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KN Toosi University of Technology Chapter 8. Network Theorems By: FARHAD FARADJI, Ph.D. Assistant Professor, Electrical and Computer Engineering, K. N. Toosi University of Technology http://wp.kntu.ac.ir/faradji/ElectricCircuits2.htm References: Basic Circuit Theory, by Ch. A. Desoer and E. S. Kuh, 1969 KNTU Chapter Contents 0. Introduction 1. The substitution theorem 2. The superposition theorem 3. Thevenin-Norton equivalent network ton e quivalent n etwork ttheorem heore em 4. The reciprocityy theorem Electric Circuits 2 Chapter 8. Network Theorems 2 KNTU 0. Introduction o In this chapter, we study 4 very general and useful network theorems: ¾ substitution theorem, ¾ superposition theorem, ¾ Thevenin-Norton Nortton e equivalent quivvale ent n network etwork theorem, theeorem, ¾ reciprocity y ttheorem. heorem. o Principal assumption mption underlying alll these these network network theo theorems orem is uniqueness of solution forr netw network under work u nder consideration. consideration.. o Substitution theorem: heorem: ¾ holds for all networks with a unique solution, ¾ can be applied to: y y linear and nonlinear networks, time-invariant and time-varying networks. o Other 3 theorems apply only to linear networks. Electric Circuits 2 Chapter 8. Network Theorems 3 KNTU 0. Introduction A linear network: ¾ by definition, consists of: linear elements: 9 linear r resistors, 9 linear r iinductors, nductors, 9 linear r coup coupled pled inductors, indu uctors, 9 linear r capacitors, 9 linear r tran transformers, nsformers, 9 linear r depe dependent endent sources sources independent sources which are inputs to network. Electric Circuits 2 Chapter 8. Network Theorems 4 KNTU 0. Introduction Superposition theorem and Thevenin-Norton equivalent network theorem apply to all linear networks: ¾ time-invariant or ¾ time-varying. ng. Reciprocity theorem eorem applies appliees to to a more mo ore restricted resttricted class classs of of linear lin networks: ¾ elements must be time-invariant. ¾ elements can not not include: include: dependent ndentt ssources, ource es, independent sources, and gyrators. Electric Circuits 2 Chapter 8. Network Theorems 5 KNTU Chapter Contents 0. Introduction 1. The substitution theorem 2. The superposition theorem 3. Thevenin-Norton equivalent network ton e quivalent n etwork ttheorem heore em 4. The reciprocityy theorem Electric Circuits 2 Chapter 8. Network Theorems 6 KNTU 1. The substitution theorem SUBSTITUTION THEOREM ¾ Consider an arbitrary network containing some independent sources. Suppose that for these sources and for ggiven initial conditions network has a unique solution branch sollution n for for aallll iits ts b ranch voltages voltagges and and branch branch currents. Consider a particular other branches. rticularr branch branch h (k), (kk), which is not not ccoupled ouple ed to to o t currrent and and voltagee waveforms waveforrms of of branch branch kk. Let jk and vk bee current Suppose that branc branch byy either: ch k iiss rreplaced eplaced db an independent pendent current source with waveform waveeform jk or an independent voltage source with waveform vk. If modified network has a unique solution for all its branch currents and voltages: these branch currents and voltages are identical with those of original network. Electric Circuits 2 Chapter 8. Network Theorems 7 KNTU 1. The substitution theorem Example 1: y Consider circuit shown. y Problem is to find voltage g V and current I o off tunnel diode ggiven iven E, E, R, R, and an nd tunnel-diode characteristic. y Solution is obtained taineed ggraphically. raphicallyy. y According to substitution ubstittuttion th theorem, heorem, we may replace replaace tunnel diode either: ¾ by a current source I or ¾ by a voltage source V. y In both cases, solutions are same as that obtained originally. Electric Circuits 2 Chapter 8. Network Theorems 8 KNTU 1. The substitution theorem Example 2: o Consider any LTI network which: ¾ is in zero state at time 0 ¾ has no independent ep pende ent ssources. ourcces. o Assume that there here are are 2 acc accessible cessiible termin terminals nals to network which hich form a port. o If we use substitution theorem: Electric Circuits 2 Chapter 8. Network Theorems 9 KNTU Chapter Contents 0. Introduction 1. The substitution theorem 2. The superposition