THE MATHEMATICS OF CIRCUIT ANALYSIS

By RICHARD PEREZ

CONTENTS ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS INTRODUCTION VOLTAGE AND CURRENT VOLTAGE SOURCES CURRENT SOURCES ELECTRICAL PASSIVE ELEMENTS RESISTANCE AND CONDUCTANCE CAPACITANCE INDUCTANCE KIRCHHOFF’S LAWS SERIES AND PARALLEL COMBINATIONS OF PASSIVE ELEMENTS OTHER TYPES OF CIRCUIT ELEMENT CONNECTIONS ELECTRICAL NETWORK ANALYSIS – NODAL ANALYSIS RESISTIVE VOLTAGE AND CURRENT DIVIDERS CONTROLLED SOURCES TH´EVENIN’S THEOREM NORTON’S THEOREM SUPERPOSITION THEOREM BIBLIOGRAPHY PROBLEMS COMPLEX NUMBERS: AN INTRODUCTION DE?NITION ELEMENTARY ALGEBRA ADDITION SUBTRACTION MULTIPLICATION DIVISION COMPLEX EQUATIONS QUADRATIC EQUATIONS POLAR REPRESENTATION MULTIPLICATION AND DIVISION THE EXPONENTIAL FORM TRIGONOMETRIC FUNCTIONS AND THE EXPONENTIAL FORM POWERS AND ROOTS FREQUENCY DOMAIN ELECTRICAL SIGNAL AND CIRCUIT ANALYSIS SINUSOIDAL AC ELECTRICAL ANALYSIS EFFECTIVE ELECTRICAL VALUES I–V CHARACTERISTICS FOR PASSIVE ELEMENTS PHASOR ANALYSIS THE GENERALISED IMPEDANCE MAXIMUM POWER TRANSFER GENERALISED FREQUENCY DOMAIN ANALYSIS THE FOURIER SERIES FOURIER COEF?CIENTS, PHASORS AND LINE SPECTRA ELECTRICAL SIGNAL AND CIRCUIT BANDWIDTHS LINEAR DISTORTION BODE PLOTS THE FOURIER TRANSFORM TRANSFER FUNCTION AND IMPULSE RESPONSE THE CONVOLUTION OPERATION NATURAL AND FORCED RESPONSES CIRCUIT ANALYSIS TIME DOMAIN ANALYSIS TRANSIENT ANALYSIS USING FOURIER TRANSFORMS DIFFERENTIATION THEOREM INTEGRATION THEOREM THE LAPLACE TRANSFORM THEOREMS OF THE LAPLACE TRANSFORM PARTIAL-FRACTION EXPANSION ANALYSIS USING LAPLACE TRANSFORMS SOLVING DIFFERENTIAL EQUATIONS NATURAL RESPONSE RESPONSE TO THE STEP FUNCTION CONTENTS ELECTRICAL TWO-PORT NETWORK ANALYSIS ELECTRICAL REPRESENTATIONS ELECTRICAL IMPEDANCE REPRESENTATION ELECTRICAL ADMITTANCE REPRESENTATION ELECTRICAL CHAIN REPRESENTATION CONVERSION BETWEEN ELECTRICAL REPRESENTATIONS MILLER’S THEOREM COMPUTER-AIDED ELECTRICAL ANALYSIS BASIC ELECTRONIC AMPLI?ER BUILDING BLOCKS MODELLING THE AMPLI?CATION PROCESS OPERATIONAL AMPLI?ERS OPEN-LOOP AND FEEDBACK CONCEPTS OTHER EXAMPLES AND APPLICATIONS ACTIVE DEVICES THE JUNCTION OR P–N DIODE THE BIPOLAR JUNCTION TRANSISTOR THE INSULATED GATE ?ELD-EFFECT TRANSISTOR THE COMMON-EMITTER AMPLI?ER THE DIFFERENTIAL PAIR AMPLI?ER RF CIRCUIT ANALYSIS TECHNIQUES LUMPED VERSUS DISTRIBUTED ELECTRICAL MODEL FOR IDEAL TRANSMISSION LINES VOLTAGE STANDING WAVE RATIO – VSWR THE ?/4 TRANSFORMER LOSSY TRANSMISSION LINES MICROSTRIP TRANSMISSION LINES SCATTERING PARAMETERS S-PARAMETERS AND POWER WAVES POWER WAVES AND GENERALISED S-PARAMETERS CONVERSIONS BETWEEN DIFFERENT TWO-PORT PARAMETERS THE SMITH CHART THE IMPEDANCE AND THE RE?ECTION COEF?CIENT PLANES REPRESENTATION OF IMPEDANCES INTRODUCTION TO IMPEDANCE MATCHING NOISE IN ELECTRONIC CIRCUITS RANDOM VARIABLES MOMENTS OF A RANDOM VARIABLE THE CHARACTERISTIC FUNCTION THE CENTRAL LIMIT THEOREM BIVARIATE GAUSSIAN DISTRIBUTIONS STOCHASTIC PROCESSES ENSEMBLE AVERAGES STATIONARY RANDOM PROCESSES ERGODIC RANDOM PROCESSES POWER SPECTRUM CROSS-POWER SPECTRUM GAUSSIAN RANDOM PROCESSES FILTERED