9. FREQUENCY RESPONSE

CIRCUITS by Ulaby & Maharbiz

Overview

Transfer FunctionTransfer function of a circuit or system describes the output response to an input excitation as a function of the angular frequency ω.

Voltage GainOther Transfer Functions

Magnitude Phase

Filters

RC Low Pass

To determine corner frequency:

RC High Pass

Filter Terminology

Zin1 = R + jωL.

Im [Zin1] = 0 when ω = 0

Im [Zin2] = 0 requires that ZL = −ZC

or, equivalently, ω2 = 1/LC

Scaling Scaling is used to configure a prototype version of the intended practical scaled circuit such that in the prototype circuit, element values are on the order of ohms, henrys and farads.

dB Scale

RL Filter --Magnitude

Log scale for ω and dB scale for M

RL Filter--Phase

Log scale for ω and linear scale for φ(ω)

Bode Plots: Straight line approximations

Bode Magnitude Slope= 20N dB per decade

Bode Phase Slope= 45N degrees per decade

1 decade 1 decade

Bode Plots

Bode Magnitude Slope= 40dB per decade

Bode Phase Slope= 90 degrees per decade

Bode Factors

Example 9-4: Bode Plots

Standard form

Numerator: simple zero of second order with corner frequency 5 rad/s

Denominator: pole @ origin, and simple pole with corner frequency 50 rad/s

Example 9-5: More Bode Plots

Example 9-6:Given Bode Plot, Obtain Expression

Bandpass RLC Filter

Bandpass RLC Filter (cont.)Quality Factor Q: characterizes degree of selectivity of a circuit

where Wstor is the maximum energy that can be stored in the circuit at resonance (ω = ω0), and Wdiss is the energy dissipated by the circuit during a single period T.

Bandpass RLC Filter (cont.) Derivation of Q

Resonant frequency

Bandwidth

Bandpass Filter

Example 9-7: Bandpass Filter Design

Highpass Filter Lowpass Filter

Bandreject Filter

Filter Order

Active Filters ̶ Lowpass

Active Filters ̶ Highpass

Cascading Active Filters

Example 9-10: Third-Order Lowpass Filter

Cont.

Example 9-11 cont.

Cont.

Signal Modulation

Superheterodyne receiver

Frequency of received signal is “down-converted” to a lower intermediate frequency, while retaining the modulation ( which contains the message information) intact

Multisim Analysis of RLC Circuit

Multisim Analysis of Active Filters

Tech Brief 17: Bandwidth and Data Rate

Signal-to-noise ratio

Tech Brief: Bandwidth and Data Rate

Channel capacity (data rate) in bits/s

Bandwidth in Hz

Shannon-Hartley Theorem

Note: A high data rate can be achieved even if the signal power is smaller than the noise, so long as sufficient bandwidth is available.

Summary