1. Filters and the Bode Plot
Chapter 22
Filters and the Bode Plot

2. Gain
Power gain is the ratio of output signal power to input signal power.
Voltage gain is the ratio of output signal voltage to input signal voltage.

3. 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 is called an attenuator.
Since gains may be very large or very small, it may be inconvenient to express the gain as a simple ratio.

4. The Decibel
The bel is a logarithmic unit that represents a tenfold increase or decrease in power.
Because the bel is such a large unit, the decibel is often used.

5. The Decibel
To express voltage gain in decibels:

6. Multistage Systems
To find the total gain of a system having more than one stage, each with a gain of An , multiply the gains together.
AT = A1A2A3
If the gains are expressed in decibels (which are logarithmic), then gains will add instead of multiplying.
AT(dB) = A1(dB) + A2(dB) + A3(dB)

7. Transfer Functions
The transfer function is the ratio of the output voltage phasor to the input voltage phasor for any frequency.
The amplitude of the transfer function is the voltage gain.
The phase angle, , represents the phase shift between the input and output voltage phasors.
If the circuit contains capacitors or inductors, the transfer function will be frequency dependent.

8. Transfer Functions
To examine the operation of a circuit over a wide range of frequencies, we draw a frequency response curve.
Any circuit which passes a particular range of frequencies is called a filter circuit.
Common types of filters are low-pass, high-pass, band-pass, and band-reject filters.

9. Low-Pass Filter
A low-pass filter has a greater gain at low frequencies; at higher frequencies the gain decreases.
The cutoff frequency occurs when the gain drops to of the maximum gain.
At cutoff the voltage gain is -3dB and the phase angle is -45°.

10. Bode Plots
A Bode plot is a straight-line approximation of the frequency response of a circuit.
The abscissa will be the frequency on a logarithmic scale.
The ordinate will be the gain expressed in dB.
The actual response will approach straight lines called asymptotes.

11. Bode Plots
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.

12. Writing Transfer Functions
A properly written transfer function allows us to easily sketch the frequency response of a circuit.
First, determine the voltage gain when  = 0 and   .
Use the voltage divider rule to write the general expression for the transfer function in terms of the frequency.

13. Writing Transfer Functions
Simplify the results into a form containing only terms of j or (1 + j).
Determine the break frequency(ies) at  = 1/.
Sketch the straight-line approximation by separately considering the effects of each term of the transfer function.
Sketch the actual response from the approximation.

14. The RC Low-Pass Filter
A series RC circuit with the output taken across the capacitor is a low-pass filter.
At low frequencies, the reactance is high, so the output voltage is essentially equal to the input.
At high frequencies, the output voltage approaches zero.

15. The RC Low-Pass Filter
By applying the voltage divider rule, we can determine the transfer function
The cutoff frequency is

16. The RL Low-Pass Filter
A low-pass filter may be made up of a resistor and an inductor, with the output taken across the resistor.
The transfer function is
The cutoff frequency is

17. The RC High-Pass Filter
A simple RC circuit with the output taken across the resistor is a high-pass filter.
The transfer function is given by
The phase shift is  = 90° - arctan /c.
The cutoff frequency is

18. The RL High-Pass Filter
An RL circuit is a high-pass filter if the output is taken across the inductor.
The transfer function is
The cutoff frequency is

19. The Band-Pass Filter
A band-pass filter will permit frequencies within a certain range to pass from the input of a circuit to the 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.

20. The Band-Reject Filter
A band-reject filter passes all frequencies except for a narrow band.
It can be constructed from a RLC series circuit, taking the output across the inductor and capacitor.
It can also be constructed from a circuit containing a RC parallel combination in series with a resistor, taking the output across the resistor.