RANDOM SIGNALS THERMAL NOISE ELECTRONIC SHOT-NOISE 1/F NOISE NOISE MODELS FOR PASSIVE DEVICES NOISE MODELS FOR ACTIVE DEVICES THE EQUIVALENT INPUT NOISE SOURCES THE NOISE ?GURE COMPUTER-AIDED NOISE MODELLING AND ANALYSIS NOISE REPRESENTATIONS CALCULATION OF THE CORRELATION MATRICES ELEMENTARY TWO-PORT INTERCONNECTIONS TRANSFORMATION MATRICES VOLTAGE AND CURRENT CHARGES PASSING A GIVEN POINT OF AN ELECTRICAL CIRCUIT THIS DE?NITION CAN BE EXPRESSED AS FOLLOWS: I(T) = D Q(T) D T (1 1) 1THE TERM ‘ELECTRONIC SIGNALS’ IS SOMETIMES USED TO DESCRIBE LOW-POWER SIGNALS IN THIS BOOK, THE TERMS ‘ELECTRICAL’ AND ‘ELECTRONIC’ SIGNALS ARE USED INTERCHANGEABLY TO DESCRIBE SIGNALS PRO- CESSED BY A CIRCUIT 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS WHERE I(T) REPRESENTS THE ELECTRICAL CURRENT AS A FUNCTION OF TIME REPRESENTED BY T THE UNIT FOR THE CURRENT IS THE AMPERE (A) Q(T) REPRESENTS THE QUANTITY OF ?OWING ELECTRICAL CHARGE AS A FUNCTION OF TIME AND ITS UNIT IS THE COULOMB (C) THE ELEMENTARY ELECTRICAL CHARGE IS THE CHARGE OF THE ELECTRON WHICH IS EQUAL TO 1 6 × 10?19 C AT THIS STAGE IT IS RELEVANT TO MENTION THAT IN THIS CHAPTER WE REPRESENT CON- STANT QUANTITIES BY UPPERCASE LETTERS WHILE QUANTITIES THAT VARY WITH TIME ARE REPRESENTED BY THE LOWER CASE HENCE, A CONSTANT ELECTRICAL CURRENT IS REPRE- SENTED BY I WHILE AN ELECTRICAL CURRENT VARYING WITH TIME IS REPRESENTED BY I(T) ELECTRICAL CURRENT HAS A A VERY INTUITIVE HYDRAULIC ANALOGUE; WATER ?OW FIG- URE 1 1 A) SHOWS A VOLTAGE SOURCE WHICH IS CONNECTED TO A RESISTANCE, R, CREATING !#" $ % & ' () " & % () *" +,' $ - $ +,' +,' (/ *" VOLTAGE SOURCES FIGURE 1 2 A) SHOWS THE SYMBOLS USED TO REPRESENT VOLTAGE SOURCES THE PLUS 9;: = > ? FIGURE 1 2: IDEAL VOLTAGE SOURCE A) SYMBOLS B) V –I CHARACTERISTIC SIGN, THE ANODE TERMINAL, INDICATES THE HIGHER POTENTIAL AND THE MINUS SIGN, THE CATHODE TERMINAL, INDICATES THE LOWER POTENTIAL THE POSITIVE ?OW OF CURRENT SUPPLIED BY A VOLTAGE SOURCE IS FROM THE ANODE, THROUGH THE EXTERIOR CIRCUIT, SUCH AS THE RESISTANCE IN ?GURE 1 1 A), TO THE CATHODE NOTE THAT THE POSITIVE CURRENT ?OW IS CONVENTIONALLY TAKEN TO BE IN THE OPPOSITE DIRECTION TO THE ?OW OF ELECTRONS AN IDEAL CONSTANT VOLTAGE SOURCE HAS A VOLTAGE–CURRENT, V –I, 2IT IS COMMON PRACTICE TO USE THE LETTER V TO REPRESENT THE VOLTAGE, AS WELL AS ITS UNIT THIS PRACTICE IS FOLLOWED IN THIS BOOK 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS CHARACTERISTIC LIKE THAT ILLUSTRATED IN ?GURE 1 2 B) FROM THIS ?GURE WE OBSERVE THAT AN IDEAL VOLTAGE SOURCE IS ABLE TO MAINTAIN A CONSTANT VOLTAGE V ACROSS ITS TERMINALS REGARDLESS OF THE VALUE OF THE CURRENT SUPPLIED TO (POSITIVE CURRENT) OR THE CURRENT ABSORBED FROM (NEGATIVE CURRENT) AN ELECTRICAL CIRCUIT WHEN A VOLTAGE SOURCE, SUCH AS THAT SHOWN IN ?GURE 1 1 A), PROVIDES A CONSTANT VOLTAGE AT ITS TERMINALS IT IS CALLED A DIRECT CURRENT (OR DC) VOLTAGE SOURCE NO PRACTICAL DC VOLTAGE SOURCE IS ABLE TO MAINTAIN THE SAME VOLTAGE ACROSS ITS TERMINALS WHEN THE CURRENT INCREASES A TYPICAL V –I CHARACTERISTIC OF A PRACTICAL VOLTAGE SOURCE IS AS SHOWN IN ?GURE 1 3 A) FROM THIS ?GURE WE OBSERVE THAT AS THE CURRENT I INCREASES UP TO A VALUE IX THE VOLTAGE DROPS FROM VS TO VX IN A LINEAR MANNER A PRACTICAL VOLTAGE SOURCE CAN BE MODELLED ACCORDING TO THE CIRCUIT OF ?GURE 1 3 B) WHICH CONSISTS OF AN IDEAL VOLTAGE SOURCE AND A RESISTANCE RS WHOSE VALUE IS GIVEN BY: RS = VS ?VX IX (1 3) THIS RESISTANCE IS CALLED THE ‘SOURCE OUTPUT RESISTANCE’ EXAMPLES OF DC VOLT- B) A) V I CURRENT SOURCES FIGURE 1 4 A) SHOWS THE SYMBOL FOR THE IDEAL CURRENT SOURCE3 THE ARROW INDI- CATES THE POSITIVE ?OW OF THE CURRENT FIGURE 1 4 B) SHOWS THE CURRENT–VOLTAGE, 3ALTHOUGH THE SYMBOL OF A CURRENT SOURCE IS SHOWN WITH ITS TERMINALS IN AN OPEN-CIRCUIT SITUA- TION, THE PRACTICAL OPERATION OF A CURRENT SOURCE REQUIRES AN ELECTRICAL PATH BETWEEN ITS TERMINALS OR THE OUTPUT VOLTAGE WILL BECOME IN?NITE 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS I–V , CHARACTERISTIC OF AN IDEAL CURRENT SOURCE FROM THIS ?GURE IT IS CLEAR THAT AN IDEAL CURRENT SOURCE IS ABLE TO PROVIDE A GIVEN CURRENT REGARDLESS OF THE VOLTAGE AT ITS TERMINALS PRACTICAL CURRENT SOURCES HAVE AN I–V CHARACTERISTIC LIKE THAT REPRESENTED IN ?GURE 1 5 A) AS THE VOLTAGE ACROSS THE CURRENT SOURCE INCREASES UP TO A VALUE VX THE CURRENT TENDS TO DECREASE IN A LINEAR FASHION FIGURE 1 5 B) @ @6A BDC E F G C FIGURE 1 4: IDEAL CURRENT SOURCE A) SYMBOL B) I–V CHARACTERISTIC SHOWS THE EQUIVALENT CIRCUIT FOR A PRACTICAL CURRENT SOURCE INCLUDING A RESISTANCE RS WHICH IS ONCE AGAIN CALLED THE ‘SOURCE OUTPUT RESISTANCE’ THE VALUE OF THIS RESISTANCE IS: RS = VX IS ?IX (1 5) EXAMPLES OF SIMPLE CURRENT SOURCES ARE DIF?CULT TO PROVIDE AT THIS STAGE MOST CURRENT SOURCES ARE IMPLEMENTED USING ACTIVE DEVICES SUCH AS TRANSISTORS AC- TIVE DEVICES ARE STUDIED IN DETAIL IN CHAPTER 6 WHERE IT IS SHOWN THAT, FOR EX- AMPLE, THE ?ELD-EFFECT TRANSISTOR, IN SPECI?C CON?GURATIONS, DISPLAYS CURRENT SOURCE BEHAVIOUR ——————————————————————————————— I IX IS V IS A) VX RS B) V + ? FIGURE 1 5: PRACTICAL CUR- RENT SOURCE A) I–V CHARAC- TERISTIC B) EQUIVALENT CIRCUIT EXAMPLE 1 2 2 DETERMINE THE OUTPUT RESISTANCE OF A CURRENT SOURCE WHOSE OUT- PUT CURRENT FALLS FROM 2 A TO 1 99 A WHEN ITS OUTPUT VOLTAGE INCREASES FROM TO 100 V SOLUTION: THE OUTPUT RESISTANCE IS CALCULATED ACCORDING TO: RS = VX IS ?IX = 104 ? = 10 K? ——————————————————————————————— POWER SUPPLIED BY A SOURCE AS MENTIONED PREVIOUSLY, THE MAIN PURPOSE OF A VOLTAGE OR CURRENT SOURCE IS TO PROVIDE POWER TO A CIRCUIT THE INSTANTANEOUS POWER DELIVERED BY EITHER SOURCE IS GIVEN BY THE PRODUCT OF THE CURRENT SUPPLIED WITH THE VOLTAGE AT ITS TERMINALS, THAT IS, PS(T) = V(T) I(T) (1 6) THE UNIT FOR POWER IS THE WATT (W) WHEN THE VOLTAGE IS EXPRESSED IN VOLTS (V) AND THE CURRENT IS EXPRESSED IN AMPERES (A) IF THE VOLTAGE AND CURRENT ARE CONSTANT THEN EQN 1 6 CAN BE WRITTEN AS: PS = V I (1 7) OFTEN IT IS OF INTEREST TO CALCULATE THE AVERAGE POWER, PAVS, SUPPLIED BY A SOURCE DURING A PERIOD OF TIME T THIS AVERAGE POWER CAN BE CALCULATED BY THE SUC- CESSIVE ADDITION OF ALL VALUES OF THE INSTANTANEOUS POWER, PS(T), DURING THE TIME 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS INTERVAL T AND THEN DIVIDING THE OUTCOME BY THE TIME INTERVAL T THAT IS, PAVS CAN BE CALCULATED AS FOLLOWS4: PAVS = T Z TO+T TO PS(T) DT = = 36 + 24 COS(2 ? 