Chapter 22

Filters and the Bode Plot

Gain

Power gain is ratio of output power to input power

Gain

Voltage gain is ratio of output voltage to input voltage

Gain

Any circuit in which the output signal power is greater than the input signal

Power is referred to as an amplifier

Any circuit in which the output signal power is less than the input signal power

Called an attenuator

Gain

Gains are very large or very small

Inconvenient to express gain as a simple ratio

The Decibel

Bel is a logarithmic unit that represents a tenfold increase or decrease in power

The Decibel

Because the bel is such a large unit, the decibel (dB) is often used

The Decibel

To express voltage gain in decibels:

Multistage Systems

To find total gain of a system having more than one stage, each with a gain of A_n

Multiply gains together

A_T = A₁A₂A₃∙∙∙

Multistage Systems

If gains are expressed in decibels (which are logarithmic)

Gains will add instead of multiplying

A_T(dB) = A_1(dB)+ A_2(dB)+ A_3(dB)∙∙∙

Voltage Transfer Functions

Ratio of output voltage phasor to input voltage phasor for any frequency

Amplitude of transfer function is voltage gain

Voltage Transfer Functions

Phase angle q

Represents phase shift between input and output voltage phasors

If the circuit contains capacitors or inductors

Transfer function will be frequency dependent

Transfer Functions

To examine the operation of a circuit over a wide range of frequencies

Draw a frequency response curve

Any circuit which is said to pass a particular range of frequencies

Called a filter circuit

Transfer Functions

By passing a range of frequencies

Filter output response is high enough at these frequencies to be usable

Common types of filters

Low-pass, high-pass, band-pass, and band-reject filters

Low-Pass Filter

Has a greater gain at low frequencies

At higher frequencies the gain decreases

Cutoff frequency

Occurs when gain drops to ½ power point

This is 0.707 of the maximum voltage gain

At cutoff

Voltage gain is –3dB; phase angle is –45°

Bode Plots

A Bode plot is a straight-line approximation to the frequency response of a particular filter

Abscissa will be the frequency in Hz on a logarithmic scale (base 10)

Bode Plots

Ordinate will be gain in dB on a linear scale

Asymptotes

Actual response will approach the straight lines of the Bode approximation

Bode Plots

A decade represents a tenfold increase or decrease in frequency

An octave represents a two-fold increase or decrease in frequency

Bode Plots

Slopes are expressed in either dB/decade or dB/octave

A simple RC or RL circuit will have a slope of 20 dB/decade or 6 dB/octave

Writing Voltage Transfer Functions

A properly written transfer function allows us to easily sketch the frequency response of a circuit

First, determine voltage gain when w = 0 and w ® ¥ (approaches infinity)

Writing Voltage Transfer Functions

Use voltage divider rule to write the general expression for transfer function in terms of the frequency

Writing Voltage Transfer Functions

Simplify results into a form containing only terms of jwt or (1 + jwt)

Determine break frequencies at w = 1/t

Writing Voltage Transfer Functions

Sketch straight-line approximation by separately considering the effects of each term of transfer function

Sketch actual response freehand from the approximation

The RC Low-Pass Filter

A series RC circuit with output taken across capacitor is a low-pass filter

At low frequencies

Reactance is high

Output voltage is essentially equal to input

The RC Low-Pass Filter

At high frequencies

Output voltage approaches zero

The RC Low-Pass Filter

By applying voltage divider rule

Determine transfer function

The RC Low-Pass Filter

The cutoff frequency is

The RL Low-Pass Filter

Low-pass filter may be made up of a resistor and an inductor

Output taken across the resistor

Transfer function is

The RL Low-Pass Filter

Cutoff frequency is

The RC High-Pass Filter

Simple RC circuit with output taken across resistor is a high-pass filter

Transfer function is given by

The RC High-Pass Filter

Phase shift is q = 90° – tan^-1(w/w_c)

Cutoff frequency is

The RL High-Pass Filter

RL circuit is a high-pass filter if output is taken across the inductor

Transfer function is

The RL High-Pass Filter

Cutoff frequency is

The Band-Pass Filter

Permits frequencies within a certain range to pass from input to output

All frequencies outside this range will be attenuated

The Band-Pass Filter

One way to build a band-pass filter is to cascade a low-pass filter with a high-pass filter

