1. Frequency Characteristics of AC Circuits
Introduction
A High-Pass RC Network
A Low-Pass RC Network
A Low-Pass RL Network
A High-Pass RL Network
A Comparison of RC and RL Networks
Bode Diagrams
Combining the Effects of Several Stages
RLC Circuits and Resonance

2. Introduction
Filters are circuits that are capable of passing signals within a band of frequencies while rejecting or blocking signals of frequencies outside this band. This property of filters is also called “frequency selectivity”.
Filter can be passive or active filter.
Passive filters: The circuits built using RC, RL, or RLC circuits.
Active filters : The circuits that employ one or more
op-amps in the design an addition to
resistors and capacitors

3. Advantages of Active Filters over Passive Filters
Active filters can be designed to provide required gain, and hence no attenuation as in the case of passive filters
No loading problem, because of high input resistance and low output resistance of op-amp.
Active Filters are cost effective as a wide variety of economical op-amps are available.

4. Applications
Active filters are mainly used in communication and signal processing circuits.
They are also employed in a wide range of applications such as entertainment, medical electronics, etc.

5. Earlier we looked at the bandwidth and frequency response of amplifiers
Having now looked at the AC behaviour of components we can consider these in more detail
The reactance of both inductors and capacitance is frequency dependent and we know that

6. We will start by considering very simple circuits
Consider the potential divider shown here
from our earlier consideration of the circuit
rearranging, the gain of the circuit is
this is also called the transfer function of the circuit

7. A High-Pass RC Network Consider the following circuit
which is shown re-drawn in a more usual form

8. Clearly the transfer function is
At high frequencies
 is large, voltage gain  1
At low frequencies
 is small, voltage gain  0

9. Since the denominator has real and imaginary parts, the magnitude of the voltage gain is
When
This is a halving of power, or a fall in gain of 3 dB

10. The half power point is the cut-off frequency of the circuit
the angular frequency C at which this occurs is given by
where  is the time constant of the CR network.
Also

11. Substituting  =2f and CR = 1/ 2fC in the earlier equation gives
This is the general form of the gain of the circuit
It is clear that both the magnitude of the gain and the phase angle vary with frequency

12. Consider the behaviour of the circuit at different frequencies:
When f >> fc
fc/f << 1, the voltage gain  1
When f = fc
When f << fc

13. The behaviour in these three regions can be illustrated using phasor diagrams
At low frequencies the gain is linearly related to frequency.
It falls at -6dB/octave (- 20dB/decade)

14. Frequency response of the high-pass network
the gain response has two asymptotes that meet at the cut-off frequency
figures of this form are called Bode diagrams

15. High-Pass Filter Response
A high-pass filter is a filter that significantly attenuates or rejects all frequencies below fc and passes all frequencies above fc.
The passband of a high-pass filter is all frequencies above the critical frequency. Vo
Actual response
Ideal response
Ideally, the response rises abruptly at the critical frequency, fL

16. The critical frequency of a high-pass RC filter occurs when
XC = R and can be calculated using the formula below:

17. A Low-Pass RC Network Transposing the C and R gives
At high frequencies
 is large, voltage gain  0
At low frequencies
 is small, voltage gain  1

18. A Low-Pass RC Network A similar analysis to before gives
Therefore when, when CR = 1
Which is the cut-off frequency

19. Therefore the angular frequency C at which this occurs is given by
where  is the time constant of the CR network, and as before

20. Substituting  =2f and CR = 1/ 2fC in the earlier equation gives
This is similar, but not the same, as the transfer function for the high-pass network

21. Consider the behaviour of this circuit at different frequencies:
When f << fc
f/fc << 1, the voltage gain  1
When f = fc
When f >> fc

22. The behaviour in these three regions can again be illustrated using phasor diagrams
At high frequencies the gain is linearly related to frequency. It falls at 6dB/octave (20dB/decade)

23. Frequency response of the low-pass network
the gain response has two asymptotes that meet at the cut-off frequency
you might like to compare this with the Bode Diagram for a high-pass network

24. BASIC FILTER RESPONSES
Low-Pass Filter Response
A low-pass filter is a filter that passes frequencies from 0Hz to critical frequency, fc and significantly attenuates all other frequencies.
roll-off rate Vo
Actual response
Ideal response
Ideally, the response drops abruptly at the critical frequency, fH

25. Transition region shows the area where the fall-off occurs.
Passband of a filter is the range of frequencies that are allowed to pass through the filter with minimum attenuation (usually defined as less than -3 dB of attenuation).
Transition region shows the area where the fall-off occurs.
roll-off rate
Stopband is the range of frequencies that have the most attenuation.
Critical frequency, fc, (also called the cutoff frequency) defines the end of the passband and normally specified at the point where the response drops – 3 dB (70.7%) from the passband response.

26. Vo
At low frequencies, XC is very high and the capacitor circuit can be considered as open circuit. Under this condition, Vo = Vin or AV = 1 (unity).
At very high frequencies, XC is very low and the Vo is small as compared with Vin. Hence the gain falls and drops off gradually as the frequency is increased.

27. The bandwidth of an ideal low-pass filter is equal to fc:
The critical frequency of a low-pass RC filter occurs when
XC = R and can be calculated using the formula below:

28. A Low-Pass RL Network
Low-pass networks can also be produced using RL circuits
these behave similarly to the corresponding CR circuit
the voltage gain is
the cut-off frequency is

29. A High-Pass RL Network
High-pass networks can also be produced using RL circuits
these behave similarly to the corresponding CR circuit
the voltage gain is
the cut-off frequency is

30. A Comparison of RC and RL Networks
Circuits using RC and RL techniques have similar characteristics

31. Bode Diagrams Straight-line approximations

32. Creating more detailed Bode diagrams

33. Combining the Effects of Several Stages

34. Multiple high- and low-pass elements may also be combined

35. RLC Circuits and Resonance
Series RLC circuits
the impedance is given by
if the magnitude of the reactance of the inductor and capacitor are equal, the imaginary part is zero, and the impedance is simply R
this occurs when

36. Resonant frequency This situation is referred to as resonance
the frequency at which is occurs is the resonant frequency
in the series resonant circuit, the impedance is at a minimum at resonance
the current is at a maximum at resonance

37. The quality factor, Q
The resonant effect can be quantified by the quality factor, Q
quality factor, Q: is the ratio of the energy dissipated to the energy stored in each cycle
it can be shown that
and

38. RLC circuit is an acceptor circuit
The series RLC circuit is an acceptor circuit
the narrowness of bandwidth is determined by the Q
combining this equation with the earlier one gives

39. SUMMARY
The bandwidth of a low-pass filter is the same as the upper critical frequency.
The bandwidth of a high-pass filter extends from the lower critical frequency up to the inherent limits of the circuit.
The band-pass passes frequencies between the lower critical frequency and the upper critical frequency.
A band-stop filter rejects frequencies within the upper critical frequency and upper critical frequency.

40. Passive Analog Filters
Background:
Four types of filters - “Ideal”
lowpass
highpass
bandpass
bandstop

41. Passive Analog Filters
Background:
Realistic Filters:
lowpass
highpass
bandpass
bandstop