100 T) (MW) THIS IS PLOTTED IN ?GURE 1 6 THE AVERAGE POWER IS CALCULATED ACCORDING TO EQN H'I JK KL M#N OP Q R S;TU L H'L VXW R QUYR S[Z\U FIGURE 1 6: INSTANTANEOUS AND AVERAGE POWER 1 8: PAVS = 3 T + RESISTANCE AND CONDUCTANCE THE RESISTANCE5 HAS BEEN PRESENTED IN THE PREVIOUS SECTION OHM’S LAW RELATES THE VOLTAGE AT THE TERMINALS OF A RESISTOR WITH THE CURRENT WHICH ?OWS THROUGH IT ACCORDING TO EQN 1 2 THE HYDRAULIC ANALOGUE FOR A RESISTANCE HAS ALSO BEEN PRESENTED ABOVE IN ?GURE 1 1 IT IS WORTH MENTIONING THAT IF THE VOLTAGE VARIES 4RECALL THAT THE INTEGRAL OPERATION IS BASICALLY AN ADDITION OPERATION 5STRICTLY SPEAKING, THE SUF?X -OR DESIGNATES THE NAME OF THE ELEMENT (LIKE RESISTOR) WHILE THE SUF?X -ANCE DESIGNATES THE ELEMENT PROPERTY (LIKE RESISTANCE) OFTEN THESE TWO ARE USED INTER- CHANGEABLY (1 13) = V R (1 14) THESE TWO EQNS (1 13 AND 1 14) APPEAR TO BE CONTRADICTORY IN TERMS OF THE ROLE THE RESISTANCE PLAYS IN DETERMINING THE LEVEL OF POWER DISSIPATION DOES THE DISSIPATED POWER INCREASE WITH INCREASING THE RESISTANCE (EQN 1 13) OR DOES IT DECREASE (EQN 1 14)? THE ANSWER TO THIS QUESTION RELATES TO THE WAY WE VIEW THE CIRCUIT AND TO WHAT QUANTITY WE MEASURE ACROSS THE RESISTOR LET US CONSIDER THE CASE WHERE A RESISTOR IS CONNECTED ACROSS THE TERMINALS OF AN IDEAL VOLTAGE SOURCE HERE, THE STIMULUS IS THE VOLTAGE THAT RESULTS IN A CURRENT THROUGH THE RESISTOR IN THIS SITUATION THE LARGER THE RESISTANCE THE SMALLER THE CURRENT IS AND, 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS ACCORDING TO EQN 1 12, THERE IS LESS POWER DISSIPATION IN THE RESISTANCE NOTE THAT, IN THE EXTREME SITUATION OF R ??THE RESISTANCE BEHAVES AS AN OPEN CIRCUIT (THE CURRENT IS ZERO) AND THERE IS NO POWER DISSIPATION ON THE OTHER HAND, IF THE RESISTANCE IS DRIVEN BY AN IDEAL CURRENT SOURCE, THEN THE VOLTAGE ACROSS THE RESISTOR IS THE RESULTING EFFECT HENCE, THE LARGER THE RESISTANCE, THE LARGER THE VOLTAGE DEVELOPED ACROSS ITS TERMINALS AND, ACCORDING TO EQN 1 12, THE POWER DISSIPATION INCREASES FOR TIME-VARYING SOURCES THE INSTANTANEOUS POWER DISSIPATED IN A RESISTOR IS GIVEN BY: PR(T) = V(T) I(T) (1 15) = R I2(T) (1 16) = V2(T) R (1 17) WHERE V(T) REPRESENTS THE VOLTAGE ACROSS THE RESISTANCE AND I(T) IS THE CURRENT THAT ?OWS THROUGH IT THE AVERAGE POWER DISSIPATED IN A PERIOD OF TIME T CAN BE EXPRESSED AS FOLLOWS: PAVR = T Z TO+T TO PR(T) DT (1 18) = R T Z TO+T TO I2(T) DT (1 19) = R T Z TO+T TO V2(T) DT (1 20) WHERE TO IS A CHOSEN INSTANT OF TIME WHEN THE RESISTOR IS DRIVEN BY A PERIODIC SIGNAL, T IS NORMALLY CHOSEN TO BE ITS PERIOD ——————————————————————————————— EXAMPLE 1 3 2 AN AC VOLTAGE V(T) = A SIN(? T), WITH A = 10 V, IS APPLIED TO A RESISTANCE R = 50 ? DETERMINE THE AVERAGE POWER DISSIPATED SOLUTION: ACCORDING TO EQN 1 20 WE CAN WRITE: PAVR = T R Z T A SIN2(? T) DT = T V A R Z T 1 ?COS(2 ? T) DT = V 2 ? SIN(2 ? T) ¸T SINCE THE PERIOD OF THE AC WAVEFORM IS T = 2?