theorem 3. Thevenin-Norton equivalent network ton e quivalent n etwork ttheorem heore em 4. The reciprocityy theorem Electric Circuits 2 Chapter 8. Network Theorems 10 KNTU 2. The superposition theorem SUPERPOSITION THEOREM Let ॉ be a linear network: each element is either an independent source or a linear element. Elements mayy be be time-varying. time-vaaryin ng. We further assume has zero-state sume tthat hatt ॉ h as a unique ze ero-sstate response respons to independent source waveforms, whatever be. wh hateverr tthey hey may be e. Let response of ॉ be: be: either current urrentt in in a specific branch or voltage across any specific node pair or more generally any linear combination of currents and voltages. Under these conditions, zero-state response of ॉ due to all independent sources acting simultaneously is equal to: sum of zero-state responses due to each independent source acting one at a time. Electric Circuits 2 Chapter 8. Network Theorems 11 KNTU 2. The superposition theorem When a voltage source is set equal to 0, it becomes a short circuit. When a current source is set equal to 0, it becomes an open circuit. Superposition theorem can be expressed p in terms of concept of linear function. Consider waveforms efformss o off iindependent nde epend dent sour sources rces to to be ccomponents ompo of a vector which we call vector-input waveform. waveforrm. If vector-input waveform wavveform is: is: Zero-state response v(·) is a function of vector-input waveform: Electric Circuits 2 Chapter 8. Network Theorems 12 KNTU 2. The superposition theorem ¾ If function f is linea linear, ar, then then superposition superposition theorem theo orem holds: holds: Electric Circuits 2 Chapter 8. Network Theorems 13 KNTU 2. The superposition theorem y In general, superposition does not apply to nonlinear networks. y If a particular network has: ¾ all linear elements ¾ except for a few few nonlinear nonliinear ones o nes it is possible to o make makke superposition su uperrposiition hold by by ccareful areful sselection electio of: ¾ element values, alues, ¾ source location, ation n, ¾ source waveform, veform and ¾ response. Electric Circuits 2 Chapter 8. Network Theorems 14 KNTU 2. The superposition theorem o Superposition theorem has been stated exclusively in terms of zero-state response of a linear network. o Sinusoidal steady state is limiting condition (as ݐ՜ λ) of zero-state nputt, it ffollows ollo ows that t: response to a sinusoidal in input, that: ¾ superposition tion ttheorem heorem applies applies iin n particular particu ulaar to to sinusoidal siinusoid steady state. COROLLARY 1 o Let ॉ be an LTI TI netw network. workk. Suppose that all independent sources are sinusoidal (not necessarily of same frequency). Then steady-state response of ॉ due to all independent sources acting simultaneously is equal to: sum of sinusoidal steady-state responses due to each independent source acting one at a time. Electric Circuits 2 Chapter 8. Network Theorems 15 KNTU 2. The superposition theorem COROLLARY 2 Let ॉ be an LTI network. g across anyy node p Let response be voltage pair or current through any branch of ॉ. More specifically, off zero-state alllyy, call caall X(s) X(s) Laplace Lapllace transform transfo orm o zero o-state response due to all independent nt sources acting simultaneously. sim multaneo ously. Then: Ik(s), k = 1, 2, ..., m, are Laplace transforms of m inputs. Hk(s), k = 1, 2, ..., m, are respective network functions from m inputs to specified output. Electric Circuits 2 Chapter 8. Network Theorems 16 KNTU Chapter Contents 0. Introduction 1. The substitution theorem 2. The superposition theorem 3. Thevenin-Norton equivalent network ton e quivalent n etworrk ttheorem heore em 4. The reciprocityy theorem Electric Circuits 2 Chapter 8. Network Theorems 17 KNTU 3. Thevenin-Norton equivalent network theorem THEVENIN-NORTON THEOREM Let linear network ॉ, be connected by 2 of its terminals 1 and 1’ to an arbitrary load. Network elements men nts m may ay b bee tim time-varying. me-varying. We further assume has sume tthat hatt ॉ h as a unique solution: so olution: when it is terminated by load load, d, aand nd when load ad iss replaced replaced by by an an iindependent ndepend dent ssource. ource. Let ॉ0 be network work ob obtained btained d from ॉ by setting: all independent sources to 0 and all initial conditions to 0. Electric