# Frequency Characteristics of AC Circuits

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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

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

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.

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.

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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

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

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)

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

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

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.

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.

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:

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

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

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

Bode Diagrams Straight-line approximations

Creating more detailed Bode diagrams

Combining the Effects of Several Stages

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

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

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

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

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

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.

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

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