/?, THE LAST EQN CAN BE WRITTEN AS: PAVR = R V (1 21) = 1 W ——————————————————————————————— CAPACITANCE FIGURE 1 8 A) SHOWS A CAPACITOR CONNECTED TO A VOLTAGE SOURCE A CAPACITOR IS USUALLY IMPLEMENTED USING TWO METAL PLATES SEPARATED BY AN INSULATOR THIS MEANS THAT THE CAPACITOR DOES NOT ALLOW THE PASSAGE OF DIRECT CURRENT HOW- EVER, WHEN A VOLTAGE IS APPLIED ACROSS THE TERMINALS OF AN UNCHARGED CAPACITOR ELECTRONIC CHARGE CAN BE ADDED TO ONE OF THE METAL PLATES AND REMOVED FROM THE OTHER THIS CHARGE IS PROPORTIONAL TO THE APPLIED VOLTAGE AND DE?NES THE CAPACITANCE ACCORDING TO THE FOLLOWING EQN: Q(T) = C V(T) (1 22) WHERE C IS THE CAPACITANCE AND V(T) IS THE VOLTAGE APPLIED ACROSS ITS TERMINALS THE CAPACITANCE CAN BE SEEN TO HAVE THE ABILITY TO ACCUMULATE CHARGE THE UNIT FOR THE CAPACITANCE IS THE FARAD (F) WHEN, IN EQN 1 22, V(T) IS EXPRESSED IN VOLTS AND Q(T) IS EXPRESSED IN COULOMBS IN OTHER WORDS, A CAPACITOR OF ONE FARAD WILL STORE ONE COULOMB OF CHARGE IF A POTENTIAL DIFFERENCE OF ONE VOLT EXISTS ACROSS ITS PLATES FROM EQNS 1 1 AND 1 22 WE CAN RELATE THE CURRENT THROUGH A CAPACITOR WITH THE VOLTAGE ACROSS ITS TERMINALS ACCORDING TO THE FOLLOWING EQN: I(T) = C D V(T) D T (1 23) DXE FHGIJK6LMKJ NNOMKJ P)Q GNIR S TDJ T D KGU J G E V GLMGS R IWK X Y'Z { E { Y'Z { E | Z { E { } ~ | Z { E FIGURE 1 8: THE CAPACITOR A) IN AN ELECTRICAL CIRCUIT B) HYDRAULIC ANALOGUE FIGURE 1 8 B) SHOWS A HYDRAULIC ANALOGUE FOR THE CAPACITOR WHICH IS AN ELAS- TIC MEMBRANE COVERING THE SECTION OF A WATER PIPE IN THIS ANALOGY THE VOLTAGE BECOMES THE WATER PRESSURE, THE CAPACITOR BECOMES THE MEMBRANE ELASTICITY, AND THE CHARGE BECOMES THE WATER VOLUME DISPLACED BY THE MEMBRANE LET US CONSIDER THAT THERE IS WATER ON BOTH SIDES OF THE ELASTIC MEMBRANE THIS MEM- BRANE DOES NOT ALLOW THE DIRECT CROSSING OF WATER BETWEEN THE TWO SIDES OF THE PIPE HOWEVER, IF THERE IS A PULSED INCREASE OF THE WATER PRESSURE IN ONE SIDE OF THE MEMBRANE, AS ILLUSTRATED IN ?G 1 8 B), THE STRETCHING OF THIS MEMBRANE CAUSES AN EFFECTIVE TRAVELLING OF THE PULSE FROM ONE SIDE OF THE MEMBRANE TO THE OTHER (WITH NO WATER PASSING THROUGH THE MEMBRANE!) THIS PROCESS IS CONCEP- TUALLY SIMILAR TO THE ?OW OF CHARGES (CURRENT) IN THE METAL PLATES OF A CAPACITOR NOTE THAT IF THERE IS TOO MUCH PRESSURE ON THE MEMBRANE IT WILL EVENTUALLY BREAKDOWN AND THE SAME CAN HAPPEN TO A CAPACITOR IF TOO MUCH VOLTAGE IS AP- PLIED TO IT INTEGRATING EQN 1 23 WE OBTAIN AN EXPRESSION FOR THE VOLTAGE IN TERMS OF THE CURRENT, THAT IS: V(T) = C Z T I(T) DT + VCO (1 24) WHERE VCO REPRESENTS THE INITIAL VOLTAGE ACROSS THE TERMINALS OF THE CAPACITOR AT T = 0 UNLIKE THE RESISTOR THE CAPACITOR DOES NOT DISSIPATE ANY POWER IN FACT BE- CAUSE THIS CIRCUIT ELEMENT IS ABLE TO ACCUMULATE CHARGE ON ITS METAL PLATES IT STORES ENERGY TO ILLUSTRATE THIS WE RETURN TO THE HYDRAULIC ANALOGUE; IF TWO SHUTTERS ARE INSERTED ACROSS EACH SIDE OF THE PIPE IN ORDER TO STOP THE STRETCHED 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS MEMBRANE RELAXING TO ITS ORIGINAL STATE THIS IS EQUIVALENT TO THE DISCONNECTION