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Transcript
2. Analogue Theory and Circuit Analysis 2.1 Steady-State (DC) Circuits 2.2 Time-Dependent Circuits DeSiaMore Powered by DeSiaMore 1 Electrical systems have two main objectives: • To gather, store, process, transport, and present information • To distribute and convert energy between various forms DeSiaMore Powered by DeSiaMore 2 Electrical Engineering Subdivisions • • • • DeSiaMore Communication systems Computer systems Control systems Electromagnetics • Electronics • Power systems • Signal processing Powered by DeSiaMore 3 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). DeSiaMore Powered by DeSiaMore 4 Electrical Current dq(t ) i (t ) dt t q(t ) i (t )dt q(t0 ) t0 DeSiaMore Powered by DeSiaMore 5 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. DeSiaMore Powered by DeSiaMore 6 . DeSiaMore Powered by DeSiaMore 7 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). DeSiaMore Powered by DeSiaMore 8 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. DeSiaMore Powered by DeSiaMore 9 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. DeSiaMore Powered by DeSiaMore 10 CAPACITANCE ic dvc ic C dt vc C[Farads] DeSiaMore d I n D C S te a d y S ta te ; 0 d t iC 0 O p e n C ir c u it S S Powered by DeSiaMore 11 CAPACITANCE q Cv t qt i t dt qt0 t0 dv iC dt DeSiaMore t 1 v t i t dt v t0 C t0 Powered by DeSiaMore 12 INDUCTANCE iL diL vL L dt v DeSiaMore L[Henries] L d I n D C S te a d y S ta te ; 0 d t v 0 S h o r tC ir c u it L S S Powered by DeSiaMore 13 INDUCTANCE di v t L dt t 1 i t v t dt i t0 L t0 1 2 wt Li t 2 DeSiaMore Powered by DeSiaMore 14 SWITCHED CIRCUITS • • • • • Circuits that Contain Switches Switches Open or Close at t = t0 to = Switching Time Often choose to = 0 Want to Find i’s and v’s in Circuit Before and After Switching Occurs • i(to-), v(t0-); i(to+), v(t0+) • Initial Conditions of Circuit DeSiaMore Powered by DeSiaMore 15 INITIAL CONDITIONS • • • • • • • C’s and L’s Store Electrical Energy vC Cannot Change Instantaneously iL Cannot Change Instantaneously In DC Steady State; C => Open Circuit In DC Steady State; L => Short Circuit Use to Find i(to-), v(t0-); i(to+), v(t0+) Let’s do an Example DeSiaMore Powered by DeSiaMore 16 EXAMPLE i1 S w itc h O p e n sa tt 0 v1 12 V 2 i3 v3 i2 2 4 1 F v2 iC vC A s s u m e S w i t c h h a s b e e n C l o s e dF i n d I n i t i a lC o n d i t i o n s + f o r a l o n g t i m e b e f o r e t 0 i ' s a n d v ' s a tt 0 a n d t 0 DeSiaMore Powered by DeSiaMore 17 EXAMPLE At t 0 : i3 C S te a d yS ta te i1 D v1 Sw itchC losed v3 2 iC i2 2 12 V 4 v C C v2 Open Ckt 1 2 i (0 ) 0 i (0 ) 0 3 C i ( 0 ) i ( 0 ) 3 A 1 2 2 2 v (0 ) 0 3 v ( 0 ) v ( 0 ) 3 x 2 6 V 1 2 v ( 0 )v ( 0 )v ( 0 V C 2 3 )6 DeSiaMore Powered by DeSiaMore 18 EXAMPLE At t 0 : i1 i( 0) 0 v ( 0) 1 1 v1 Sw itchO pen 2 i2 2 12 V i3 i C v3 4 v 2 1 F vC 6 v ( 0 ) v ( 0 6 V i ( 0 ) i ( 0 ) i ( 0 ) 1 A C C ) 4 2 v ( 0 ) 2 x 1 2 V v ( 0 4 x (1 ) 4 V 2 3 ) 2 DeSiaMore 3 C Powered by DeSiaMore 19 EXAMPLE In itia lC o n d itio n s t 0 i1 3 A t 0 i1 0 A i2 3 A i2 1 A i3 0 A i3 1 A iC 0 A iC 1 A v1 6 V v1 0 V 6 V v2 2 V v 2 v3 0 V v DeSiaMore C 6 V v3 4 V v C 6 V Powered by DeSiaMore 20 1ST ORDER SWITCHED DC CIRCUITS W illL ookat1 O rderC ircuits(C ircuitsw ith 1Cor1L )w ithSw itchedD CInputsT om orrow W illU seInitialC onditionstoH elpU sSolvethe st st 1O rderD ifferentialE quationR elatingthe O utputtotheInput T odayW eW illL ookata1 O rderC ircuitusing PSpice st DeSiaMore Powered by DeSiaMore 21 ACTIVITY 13-1 R 100 V DeSiaMore 20 nF Powered by DeSiaMore vC 