Circuits 2 Chapter 8. Network Theorems 18 KNTU 3. Thevenin-Norton equivalent network theorem THEVENIN-NORTON THEOREM Let eoc be open-circuit voltage of ॉ at terminals 1 and 1’, as shown. Let isc be short-circuit t-ccircuiit current currrent of of ॉ nto 1 shown. flowing out of 1 iinto 1’’ aass shown. Under these conditions, onditions, whatever load may be, d ma ay b e, vvoltage oltage v and current i remain remaiin unchanged unchanged d when networkk ॉ is replaced by either its Thevenin equivalent or by its Norton equivalent network. Electric Circuits 2 Chapter 8. Network Theorems 19 KNTU 3. Thevenin-Norton equivalent network theorem THEVENIN-NORTON THEOREM ¾ Relaxed network ॉ0 is obtained from ॉ by setting: 9 all its independent pend den nt sources sourrcess to to 0, 0, 9 all initial conditions onditions to to 0. 0. ¾ ॉ0 could be referred eferred to as d zer ro-sstate zero-input and zero-state tworkk. equivalent network. Electric Circuits 2 Chapter 8. Network Theorems 20 KNTU 3. Thevenin-Norton equivalent network theorem Example: o Consider resistive circuit shown. o Determine voltage g across tunnel diode. o We can: 9 use Thevenin niin th theorem heoreem and and 9 consider tunnel unnel diode as load. o First we determine minee Thevenin Thevenin equivalent equivalent network netw work of one-port faced ced by tunnel tunnell diode: o Terminal characteristic of Thevenin equivalent network is: o Any intersection of characteristics gives one solution. Electric Circuits 2 Chapter 8. Network Theorems 21 KNTU 3. Thevenin-Norton equivalent network theorem COOLLARY Let linear “time-invariant” network ॉ be connected by 2 of its terminals, 1 and 1’, to an arbitrary load. Let Eoc(s) be Lap Laplace place ttransform ran nsforrm o off o open-circuit pen-circu u it voltage eoc(t) aatt terminals terminals 1 and and 1’ 1’ (voltage when no current flows into ॉ thr through rough 1 and d 1’). 1’). Let Isc(s) be Laplace aplacce ttransform ransform of of current current isc((t) t) flowing out of 1 and d into into 1’ 1’ when when lload oad d is shorted. shortted. Let Zeq = 1/Yeq be impedance (seen between terminals 1 and 1’) of network obtained from ॉ by setting: all independent sources to 0 and all initial conditions to 0. Electric Circuits 2 Chapter 8. Network Theorems 22 KNTU 3. Thevenin-Norton equivalent network theorem COOLLARY Under these conditions, whatever load may be, V(s) across 1 and 1’ and I(s) through 1 and 1’ remain unchanged by: nged when network netw workk ॉ is is replaced replacced b y: either itss Thevenin equivalent network Thevenin e quivalent n etworkk or its Norton orton eequivalent quiivale ent network. Furthermore: Above formula is especially useful in computing Thevenin equivalent network in circuits with dependent sources. Electric Circuits 2 Chapter 8. Network Theorems 23 KNTU 3. Thevenin-Norton equivalent network theorem ¾ In general, Thevenin equivalent impedance can always be obtained by: 1. 2. 3. setting to 0 all independent sources of ॉ, connecting a “test” current source It(s) to terminals 1 and 1’, and using node de analysis anallysis tto o calculate caalculate Vt(s), Laplace Laplace ttransform ransfo of zero-state responsee tto o It((s). s ). Then: Electric Circuits 2 Chapter 8. Network Theorems 24 KNTU Chapter Contents 0. Introduction 1. The substitution theorem 2. The superposition theorem 3. Thevenin-Norton equivalent network ton e quivalent n etwork ttheorem heore em 4. The reciprocityy theorem Electric Circuits 2 Chapter 8. Network Theorems 25 KNTU 4. The reciprocity theorem RECIPROCITY THEOREM y Consider an LTI network ॉܴ which consists of: ¾ resistors, ¾ inductors, ¾ coupled inductors, ductors, ¾ capacitors, and ¾ transformers ers only. on nly. y Elements below w aree rules ruless o out: ut : ¾ gyrators, ¾ dependent sources, and ¾ independent sources. y ॉܴ is in zero state and is not degenerate. y Connect 4 wires to ॉܴ thus obtaining 2 pairs of terminals 11’ and 22’. Electric Circuits 2 Chapter 8. Network Theorems 26 KNTU 4. The reciprocity theorem Statement 1 o Connect a voltage source e0 to terminals 11’ and observe zero-state current response j2 in a short circuit connected 22’.. ted to 22 o terminals term minals 22’ 22’ Next, connect ssame ame vvoltage oltage ssource ource e0 tto ero-staate current currrent response ݆1 iin and observe zero-state n a short shorrt ted to 11’. circuit connected eoreem asserts asserts tthat: hat: Reciprocity theorem and nd ology aand nd e lement values of ॉܴ an whatever topology element whatever waveform e0 of source: Electric Circuits 2 Chapter 8. Network Theorems 27 KNTU 4. The reciprocity theorem Statement 2 Connect a current source i0 to terminals 11’ and observe zero-state voltage response v2 across open-circuited d terminals 22’. 