OF A CHARGED CAPACITOR FROM THE CIRCUIT THE ENERGY STORED IN A CAPACITOR CAN BE WRITTEN AS EC = INDUCTANCE FIGURE 1 9 A) SHOWS AN INDUCTOR CONNECTED TO A CURRENT SOURCE THE INDUCTOR IS USUALLY FORMED BY A COIL OF A METAL WIRE THE PASSAGE OF CURRENT, I(T), IN A METAL WIRE INDUCES A MAGNETIC ?UX WHICH IN TURN (ACCORDING TO FARADAY’S LAW) RESULTS IN A VOLTAGE, V(T), DEVELOPING ACROSS THE TERMINALS, SUCH THAT: V(T) = L D I(T) D T (1 26) WHERE L IS THE INDUCTANCE OF THE WIRE THE UNIT FOR THE INDUCTANCE IS THE HENRY (H) WHEN, IN EQN 1 26, V(T) IS EXPRESSED IN VOLTS AND THE CURRENT RATE OF CHANGE, DI(T)/DT, IS EXPRESSED IN AMPERES PER SECOND THUS, AN INDUCTANCE OF ONE HENRY WILL HAVE A POTENTIAL DIFFERENCE OF ONE VOLT ACROSS ITS TERMINALS WHEN THE CURRENT PASSING THROUGH IT IS CHANGING AT A RATE OF ONE AMPERE PER SECOND FIGURE 1 9 B) SHOWS A HYDRAULIC ANALOGUE FOR THE INDUCTOR WHICH CONSISTS OF PADDLES CONNECTED TO A ?YWHEEL IN THIS ANALOGY THE VOLTAGE BECOMES THE WATER PRESSURE AND THE CURRENT BECOMES THE RATE OF WATER ?OW THE INDUCTANCE BECOMES THE ?YWHEEL MOMENT OF INERTIA THE ?YWHEEL REQUIRES WATER PRESSURE TO CHANGE THE SPEED OF THE PADDLES WHICH, IN TURN, CHANGE THE RATE OF THE WATER ?OW IN ?GURE 1 9 B) WE ILLUSTRATE THE SITUATION WHERE THE APPLICATION OF A PULSE OF WATER PRESSURE CAUSES AN INCREASE IN THE SPEED OF THE WATER ?OW WHICH WILL BE MAINTAINED CONSTANT BY THE ?YWHEEL INERTIA UNTIL A DIFFERENT LEVEL OF WATER PRESSURE IS APPLIED IN ELECTRICAL TERMS THIS MEANS THAT THE VOLTAGE DIFFERENCE AT THE TERMINALS OF AN INDUCTOR IS PROPORTIONAL TO THE RATE OF VARIATION OF THE CURRENT THAT ?OWS THROUGH IT €6 ‚Xƒ„D ‡†† ˆ ‰ Š‹‡Œ‡#Ž  ˆ ‹‡Ž†  ‘“’ ” ‰ • ‘“’ ” ‰ ” ” – ’ ” ‰ – ’ ”‰ — ˆ †˜š™›„ œ ‰ — ˆ  †˜[‡˜'†ž'ž ‡˜'† FIGURE 1 9: THE INDUCTANCE A) IN AN ELECTRICAL CIRCUIT B) HYDRAULIC ANALOGUE INTEGRATING EQN 1 26 WE OBTAIN AN EXPRESSION FOR THE CURRENT THAT ?OWS IN THE INDUCTOR IN TERMS OF THE VOLTAGE AT ITS TERMINALS, THAT IS: I(T) = L Z T V(T) DT + ILO (1 27) WHERE ILO REPRESENTS THE INITIAL CURRENT IN THE INDUCTOR AT T = 0 LIKE THE CAPACITOR, THE INDUCTOR DOES NOT DISSIPATE ENERGY AND IS CAPABLE OF STORING ENERGY HOWEVER, THE ENERGY IS NOW STORED IN TERMS OF THE MAGNETIC ?UX CREATED BY THE CURRENT THIS ENERGY CAN BE WRITTEN AS EL = 2 L I2 (JOULE) (1 28) WHERE I IS THE CURRENT THROUGH THE INDUCTOR WHICH CAN ALSO BE EXPRESSED AS: I1 + I2 + (?I3) = = 3 + 4 + 3 + = 12 A NOTE THAT THE CURRENT I1 IS NEGATIVE WHICH MEANS THAT THE DIRECTION OF THE CURRENT IS THE OPPOSITE OF THAT SHOWN IN THE ?GURE ¬­ ® ¬¯ ° ± ²“³ ´6³ µ ³ ¶ ³ “³ ¬ ¸ FIGURE 1 11: CIRCUIT FOR THE APPLICATION OF THE CURRENT LAW ——————————————————————————————— THE VOLTAGE LAW THE VOLTAGE LAW STATES THAT THE SUM OF ALL VOLTAGES AROUND ANY CLOSED ELECTRICAL LOOP, TAKING INTO ACCOUNT POLARITIES6, IS ZERO FIGURE 1 12 SHOWS A CIRCUIT WITH 6IN THIS BOOK WE USE CURVED ARROWS TO INDICATE THE POTENTIAL DIFFERENCE BETWEEN TWO POINTS IN A CIRCUIT, WITH THE ARROW HEAD POINTING TO THE LOWER POTENTIAL 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS TWO CLOSED LOOPS BY APPLYING THE VOLTAGE LAW WE CAN