22 ACTIVITY 13-1 • Charge a 20 nF Capacitor to 100 V thru a Variable Resistor, Rvar: • Let’s Use a Switch that Closes at t = 0 • Rvar = 250k, 500k, 1 M • Circuit File Has Been Run: • C:/Files/Desktop/CE-Studio/Circuits/act_52.dat • But Let’s Practice Using Schematics and Take a Quick Look DeSiaMore Powered by DeSiaMore 23 ACTIVITY 13-1 DeSiaMore 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 Powered by DeSiaMore 24 ACTIVITY 13-1 P r in tG r a p h so fv s .tim e Cv F illin T a b le f o rA c tiv ity 1 3 1 H a n d I n f o rG r a d in g DeSiaMore Powered by DeSiaMore 25 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 DeSiaMore Powered by DeSiaMore 26 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 DeSiaMore Powered by DeSiaMore 27 Charging Capacitors and Energising Inductors • Capacitor Charging Consider the circuit shown here – Applying Kirchhoff’s voltage law iR v V – Now, in a capacitor dv i C dt – which substituting gives d v CR v V d t DeSiaMore Powered by DeSiaMore 28 The above is a first-order differential equation with constant coefficients Assuming VC = 0 at t = 0, this can be solved to give t t CR v V ( 1 e ) V ( 1 e ) Since i = Cdv/dt this gives (assuming VC = 0 at t = 0) – where I = V/R DeSiaMore t t CR i I e I e Powered by DeSiaMore 29 Thus both the voltage and current have an exponential form DeSiaMore Powered by DeSiaMore 30 • Inductor energising A similar analysis of this circuit gives Rt t L v V e V e Rt t L i I ( 1 e ) I ( 1 e ) where I = V/R – DeSiaMore Powered by DeSiaMore 31 Thus, again, both the voltage and current have an exponential form DeSiaMore Powered by DeSiaMore 32 Discharging Capacitors and De-energising Inductors • Capacitor discharging Consider this circuit for discharging a capacitor – At t = 0, VC = V – From Kirchhoff’s voltage law iR v 0 – giving DeSiaMore d v CR v 0 d t Powered by DeSiaMore 33 Solving this as before gives t t CR v V e V e t t CR i I e I e – where I = V/R – DeSiaMore Powered by DeSiaMore 34 In this case, both the voltage and the current take the form of decaying exponentials DeSiaMore Powered by DeSiaMore 35 • Inductor de-energising A similar analysis of this circuit gives Rt t L v V e V e Rt t i IeL Ie – where I = V/R – see Section 18.3.1 for this analysis DeSiaMore Powered by DeSiaMore 36 And once again, both the voltage and the current take the form of decaying exponentials DeSiaMore Powered by DeSiaMore 37 A comparison of the four circuits DeSiaMore Powered by DeSiaMore 38 Response of First-Order Systems Initial and final value formulae – increasing or decreasing exponential waveforms (for either voltage or current) are given by: – – – – – where Vi and Ii are the initial values of the voltage and current where Vf and If 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 DeSiaMore Powered by DeSiaMore 39 • 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. • DeSiaMore Powered by DeSiaMore 40 • Here the initial value is 5 V and the final value is 10 V. The time constant of the circuit equals CR = 10 103 20 10-6 = 0.2s. Therefore, from above, for t 0 • t/ vV ( V V ) e f i f t/0 .2 10 (510 )e t/0 .2 10 5 e volts DeSiaMore Powered by DeSiaMore 41 • The nature of exponential curves DeSiaMore Powered by DeSiaMore 42 • Response of first-order systems to a square waveform DeSiaMore Powered by DeSiaMore 43 • Response of first-order systems to a square waveform of different frequencies DeSiaMore Powered by DeSiaMore 44 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 firstorder 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 DeSiaMore Powered by DeSiaMore 45