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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.1 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 Filters Stray Capacitance and Inductance Chapter 17

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.2 Introduction 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 17.1

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.3 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.4 A High-Pass RC Network Consider the following circuit –which is shown re-drawn in a more usual form 17.2

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.5 Clearly the transfer function is At high frequencies – is large, voltage gain 1 At low frequencies – is small, voltage gain 0

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.6 Since the denominator has real and imaginary parts, the magnitude of the voltage gain is When 1/ CR = 1 This is a halving of power, or a fall in gain of 3 dB

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.7 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.8 Substituting =2 f and CR = 1/ 2 f C 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.9 Consider the behaviour of the circuit at different frequencies: When f >> f c –f c /f << 1, the voltage gain 1 When f = f c When f << f c

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.10 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)

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.11 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.12 A Low-Pass RC Network 17.3 Transposing the C and R gives At high frequencies – is large, voltage gain 0 At low frequencies – is small, voltage gain 1

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.13 A Low-Pass RC Network 17.3 A similar analysis to before gives Therefore when, when CR = 1 Which is the cut-off frequency

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.14 Therefore –the angular frequency C at which this occurs is given by –where is the time constant of the CR network, and as before

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.15 Substituting =2 f and CR = 1/ 2 f C in the earlier equation gives This is similar, but not the same, as the transfer function for the high-pass network

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.16 Consider the behaviour of this circuit at different frequencies: When f << f c –f/f c << 1, the voltage gain 1 When f = f c When f >> f c

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.17 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)

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.18 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.19 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 17.4

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.20 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 17.5

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.21 A Comparison of RC and RL Networks Circuits using RC and RL techniques have similar characteristics –for a more detailed comparison, see Figure 17.10 in the course text 17.6

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.22 Bode Diagrams Straight-line approximations 17.7

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.23 Creating more detailed Bode diagrams

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.24 Combining the Effects of Several Stages The effects of several stages ‘add’ in bode diagrams 17.8

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.25 Multiple high- and low-pass elements may also be combined –this is illustrated in Figure 17.14 in the course text

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.26 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 17.9

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.27 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.28 The resonant effect can be quantified by the quality factor, Q –this is the ratio of the energy dissipated to the energy stored in each cycle –it can be shown that –and

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.29 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

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.30 Parallel RLC circuits –as before

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.31 The parallel arrangement is a rejector circuit –in the parallel resonant circuit, the impedance is at a maximum at resonance –the current is at a minimum at resonance –in this circuit

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.32 Filters RC Filters The RC networks considered earlier are first-order or single-pole filters –these have a maximum roll-off of 6 dB/octave –they also produce a maximum of 90 phase shift Combining multiple stages can produce filters with a greater ultimate roll-off rates (12 dB, 18 dB, etc.) but such filters have a very soft ‘knee’ 17.10

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.33 An ideal filter would have constant gain and zero phase shift for frequencies within its pass band, and zero gain for frequencies outside this range (its stop band) Real filters do not have these idealised characteristics

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.34 LC Filters Simple LC filters can be produced using series or parallel tuned circuits –these produce narrow- band filters with a centre frequency f o

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.35 Active filters –combining an op-amp with suitable resistors and capacitors can produce a range of filter characteristics –these are termed active filters

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.36 Common forms include: Butterworth –optimised for a flat response Chebyshev –optimised for a sharp ‘knee’ Bessel –optimised for its phase response see Section 17.10.3 of the course text for more information on these

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.37

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.38 Stray Capacitance and Inductance All circuits have stray capacitance and stray inductance –these unintended elements can dramatically affect circuit operation –for example: (a) C s adds an unintended low-pass filter (b) L s adds an unintended low-pass filter (c) C s produces an unintended resonant circuit and can produce instability 17.11

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.39 Key Points The reactance of capacitors and inductors is dependent on frequency Single RC or RL networks can produce an arrangement with a single upper or lower cut-off frequency. In each case the angular cut-off frequency o is given by the reciprocal of the time constant For an RC circuit = CR, for an RL circuit = L/R Resonance occurs when the reactance of the capacitive element cancels that of the inductive element Simple RC or RL networks represent single-pole filters Active filters produce high performance without inductors Stray capacitance and inductance are found in all circuits

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