WRITE THE FOLLOWING EQUATIONS FOR LOOP 1 AND LOOP 2, RESPECTIVELY: VR(T) + VL(T) ?VS(T) = VC(T) ?VR(T) = (1 30) GK (1 39) THAT IS, THE EQUIVALENT CONDUCTANCE IS THE ADDITION OF EACH CONDUCTANCE THAT COMPOSES THE PARALLEL CONNECTION 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS THE EQUIVALENT RESISTANCE, REQ, FOR THE PARALLEL COMBINATION OF TWO RESIST- ANCES CAN BE OBTAINED BY RE-WRITING EQN 1 38 AS FOLLOWS REQ = AND R2 2 DETERMINE THE EQUIVALENT RESISTANCE, REQ, FOR THE PARALLEL CONNECTION OF R1 AND R2 SOLUTION: 1 ACCORDING TO EQN 1 33 THE EQUIVALENT RESISTANCE IS REQ = 3 2 K? 2 ACCORDING TO EQN 1 41 THE EQUIVALENT RESISTANCE IS REQ = 188 ? IT SHOULD BE REMEMBERED THAT FOR THE SERIES CONNECTION OF RESISTANCES THE EQUIV- ALENT RESISTANCE IS LARGER THAN THE LARGEST RESISTANCE IN THE SERIES CHAIN WHILST FOR THE PARALLEL CONNECTION THE EQUIVALENT RESISTANCE IS SMALLER THAN THE SMALLEST RE- SISTANCE IN THE CONNECTION ——————————————————————————————— CAPACITANCE FIGURE 1 16 A) SHOWS TWO CAPACITORS IN A SERIES CONNECTION THESE CAN BE RE- PLACED BY A CAPACITOR WITH AN EQUIVALENT CAPACITANCE, CEQ, AS SHOWN IN ?GURE 1 16 B) APPLYING KIRCHHOFF’S VOLTAGE LAW TO THE CIRCUIT OF ?GURE 1 16 A) WE OBTAIN: VS(T) = V1(T) + V2(T) (1 42) SINCE THE CURRENT THAT ?OWS THROUGH BOTH CAPACITORS IS THE SAME THEN, ACCORDING Ð;ÑÒ Ó ÔÕ Ö ÷ Ð;Ø ÐDÙ Ú Û Ó Ô Õ Ö'÷ Ú Ó Ø Õ Ö ÷ Û Ú Ó Ù Õ Ö ÷ Û Ü Õ Ö ÷ Û Ú Û Ú Ó ÔÕ Ö ÷ Ü Õ Ö ÷ Ý ÷ Þ÷ FIGURE 1 16: A) SERIES COM- BINATION OF TWO CAPACITORS B) EQUIVALENT CAPACITOR TO EQN 1 24, WE CAN REWRITE THE ABOVE EQN AS FOLLOWS: VS(T) = Z T I(T) D T + I(T) D T = µ ¶ Z T I(T) D T (1 43) 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS HERE WE ASSUME THAT THE INITIAL VOLTAGE ACROSS EACH CAPACITOR IS ZERO APPLYING EQN 1 24 TO THE CAPACITOR OF THE CIRCUIT OF ?GURE 1 16 B) WE OBTAIN: VS(T) = CEQ Z T I(T) D T (1 44) FROM EQNS 1 44 AND 1 43 WE CONCLUDE THAT CEQ = (1 46) IT CAN BE SHOWN (SEE PROBLEM 1 7) THAT EQN 1 45 CAN BE GENERALISED FOR THE SERIES COMBINATION OF N CAPACITORS AS FOLLOWS CK (1 52) 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS LK (1 57) 7WE ASSUME THE INDUCTORS TO BE UNCOUPLED, THAT IS THEY ARE SUF?CIENTLY FAR APART OR MOUNTED IN SUCH A WAY SO THAT THEIR MAGNETIC ?UXES DO NOT INTERACT 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS FIGURE 1 19 A) SHOWS TWO INDUCTORS IN A PARALLEL CONNECTION APPLYING KIRCHHOFF’S CURRENT LAW TO THE CIRCUIT OF ?GURE 1 19 A) WE OBTAIN: IS(T) = I1(T) + I2(T) (1 58) SINCE THE VOLTAGE ACROSS EACH INDUCTOR IS THE SAME, WE CAN APPLY EQN 1 27 TO EQN 1 58 AS FOLLOWS: IS(T) = V(T) D T + V(T) D T = µ V(T) D T (1 59) WHERE WE ASSUME THAT THE INITIAL CURRENT IN EACH INDUCTOR IS ZERO APPLYING EQN 1 27 TO THE EQUIVALENT INDUCTOR OF ?GURE 1 15 B) WE OBTAIN: IS(T) = LEQ Z T V(T) D T (1 60) COMPARING THE LAST EQN WITH EQN 1 59 WE CONCLUDE THAT LEQ = (1 62) IT CAN BE SHOWN (SEE PROBLEM 1 11) THAT THE ABOVE EQN CAN BE GENERALISED FOR (*)+ ,- / + ,- FIGURE 1 19: A) PARALLEL COMBINATION OF TWO INDUC- TORS B) EQUIVALENT INDUCTOR THE PARALLEL COMBINATION OF N INDUCTORS AS FOLLOWS LK (1 