A band-pass filter can also be constructed from an RLC circuit

The Band-Reject Filter

Passes all frequencies except for a narrow band

Can be constructed from an RLC series circuit

Taking output across the inductor and capacitor

The Band-Reject Filter

Can also be constructed from a circuit containing a RC parallel combination in series with a resistor

Taking output across the resistor


	Any circuit in which the output signal power is greater than the input signal
		Power is referred to as an amplifier
	Any circuit in which the output signal power is less than the input signal power
		Called an attenuator


	Gains are very large or very small
		Inconvenient to express gain as a simple ratio


	Bel is a logarithmic unit that represents a tenfold increase or decrease in power


	Because the bel is such a large unit, the decibel (dB) is often used


	To find total gain of a system having more than one stage, each with a gain of A_n
		Multiply gains together
		A_T = A₁A₂A₃∙∙∙


	If gains are expressed in decibels (which are logarithmic)
		Gains will add instead of multiplying
		A_T(dB) = A_1(dB)+ A_2(dB)+ A_3(dB)∙∙∙


	Ratio of output voltage phasor to input voltage phasor for any frequency
	Amplitude of transfer function is voltage gain


	Phase angle q
		Represents phase shift between input and output voltage phasors
	If the circuit contains capacitors or inductors
		Transfer function will be frequency dependent


	To examine the operation of a circuit over a wide range of frequencies
		Draw a frequency response curve
	Any circuit which is said to pass a particular range of frequencies
		Called a filter circuit


	By passing a range of frequencies
		Filter output response is high enough at these frequencies to be usable
	Common types of filters
		Low-pass, high-pass, band-pass, and band-reject filters


	Has a greater gain at low frequencies
		At higher frequencies the gain decreases
	Cutoff frequency
		Occurs when gain drops to ½ power point
		This is 0.707 of the maximum voltage gain
	At cutoff
		Voltage gain is –3dB; phase angle is –45°


	A Bode plot is a straight-line approximation to the frequency response of a particular filter
	Abscissa will be the frequency in Hz on a logarithmic scale (base 10)


	Ordinate will be gain in dB on a linear scale
	Asymptotes
		Actual response will approach the straight lines of the Bode approximation


	A decade represents a tenfold increase or decrease in frequency
	An octave represents a two-fold increase or decrease in frequency


	Slopes are expressed in either dB/decade or dB/octave
	A simple RC or RL circuit will have a slope of 20 dB/decade or 6 dB/octave


	A properly written transfer function allows us to easily sketch the frequency response of a circuit
	First, determine voltage gain when w = 0 and w ® ¥ (approaches infinity)


	Use voltage divider rule to write the general expression for transfer function in terms of the frequency


	Simplify results into a form containing only terms of jwt or (1 + jwt)
	Determine break frequencies at w = 1/t


	Sketch straight-line approximation by separately considering the effects of each term of transfer function
	Sketch actual response freehand from the approximation


	A series RC circuit with output taken across capacitor is a low-pass filter
	At low frequencies
		Reactance is high
		Output voltage is essentially equal to input


	Low-pass filter may be made up of a resistor and an inductor
		Output taken across the resistor
	Transfer function is


	Simple RC circuit with output taken across resistor is a high-pass filter
	Transfer function is given by


	Phase shift is q = 90° – tan^-1(w/w_c)
	Cutoff frequency is


	RL circuit is a high-pass filter if output is taken across the inductor
	Transfer function is


	Permits frequencies within a certain range to pass from input to output
	All frequencies outside this range will be attenuated


	One way to build a band-pass filter is to cascade a low-pass filter with a high-pass filter
	A band-pass filter can also be constructed from an RLC circuit


	Passes all frequencies except for a narrow band
	Can be constructed from an RLC series circuit
		Taking output across the inductor and capacitor


	Can also be constructed from a circuit containing a RC parallel combination in series with a resistor
		Taking output across the resistor