circuit. 2.The voltage and the current are in phase, so the power factor is unity. 3.The impedance Z(  ) is minimum. 4.The voltage across L and C can be much larger than the source voltage.

Half-Power Frequencies

Quality Factor: Q

Summary Voltage across L and C QV m

Parallel Resonance

When Resonance Occurs 1.The impedance is purely resistive. The LC parallel combination acts like an open circuit. 2.The voltage and the current are in phase, so the power factor is unity. 3.The admittance Y(  ) is minimum. 4.The current flowing through L and C can be much larger than the source current.

Comparisons Series circuit Parallel circuit

Passive Filters Lowpass Highpass Bandpass Bandstop

Lowpass Filter

Highpass Filter

Bandpass Filter B

Bandstop Filter rejection frequency B

Active Filters A general 1st-order active filter Active 1st-order lowpass filter

Active 1st-Order Highpass Filter

Active Bandpass Filter

Bandreject (or Notch) Filter

Applications: Radio Receiver 2055 kHz rejected 2055 kHz

Touch-Tone Telephone (1/2)

Touch-Tone Telephone (2/2)

Crossover Network (Lowpass) (Highpass)