63) ——————————————————————————————— EXAMPLE 1 4 5 CONSIDER TWO INDUCTANCES L1 = 1 5 MH AND L2 = 1 MH 1 DETERMINE THE EQUIVALENT INDUCTANCE, LEQ, FOR THE SERIES CONNECTION OF L1 AND L2 2 DETERMINE THE EQUIVALENT CAPACITANCE, LEQ, FOR THE PARALLEL CONNECTION OF L1 AND L2 SOLUTION: 1 ACCORDING TO EQN 1 56 THE EQUIVALENT INDUCTANCE IS CEQ = 2 5 MH 2 ACCORDING TO EQN 1 62 THE EQUIVALENT INDUCTANCE IS CEQ = 0 6 MH IT IS INTERESTING TO NOTE THAT LIKE RESISTORS (BUT UNLIKE CAPACITORS) THE SERIES CON- NECTION OF INDUCTORS INCREASES THE VALUE OF THE EQUIVALENT INDUCTANCE WHILE THE PARALLEL CONNECTION DECREASES IT ——————————————————————————————— 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS (1 64) ELECTRICAL NETWORK ANALYSIS – NODAL ANALYSIS THERE ARE VARIOUS ELECTRICAL ANALYSIS METHODS, ALL DERIVED FROM KIRCHHOFF’S LAWS, TO ANALYSE ELECTRICAL CIRCUITS ONE OF MOST EFFECTIVE AND COMPUTATION- ALLY EF?CIENT IS THE NODAL ANALYSIS METHOD THEREFORE, WE NOW ILLUSTRATE THE APPLICATION OF THIS METHOD TO RESISTIVE ELECTRICAL NETWORKS FIGURE 1 22 A) SHOWS A CIRCUIT FOR WHICH WE WANT TO DETERMINE THE CURRENT IA SINCE THE RESISTANCE R2 IS SHORT-CIRCUITED THE VOLTAGE ACROSS THIS RESISTANCE IS ZERO AND, ACCORDING TO OHM’S LAW, THE CURRENT THAT ?OWS THROUGH R2 IS ZERO HENCE, THE CIRCUIT OF ?GURE 1 22 A) CAN BE REPLACED BY ITS EQUIVALENT REPRESENTED IN ?GURE 1 22 B) FIRST, WE INDICATE THE VOLTAGES AT EACH NODE THESE VOLTAGES INDICATE THE POTENTIAL DIFFERENCE BETWEEN THE NODE BEING CONSIDERED AND A REF- ERENCE NODE WHICH CAN BE CHOSEN ARBITRARILY THIS NODE IS TRADITIONALLY CALLED ‘NODE ZERO’ (0) OR THE ‘GROUND TERMINAL’ AND IS OFTEN CHOSEN AS THE NODE WITH THE HIGHEST NUMBER OF ATTACHED ELECTRICAL ELEMENTS FOR THIS CIRCUIT THERE ARE THREE NODES (X, Y , Z) PLUS THE REFERENCE NODE ZERO, AS SHOWN IN ?GURE 1 22 B) THEN, WE CONSIDER, IN AN ARBITRARY MANNER, THE CURRENT DIRECTION IN EACH BRANCH, 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS (7 V) VX VX ? + + (1 74) (1 81) WE CAN CONCLUDE THAT IF R1 = R2 THEN IO = 0 5 IS ALSO, IF R1 >> R2 THEN IO TENDS TO IS ON THE OTHER HAND, IF R1 << R2 THEN IO TENDS TO ZERO 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS = R2 IS (1 + GM R1) = 81 V VS (0 1 A) ? + (1 82) = 1 1 V 1 ELEMENTARY ELECTRICAL CIRCUIT ANALYSIS NORTON’S THEOREM DEPICTED IN ?GURE 1 29 A) CONSISTING OF RESISTANCES AND INDEPENDENT SOURCES (VOLTAGE OR CURRENT OR BOTH) CAN BE REPLACED BY AN EQUIVALENT INDEPENDENT CUR- RENT SOURCE IN PARALLEL WITH AN EQUIVALENT RESISTANCE, AS SHOWN IN ?GURE 1 29 B) THE VALUE OF THE CURRENT SOURCE IS THE CURRENT ?OWING FROM X TO Y WHEN X TO Y ARE SHORT-CIRCUITED THE EQUIVALENT RESISTANCE (NORTON RESISTANCE), RNT IS THE RESISTANCE ‘LOOKING INTO’ THE TERMINALS OF THE NETWORK WHEN THE INDEPEN- DENT VOLTAGE SOURCES ARE REPLACED BY SHORT-CIRCUITS AND THE INDEPENDENT CUR- RENT SOURCES ARE REPLACED BY OPEN-CIRCUITS NORTON’S THEOREM IS THE DUAL OF TH´EVENIN’S SINCE THE EQUIVALENT VOLTAGE SOURCE, VT H, IS REPLACED BY AN EQUIVA- LENT CURRENT SOURCE, INT, AND THE SERIES RESISTANCE, RT H, IS REPLACED BY A PARALLEL RESISTANCE, RNT