22 2. o tterminals erm minals 22’ 22’ Next, connect same same current current source source i0 tto and observe zero-state ero-staate voltage volttage response ݒෝ1 aacross cross d terminals 11’. open-circuited eoreem aasserts sserts tthat: hat: Reciprocity theorem whatever topology element ology aand nd e lement values of network netwo ork ॉܴ and whatever waveform i0 of source: Electric Circuits 2 Chapter 8. Network Theorems 28 KNTU 4. The reciprocity theorem Statement 3 Connect a current source i0 to terminals 11’ and observe zero-state current response j2 in a short circuit connected onnected to 22’. 22 . o tterminals erminaals 22’ 22’ Next, connect a vvoltage oltage ssource ource e0 tto and observe zero-state ero-staate response resp ponse ݒෝ1 acrosss d terminals 11’. open-circuited eoreem aasserts sserts tthat: hat: Reciprocity theorem whatever topology ology aand nd element element values of network netwo ork ॉܴ and whatever waveform of source, if i0(t) and e0(t) are equal for all t: Electric Circuits 2 Chapter 8. Network Theorems 29 KNTU 4. The reciprocity theorem ¾ In Statement 1: 9 We observe short-circuit currents. 9 Assertion says that if voltage source e0 is interchanged zero-impedance ed for a zero o-im mped dancce ammeter, amme eter, reading of aammeter will mmeter w ill not not change. change. 9 Source and d meter are zero-impedance ed devices. evices. ¾ In Statement 2: 9 We observe open-circuit voltages. 9 Assertion says that if current source i0 is interchanged for an infinite-impedance voltmeter, readings of voltmeter will not change. 9 Source and meter are infinite-impedance devices. Electric Circuits 2 Chapter 8. Network Theorems 30 KNTU 4. The reciprocity theorem ¾ In Statement 3: 9 For both measurements, there is: o o an infinite impedance connected to 11’, a zero impedance mp pedan nce cconnected onnected to to 22’. 22’. Electric Circuits 2 Chapter 8. Network Theorems 31 KNTU 4. The reciprocity theorem Reciprocity in terms of network functions Statement 1 y We define transfer admittance from 11’ to 22’ as: We define transfer 11’ nsferr admittance admittance ffrom rom 22’ 22’ to 1 1’ aas: s: Reciprocity theorem asserts that: Electric Circuits 2 Chapter 8. Network Theorems 32 KNTU 4. The reciprocity theorem Reciprocity in terms of network functions Statement 2 o We define transfer impedance p from 11’ to 22’ as: We define transfer nsferr iimpedance mpedance ffrom rom 22’ 22’ to 11’ 11 1’ as: as: Reciprocity theorem asserts that: Electric Circuits 2 Chapter 8. Network Theorems 33 KNTU 4. The reciprocity theorem Reciprocity in terms of network functions Statement 3 We define transfer current ratio from 11’ to 22’ as: We define transfer 11’ nsferr voltage voltage ratio ratio ffrom rom 22’ 22’ to 1 1’ aas: s: Reciprocity theorem asserts that: Electric Circuits 2 Chapter 8. Network Theorems 34 KNTU 4. The reciprocity theorem o Any network which satisfies reciprocity theorem is called a reciprocal network. o Reciprocity theorem guarantees that any network made of LTI resistors, capacitors, inductors, ductors, coupled coup pled d iinductors, nd ducttors, and d ttransformers ransformers is a reciprocal network. o It is also a fact that some LTI networks tha that at contain contain d dependent e p e nd sources are reciprocal, whereas others ereaas oth hers are not.. Electric Circuits 2 Chapter 8. Network Theorems 35 KNTU 4. The reciprocity theorem Example: Statement 1: ¾ Terminal pairs 11’ and 22’ are obtained by performing pliers mingg p lie ers entries. en ntrie es. Electric Circuits 2 Chapter 8. Network Theorems 36 KNTU 4. The reciprocity theorem Example: Statement 2: ¾ For the sake of variety, let us now pick soldering-iron entries. Electric Circuits 2 Chapter 8. Network Theorems 37 KNTU 4. The reciprocity theorem Example: Statement 3: ¾ For variety, let us pick 11’ to be defined by a soldering-iron entry, pick 22’ be defined pliers p ick 2 2’ tto ob ed efined as as a p liers eentry. ntry Electric Circuits 2 Chapter 8. Network Theorems 38