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2. Analogue Theory and Circuit Analysis 2.1 Steady-State (DC) Circuits 2.2 Time-Dependent Circuits DeSiaMorePowered by DeSiaMore1
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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
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Electrical Engineering Subdivisions Communication systems Computer systems Control systems Electromagnetics Electronics Power systems Signal processing DeSiaMorePowered by DeSiaMore3
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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
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Electrical Current DeSiaMorePowered by DeSiaMore5
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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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. DeSiaMorePowered by DeSiaMore7
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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
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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
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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
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CAPACITANCE DeSiaMorePowered by DeSiaMore11
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CAPACITANCE DeSiaMorePowered by DeSiaMore12
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INDUCTANCE DeSiaMorePowered by DeSiaMore13
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INDUCTANCE DeSiaMorePowered by DeSiaMore14
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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
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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
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EXAMPLE DeSiaMorePowered by DeSiaMore17
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EXAMPLE DeSiaMorePowered by DeSiaMore18
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EXAMPLE DeSiaMorePowered by DeSiaMore19
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EXAMPLE DeSiaMorePowered by DeSiaMore20
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1 ST ORDER SWITCHED DC CIRCUITS DeSiaMorePowered by DeSiaMore21
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ACTIVITY 13-1 DeSiaMorePowered by DeSiaMore22
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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
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ACTIVITY 13-1 Circuit File v 1 0 dc 100 R 1 2 {R} C 2 0 20n ic=0.param R=250k.step param R list 250k 500k 1meg.tran.1.1 uic.probe.end DeSiaMorePowered by DeSiaMore24
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ACTIVITY 13-1 DeSiaMorePowered by DeSiaMore25
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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
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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
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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
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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
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Thus both the voltage and current have an exponential form DeSiaMorePowered by DeSiaMore30
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Inductor energising A similar analysis of this circuit gives where I = V/R – DeSiaMorePowered by DeSiaMore31
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Thus, again, both the voltage and current have an exponential form DeSiaMorePowered by DeSiaMore32
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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
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Solving this as before gives – where I = V/R – DeSiaMorePowered by DeSiaMore34
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In this case, both the voltage and the current take the form of decaying exponentials DeSiaMorePowered by DeSiaMore35
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Inductor de-energising A similar analysis of this circuit gives – where I = V/R – see Section 18.3.1 for this analysis DeSiaMorePowered by DeSiaMore36
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And once again, both the voltage and the current take the form of decaying exponentials DeSiaMorePowered by DeSiaMore37
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A comparison of the four circuits DeSiaMorePowered by DeSiaMore38
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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
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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
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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 DeSiaMorePowered by DeSiaMore41
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The nature of exponential curves DeSiaMorePowered by DeSiaMore42
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Response of first-order systems to a square waveform DeSiaMorePowered by DeSiaMore43
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Response of first-order systems to a square waveform of different frequencies DeSiaMorePowered by DeSiaMore44
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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
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