LEC 8 FREQUENCY RESPONSE

(OF CIRCUITS AND FILTERS)

Analogue Electronics and Systems

Overview

Learn to:

Derive transfer function of an ac circuit.

Generate magnitude and phase spectral plots.

Design 1st order lowpass, highpass and bandpass

and bandreject filters.

Generate Bode plots for any transfer function.

Active filters.

Transfer Function

Transfer function of a circuit or system describes the

output response to an input excitation as a function

of the angular frequency ω.

Voltage Gain

Other Transfer Functions

Magnitude Phase

Filters

Filter terminology

1. Gain factor M0 = peak value in passband.

If peak at dc => dc gain, as in lowpass filter.

If peak at ω = ∞ => high frequency gain (highpass filter)

For bandpass, called gain factor.

2. Corner (or cutoff) frequency ωc = angular

frequency when M(ωc) = 1

2of M0 =>

Half-power frequency

Bandpass and bandreject have 2 cutoff frequencies.

Filter terminology cont

3. Bandwidth, B = range of ω of filter’s passband.

4. Resonant frequency, ω0

Filter terminology cont

RC Low Pass

To determine corner frequency:

Looking at capacitor

voltage Vc gives lowpass

RC High Pass

ScalingScaling 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.

Magnitude scaling:

• TF of a circuit is based on the impedances of the elements.

• If all impedances are multiplied by Km, absolute level may or may

not change, but relative frequency response will remain the same.

Scaling : magnitude scaling

Scaling: frequency scaling

Shift TF along ω axis but keeping same shape,

Replace with ω’ = Kfω, and scale element so that

impedances remain unchanged.

Meaning eg for ZL = jωL, if ω is scaled up by Kf , L

has to be scaled down by same factor. So:

Scaling: combined magnitude and frequency

Often combine magnitude and

frequency scaling together:

Example 9-2: (a) is a 3rd order lowpass

filter with ωc = 1 rad/s. Develop a scaled version with ω’c = 106 rad/s and a resistor of 2 kΩ.

dB Scale

For the circuit shown here,

Example 9-3: RL Highpass Filter

Example: RL Filter -- Magnitude

(a) Voltage division gives

Magnitude plot

Log scale for ω and dB scale for M

The overall plot (in red) is obtained by

combining the two plots of individual

factors.

RL Filter--Phase

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

The 90o component is contributed by the

j in the numerator, and the second term is

the phase angle of the denominator.

Poles and Zeros

As an example, let’s consider:

(in standard form)

• ωz and ωp are related to circuit architecture and element values.

• ωz is called a zero of H(ω), because it is in the numerator.

• ωp is called a pole of H(ω), because it is in the denominator.

• A TF may have multiple poles and zeros.

• Converting to dB change multiplication into addition:

So that we can produce Bode plots (an approximation to the exact H(ω))

Bode Plots: Straight line approximations

Bode Magnitude Slope= 20N dB per decade

Bode Phase Slope= 45N degrees per decade

1 decade 1 decade

7 standard forms shown in Table 9.2, we

look at two in details:

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

Given this:

Rearrange to standard form:

Example 9-6: Given Bode Plot, Obtain Expression

First segment:

Second segment:

Passive Filters

Filter’s TF is characterised by:

1. Bandpass RLC Filter

Resonant frequency:Cutoff frequencies:

Bandwidth:

Bandpass RLC Filter (cont.)

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

Can you derive the

expressions for

magnitude and phase?

Filter Order

Filter order: RLC bandpass

RLC lowpass is 2nd order. But when we take Vout

across R, instead of C, we have a bandpass filter. Is

this bandpass filter a 2nd order filter?

This (BP) instead of this (LP)

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

LTspice Analysis of RLC Circuit

LTspice Analysis

of Active Filters

Summary