2. Analogue Theory and Circuit Analysis 2.1 Steady-State (DC) Circuits 2.2 Time-Dependent Circuits DeSiaMorePowered by DeSiaMore1

Electrical systems have two main objectives: To gather, store, process, transport, and present information To distribute and convert energy between various forms DeSiaMorePowered by DeSiaMore2

Electrical Engineering Subdivisions Communication systems Computer systems Control systems Electromagnetics Electronics Power systems Signal processing DeSiaMorePowered by DeSiaMore3

Electrical Current Electrical current is the time rate of flow of electrical charge through a conductor or circuit element. The units are amperes (A), which are equivalent to coulombs per second (C/s). DeSiaMorePowered by DeSiaMore4

Electrical Current DeSiaMorePowered by DeSiaMore5

Direct Current Alternating Current When a current is constant with time, we say that we have direct current, abbreviated as dc. On the other hand, a current that varies with time, reversing direction periodically, is called alternating current, abbreviated as ac. DeSiaMorePowered by DeSiaMore6

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Voltages The voltage associated with a circuit element is the energy transferred per unit of charge that flows through the element. The units of voltage are volts (V), which are equivalent to joules per coulomb (J/C). DeSiaMorePowered by DeSiaMore8

Transients The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching, are called transients. By writing circuit equations, we obtain integrodifferential equations. DeSiaMorePowered by DeSiaMore9

DC STEADY STATE The steps in determining the forced response for RLC circuits with dc sources are: 1. Replace capacitances with open circuits. 2. Replace inductances with short circuits. 3. Solve the remaining circuit. DeSiaMorePowered by DeSiaMore10

CAPACITANCE DeSiaMorePowered by DeSiaMore11

CAPACITANCE DeSiaMorePowered by DeSiaMore12

INDUCTANCE DeSiaMorePowered by DeSiaMore13

INDUCTANCE DeSiaMorePowered by DeSiaMore14

SWITCHED CIRCUITS Circuits that Contain Switches Switches Open or Close at t = t 0 t o = Switching Time Often choose t o = 0 Want to Find i’s and v’s in Circuit Before and After Switching Occurs i(t o - ), v(t 0 - ); i(t o + ), v(t 0 + ) Initial Conditions of Circuit DeSiaMorePowered by DeSiaMore15

INITIAL CONDITIONS C’s and L’s Store Electrical Energy v C Cannot Change Instantaneously i L Cannot Change Instantaneously In DC Steady State; C => Open Circuit In DC Steady State; L => Short Circuit Use to Find i(t o - ), v(t 0 - ); i(t o + ), v(t 0 + ) Let’s do an Example DeSiaMorePowered by DeSiaMore16

EXAMPLE DeSiaMorePowered by DeSiaMore17

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EXAMPLE DeSiaMorePowered by DeSiaMore19

EXAMPLE DeSiaMorePowered by DeSiaMore20

1 ST ORDER SWITCHED DC CIRCUITS DeSiaMorePowered by DeSiaMore21

ACTIVITY 13-1 DeSiaMorePowered by DeSiaMore22

ACTIVITY 13-1 Charge a 20 nF Capacitor to 100 V thru a Variable Resistor, R var : Let’s Use a Switch that Closes at t = 0 R var = 250k, 500k, 1 M Circuit File Has Been Run: C:/Files/Desktop/CE-Studio/Circuits/act_5- 2.dat But Let’s Practice Using Schematics and Take a Quick Look DeSiaMorePowered by DeSiaMore23

ACTIVITY 13-1 Circuit File v 1 0 dc 100 R 1 2 {R} C n ic=0.param R=250k.step param R list 250k 500k 1meg.tran.1.1 uic.probe.end DeSiaMorePowered by DeSiaMore24

ACTIVITY 13-1 DeSiaMorePowered by DeSiaMore25

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 DeSiaMorePowered by DeSiaMore26

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 DeSiaMorePowered by DeSiaMore27

Charging Capacitors and Energising Inductors Capacitor Charging  Consider the circuit shown here – Applying Kirchhoff’s voltage law – Now, in a capacitor – which substituting gives DeSiaMorePowered by DeSiaMore28

 The above is a first-order differential equation with constant coefficients  Assuming V C = 0 at t = 0, this can be solved to give  Since i = Cdv/dt this gives (assuming V C = 0 at t = 0) – where I = V/R DeSiaMorePowered by DeSiaMore29

 Thus both the voltage and current have an exponential form DeSiaMorePowered by DeSiaMore30

Inductor energising  A similar analysis of this circuit gives where I = V/R – DeSiaMorePowered by DeSiaMore31

 Thus, again, both the voltage and current have an exponential form DeSiaMorePowered by DeSiaMore32

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 DeSiaMorePowered by DeSiaMore33

 Solving this as before gives – where I = V/R – DeSiaMorePowered by DeSiaMore34

 In this case, both the voltage and the current take the form of decaying exponentials DeSiaMorePowered by DeSiaMore35

Inductor de-energising  A similar analysis of this circuit gives – where I = V/R – see Section for this analysis DeSiaMorePowered by DeSiaMore36

 And once again, both the voltage and the current take the form of decaying exponentials DeSiaMorePowered by DeSiaMore37

 A comparison of the four circuits DeSiaMorePowered by DeSiaMore38

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 DeSiaMorePowered by DeSiaMore39

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. DeSiaMorePowered by DeSiaMore40

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  = 0.2s. Therefore, from above, for t  0 DeSiaMorePowered by DeSiaMore41

The nature of exponential curves DeSiaMorePowered by DeSiaMore42

Response of first-order systems to a square waveform DeSiaMorePowered by DeSiaMore43

Response of first-order systems to a square waveform of different frequencies DeSiaMorePowered by DeSiaMore44

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 DeSiaMorePowered by DeSiaMore45