Frequency Characteristics of AC Circuits Chapter 17 Introduction

Frequency Characteristics of AC Circuits Chapter 17 § Introduction § A High-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. 1

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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 2

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 3

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. 4

§ Clearly the transfer function is § 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. 5

§ 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 d. B Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 6

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 7

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 8

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 9

§ 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 -6 d. B/octave (-20 d. B/decade) Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 10

§ 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. 11

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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 12

A Low-Pass RC Network 17. 3 § A similar analysis to before gives § Therefore when, when CR = 1 § Which is the cut-off frequency Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 13

§ Therefore – the angular frequency C at which this occurs is given by – where is the time constant of the CR network, and as before Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 14

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 15

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 16

§ 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 6 d. B/octave (20 d. B/decade) Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 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. 18

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 – the cut-off frequency is Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 19

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 – the cut-off frequency is Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 20

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. 21

Bode Diagrams 17. 7 § Straight-line approximations Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 22

§ Creating more detailed Bode diagrams Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 23

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. 24

§ 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. 25

RLC Circuits and Resonance 17. 9 § 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 26

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 27

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 28

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 29

§ Parallel RLC circuits – as before Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 30

§ 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 Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 31

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 d. B/octave – they also produce a maximum of 90 phase shift § Combining multiple stages can produce filters with a greater ultimate roll-off rates (12 d. B, 18 d. B, etc. ) but such filters have a very soft ‘knee’ Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 32

§ 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. 33

§ LC Filters § Simple LC filters can be produced using series or parallel tuned circuits – these produce narrowband filters with a centre frequency fo Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 34

§ 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. 35

§ 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. 36

Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 17. 37

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. 38

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. 39