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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.1 Transient Behaviour Introduction Charging Capacitors and Energising Inductors Discharging Capacitors and De-energising Inductors Response of First-Order Systems Second-Order Systems Higher-Order Systems Chapter 18

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.2 Introduction So far we have looked at the behaviour of systems in response to: –fixed DC signals –constant AC signals We now turn our attention to the operation of circuits before they reach steady-state conditions –this is referred to as the transient response We will begin by looking at simple RC and RL circuits 18.1

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.3 Charging Capacitors and Energising Inductors Capacitor Charging Consider the circuit shown here –Applying Kirchhoff’s voltage law –Now, in a capacitor –which substituting gives 18.2

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.4 The above is a first-order differential equation with constant coefficients Assuming V C = 0 at t = 0, this can be solved to give –see Section 18.2.1 of the course text for this analysis Since i = Cdv/dt this gives (assuming V C = 0 at t = 0) –where I = V/R

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.5 Thus both the voltage and current have an exponential form

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.6 Inductor energising A similar analysis of this circuit gives where I = V/R – see Section 18.2.2 for this analysis

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.7 Thus, again, both the voltage and current have an exponential form

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.8 Discharging Capacitors and De-energising Inductors Capacitor discharging Consider this circuit for discharging a capacitor –At t = 0, V C = V –From Kirchhoff’s voltage law –giving 18.3

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.9 Solving this as before gives –where I = V/R – see Section 18.3.1 for this analysis

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.10 In this case, both the voltage and the current take the form of decaying exponentials

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.11 Inductor de-energising A similar analysis of this circuit gives –where I = V/R – see Section 18.3.1 for this analysis

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.12 And once again, both the voltage and the current take the form of decaying exponentials

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.13 A comparison of the four circuits

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.14 Response of First-Order Systems Initial and final value formulae –increasing or decreasing exponential waveforms (for either voltage or current) are given by: –where V i and I i are the initial values of the voltage and current –where V f and I f are the final values of the voltage and current –the first term in each case is the steady-state response –the second term represents the transient response –the combination gives the total response of the arrangement 18.4

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.15 Example – see Example 18.3 from course text The input voltage to the following CR network undergoes a step change from 5 V to 10 V at time t = 0. Derive an expression for the resulting output voltage.

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.16 Here the initial value is 5 V and the final value is 10 V. The time constant of the circuit equals CR = 10 10 3 20 10 -6 = 0.2s. Therefore, from above, for t 0

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.17 The nature of exponential curves

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.18 Response of first-order systems to a square waveform –see Section 18.4.3

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.19 Response of first-order systems to a square waveform of different frequencies –see Section 18.4.3

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.20 Second-Order Systems Circuits containing both capacitance and inductance are normally described by second-order differential equations. These are termed second-order systems –for example, this circuit is described by the equation 18.5

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.21 When a step input is applied to a second-order system, the form of the resultant transient depends on the relative magnitudes of the coefficients of its differential equation. The general form of the response is –where n is the undamped natural frequency in rad/s and (Greek Zeta) is the damping factor

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.22 Response of second-order systems =0 undamped <1 under damped =1 critically damped >1 over damped

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.23 Higher-Order Systems Higher-order systems are those that are described by third-order or higher-order equations These often have a transient response similar to that of the second-order systems described earlier Because of the complexity of the mathematics involved, they will not be discussed further here 18.6

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Storey: Electrical & Electronic Systems © Pearson Education Limited 2004 OHT 18.24 Key Points The charging or discharging of a capacitor, and the energising and de-energising of an inductor, are each associated with exponential voltage and current waveforms Circuits that contain resistance, and either capacitance or inductance, are termed first-order systems The increasing or decreasing exponential waveforms of first-order systems can be described by the initial and final value formulae Circuits that contain both capacitance and inductance are usually second-order systems. These are characterised by their undamped natural frequency and their damping factor

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