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Frequency Characteristics of AC Circuits Chapter 17 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Introduction 17.1 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 X L L 1 XC C Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› We will start by considering very simple circuits Consider the potential divider shown here – from our earlier consideration of the circuit Z2 vo vi Z1 Z 2 – rearranging, the gain of the circuit is vo Z2 vi Z1 Z 2 – this is also called the transfer function of the circuit Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A High-Pass RC Network 17.2 Consider the following circuit – which is shown re-drawn in a more usual form Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Clearly the transfer function is vo ZR R 1 v i Z R ZC R j 1 1 j 1 C CR At high frequencies – is large, voltage gain 1 At low frequencies – is small, voltage gain 0 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Since the denominator has real and imaginary parts, the magnitude of the voltage gain is Voltage gain 1 1 1 CR 2 When 1/CR = 1 Voltage gain 2 1 1 0.707 1 1 2 This is a halving of power, or a fall in gain of 3 dB Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› The half power point is the cut-off frequency of the circuit – the angular frequency C at which this occurs is given by 1 1 cCR 1 1 c rad/s CR – where is the time constant of the CR network. Also fc c 1 Hz 2 2CR Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Substituting =2f and CR = 1/ 2fC in the earlier equation gives vo 1 1 v i 1 j 1 j CR 1 1 1 (2f ) 2fc 1 1 j fc f 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Consider the behaviour of the circuit at different frequencies: When f >> fc – fc/f << 1, the voltage gain 1 When f = fc vo 1 1 1 (1 j) (1 j) 0 .5 0 . 5 j f v i 1 j c 1 j 1 j (1 j) 2 f When f << fc vo 1 1 f j v i 1 j fc j fc fc f f Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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) Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A Low-Pass RC Network 17.3 Transposing the C and R gives 1 vo ZC 1 C v i Z R ZC R j 1 1 jCR C j At high frequencies – is large, voltage gain 0 At low frequencies – is small, voltage gain 1 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A Low-Pass RC Network 17.3 A similar analysis to before gives Voltage gain 1 1 CR 2 Therefore when, when CR = 1 Voltage gain 1 1 0.707 1 1 2 Which is the cut-off frequency Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Therefore – the angular frequency C at which this occurs is given by cCR 1 1 1 c rad/s CR – where is the time constant of the CR network, and as before 1 fc c Hz 2 2CR Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Substituting =2f and CR = 1/ 2fC in the earlier equation gives vo 1 v i 1 jCR 1 1 j c 1 1 j f fc This is similar, but not the same, as the transfer function for the high-pass network Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Consider the behaviour of this circuit at different frequencies: When f << fc – f/fc << 1, the voltage gain 1 When f = fc vo 1 j1 j 1 j 0.5 0.5 j 1 1 j v i 1 j f 2 fc When f >> fc vo f 1 1 j c f v i 1 j f f j fc fc Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 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) Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A Low-Pass RL Network 17.4 Low-pass networks can also be produced using RL circuits – these behave similarly to the corresponding CR circuit – the voltage gain is vo ZR R 1 v i Z R Z L R jL 1 j L R – the cut-off frequency is c R 1 rad/s L fc c R Hz 2 2L Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A High-Pass RL Network 17.5 High-pass networks can also be produced using RL circuits – these behave similarly to the corresponding CR circuit – the voltage gain is vo ZL jL 1 1 v i Z R Z L R jL 1 R 1 j R jL L – the cut-off frequency is c R 1 rad/s L fc c R Hz 2 2L Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› A Comparison of RC and RL Networks 17.6 Circuits using RC and RL techniques have similar characteristics – for a more detailed comparison, see Figure 17.10 in the course text Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Bode Diagrams 17.7 Straight-line approximations Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Creating more detailed Bode diagrams Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Combining the Effects of Several Stages 17.8 The effects of several stages ‘add’ in bode diagrams Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Multiple high- and low-pass elements may also be combined – this is illustrated in Figure 17.14 in the course text Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› RLC Circuits and Resonance 17.9 Series RLC circuits – the impedance is given by Z R jL 1 1 R j(L ) jC C – 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 1 L C 1 LC 2 1 LC Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› This situation is referred to as resonance – the frequency at which is occurs is the resonant frequency o 1 LC fo 1 2 LC – in the series resonant circuit, the impedance is at a minimum at resonance – the current is at a maximum at resonance Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Quality factor Q – and Q X L XC R R 1 L R C Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› The series RLC circuit is an acceptor circuit – the narrowness of bandwidth is determined by the Q Quality factor Q Resonant frequency fo Bandwidth B – combining this equation with the earlier one gives B R Hz 2L Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Parallel RLC circuits – as before o 1 LC fo 1 2 LC Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 C QR L B 1 Hz 2RC Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Filters 17.10 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’ Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› LC Filters Simple LC filters can be produced using series or parallel tuned circuits – these produce narrowband filters with a centre frequency fo fo 1 2 LC Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Active filters – combining an op-amp with suitable resistors and capacitors can produce a range of filter characteristics – these are termed active filters Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› Stray Capacitance and Inductance 17.11 All circuits have stray capacitance and stray inductance – these unintended elements can dramatically affect circuit operation – for example: (a) Cs adds an unintended low-pass filter (b) Ls adds an unintended low-pass filter (c) Cs produces an unintended resonant circuit and can produce instability Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#› 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17.‹#›