WORLD'S #1 ACADEMIC OUTLINE INTRODUCTION TO AC AND DC CIRCUIT ANALYSIS Ideal Independent Sources I Its magnitude is measured in coulombs (C) The charge carried by an electron is q.=-1.60xI o- 19 C A prolon has the same amount of charge, but with positive polarity Charges are, therefore, of two types: positive and negative Charges exist in discrete, integral multiples of ±q• Charges are conserved: They can be neither created nor destroyed CURRENT In an electric circuit, the charges move along specified, closed paths The motion of charges constitutes an electric conduction current Current is measured in amperes (coulomb/sec) That is, one ampere is the current that flows when one coulomb of charge flows per second Or, current i(t) in amperes is: () dq (I) amperes, were h t = at q() I IS t he charge In I coulombs, and I is the time in seconds - Conventions: A negative quantity of charge traveling from B to A along a conductor is equivalent to a positive quantity of charge going from A to B Hence, current is a vector - Unidirectional transfer of electric charge through a conductor constitutes the direct electric current, or dc This is in contrast to ac (alternating current) which implles a time-reversal of direction of current such as a sinusoidal wave ·VOLTAGE Voltage difference (also known as potential difference) between two points is defined as the work [wet) in joules] required to move a unit charge from one point to the other The unit of potential difference is the volt (V) Voltage (I') can be positive or negative dw (t) Note that: v (t) = ~ volt • POWER Power pet) is the rate of doing work or the rate of change of energy The electrical unit of power is the watt (W) dw (t) p (t) = ~ joule/sec or watts Note: V(t)l(t) = dw (t) dq ~ dt = dw ( t) ~ =p(t) Thus, pet) = vet) i(t) is subject to the following sign convention: Whellever th e reference direction fi)r the current ill all elelllent is ill the direction ofthe voltage drop across the element use the forllluia p=vi That is, the current must enter the positive terminal to use the above formula Otherwise use p=- vi If p(t»O , then the element absorbs power If p(t)O, Fig 43 Need to find the Thevenin equivalent across the coil Fig 44 = 20A to IL(t) = ILco+) + = Fig 43 R' Ic( t) = A.e'.' + Aze"' Resonance Goal: Need v(t) for t>O From v(t) it is possible to find: C EEJ'" 10(0 IdO+) = 'L(O, The solution is vet) Finding the natural response of a parallel RLC circuit consists of finding Fig 50 the voltage created across the parallel branches by the release of energy R L V that has been stored in the inductor or capacitor, or both The task is defined in terms of the following circuit t~O as in Fig 50 Assume the circuit has stored energy in the form IL(O-) =IL(O+) and vcCO-) = vcCO+) IRCt) = s =sz' and Sz are complex conjugates of each other Natural Response of Parallel RLC Circuits Write down the answer [Note that the final value (steady state solution) is [s" Sz real and equal] (Jc) < L~ t(sec) -240 sl = Sz Solution changes to vet) =A.e'" + Azte'.' Underdamped VL(t) Sketch ofldt) 20 (2jC) = 2c therefore - ! and 2RC S2= 2;~ - (~C) - :C If these two solutions are denoted as R VI respectively, then their sum is also a solution v = VI + V z = Ale'" + Aze s" is a solution and V z fJ Fig 52 Fig 51 LC l R + ­ L - V jwL Resonance - cont fiv lII page ~ • QUESTION: Find (a) the bandwidth BW, (b) resonant frequency 000 , and (c)Q ll(w)=~ (a)For BW, need For this example, I Fig 57, shows a more practical model of the coil because it accounts for the losses in the inductor, represented by the resistor R Recall Q(coil) = Qp = 230 Q = ~ From the sketch in Fig 53, 00 1=0, and 00 2= Y Thus, W('Bd) R = reduces to the values shown in Fig 59 B6C R T BW=~-~=r' (b) There is no finite non-zero resonant frequency since Yin(W) (c Q (w) ) = R + ~WL will be real only when 00=0 ( -t­ .tIIIII Thus, Q = 2n: 11 12 Rl.,T = "I11III Z Substitute T = : => Q (w) 2n:L RT = ~L • Consider the following two circuits: D Z(s) = Rs + jXs = ~ Y(p) FACT: Q,= I!,I ' y(t)=AIH(jW)lcos[oot + fJ+ 8(00») (for steady state) Fig 54 Definition: The amplitude of ll(jw) in decibels is l = Gp + =-x ) FACT: Qp = Adb= 20Iog lO l~pl p Objective: Find output and amplitude of H(jw) in decibels _ w = => H( 2)) Qo(system) = 50 X 10 = so = 1.2 x = ic Z k it iour(t) = Jz ~ looks like Fig 62 I)m looks like Fig 63 (1+ )WB fQ slope ~ -20 • m dB/decade Fig 63 Also, Bode plot of constant Ao is slope =\ -20 • m dB/decade 20logAo as shown in Fig 64 dB Note: Application of this Bode approximation - for a more approximation plot ~ of ll(jw) ~a.mple: (jw + 10 P H()W) = (jw+W(jW+lOO) ( I+'W/ ) J /WB' 'w/jJ [ 1000 1+) /10 For ll(jw) cos (2t - 45 ') = +2jW' 1H(jw) = that C;:I.4x 10- loF + 2010g ll+ j o/tool-1 db = 20log 1H(jw) =2010g~ 4+ = ~91r~jlJll~~II~I!rJI~JIJ211f1ll l! Ilil~I = 20logl0 + 60log II ill 2)-112 ) - 2010gl' = l Ao I+(~B) 2] + j%)()1 Now each function is in standard form Sketch each term and add as shown in Fig 65 Begin with 20dB straighl line -40dBldec slope takes us to -20dB at 00=10 The 60dBIdec slope adds to the -40dB/dec slope giving 20dBIdec slope This 20dBIdec slope takes us from -20dB at 00=10 to OdB at 00= 100 At 00= 100 the -20dB/dec slope adds to the 20dBIdec slope keeping US at OdB 1/2 Fig 65 dB Note 10 Student : Dut! to its coodem;6(l format I.!Soe thiS Guolck,turdy· guide as an outline ol!ha basics 01 Clrcuil Analysis but not as a replacement tor asSigned cours& work All ril!hts fl:SC'f\"cd N( pan of 1ius publication may be reproduced or lransmi ned in an~ foon (Ir hy any ~ans ele ... of the resistors - A short circuit in parallel with n resistors is equivalent to a short circuit - An open circuit in series with n resistors is equivalent to an open circuit - Voltage sources... apply voltage and current division , loop and nodal analysis, and Thevenin and Norton equivalent circuits to the design and analysis of linear circuits in the sinusoidal steady state IZII=IZTIII... Thevenin's Theorem o FACT: If a circuit is linear, then its v versus i curve must be a straight line Any linear circuit (Fig 25a) can be represented by its Thevenin equivalent circuit in the following