Circuits and Analog Electronics Ch 2 Basic Analysis

Circuits and Analog Electronics Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits 2. 2 Basic Nodal and Mesh Analysis 2. 3 Useful Circuit Analysis Techniques References: References Hayt-Ch 3, 4; Gao-Ch 2;

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Key Words: Words Equivalent Circuits Network Equivalent Resistance, Equivalent Independent Sources

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Network о о Two-terminal Circuits Network cо о a о d о о 5 6 15 о 5 bо о I I a о + N 1 a о + V V _ о о b b N 2

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Resistance How do we find I 1 and I 2? I I 1 I 2 R 1 R 2 + V - I 1 + I 2 = I

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Resistance i(t) + v(t) - + v(t) Req - Req is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics.

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Resistance Condition : without knowing V&I. We only know Rs Method 1 Series and parallel Resistance (source-free) Condition : without knowing Rs V&I. We only know V&I Method 2 a source I V -free Ro = Rab = V I b Method 3 source Voc source ISC

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Resistance In practice , Vs-source voltage VL- Local Let The IL-VL came : IL(A) 100 0 , Short Circuit (SC). , Open Circuit (OC). 10 VL(V)

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Resistance P 2. 1 R 2 R 1 How do we find Rab? a R 3 + - VS b Method 1 I Method 2 R 1 a R 3 V Method 3 b

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Source Transformation • Ideally: – An ideal current source has the voltage necessary to provide its rated current – An ideal voltage source supplies the current necessary to provide its rated voltage • Practice: – A real voltage source cannot supply arbitrarily large amounts of current – A real current source cannot have an arbitrarily large terminal voltage

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Source Transformation Rs Vs + - Is Rs Note: Consistency between the current source ref. direction and the voltage Source ref. terminals.

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Source How do we find I 1 and I 2? Is 1 Is 2 I 1 I 2 + R 1 R 2 V - Ieq

Ch 2 Basic Analysis Methods to Circuits 2. 1 Equivalent Circuits Equivalent Source Series Voltage Source + VS 1 - + VS 2 - + VSn - + VS - parallel Current Source IS 1 IS 2 ISn I IS RS=RS 1// RS 2//…// RSn

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Key Words: Words Branch Analysis, Nodal Analysis, Mesh (Loop) Analysis • Why? – The analysis techniques previously (voltage divider, equivalent resistance, etc. ) provide an intuitive approach to analyzing circuits – They are not systematic and cannot be easily automated by a computer • Comments: – Analysis of circuits using node or loop analysis requires solutions of systems of linear equations. – These equations can usually be written by inspection of the circuit.

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Branch Analysis P 2. 2 How do we find I 1 and I 2, I 3? KVL Mesh 1: I 1 I 2 I 3 Mesh 2: KCL

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Branch Analysis l Suppose m branches, n nodals l write KCL equation for each independent node. ——(n-1) KCL equations l write KVL equation for each independent mesh/loop ——m-(n-1) KVL equations

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Branch Analysis Here’s a quick example of a circuit that we will see later when we model the operation of transistors. For now, let’s assume ideal independent and dependent sources. Ⅰ We can write the following equations: Ⅰ: Ⅱ: Ⅲ: Ⅱ Ⅲ

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 1) Choose a reference node 500 + I 1 500 1 k V 500 - The reference node is called the ground node. I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 1) Choose a reference node 500 Ⅰ I 4 I 5 I 1 500 Ⅱ 500 + I 6 0 KCL Ⅰ: Ⅱ: Ⅲ: Ⅲ I 8 1 k V I 7 500 I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 2) Assign node voltages to the other nodes 500 V 1 I 4 1 I 5 V 2 2 500 I 6 I 8 1 k 500 I 7 V 3 3 500 0 V 1, V 2, and V 3 are unknowns for which we solve using KCL. I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 3) Apply KCL to each node other than the reference-express currents in terms of node voltages. V 1 I 4 500 V 2 500 I 7 V 3 1 I 5 2 I 6 I 8 1 k 500 0 KCL • • • Node 1: Node 2: Node 3: , , Node 1: , Node 2: , Node 3: 3 500 I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 4) Solve the resulting system of linear equations. • Node 1: V 1 500 • Node 2: I 1 1 • • 500 V 3 2 500 • Node 3: V 2 3 1 k 500 0 The left hand side of the equation: – The node voltage is multiplied by the sum of conductances of all resistors connected to the node. – The neighbourly node voltages are multiplied by the conductance of the resistor(s) connecting to the two nodes and to be subtracted. The right hand side of the equation: – The right side of the equation is the sum of currents from sources entering the node. I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 4) Solve the resulting system of linear equations. • Node 1: V 1 500 • Node 2: I 1 1 Matrix Notation(Symmetric) 500 V 3 2 500 • Node 3: V 2 3 1 k 500 I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis 4) Solve the resulting system of linear equations. • Node 1: V 1 500 • Node 2: I 1 1 G 11 V 1+G 12 V 2 +G 13 V 3 =I 11 G 21 V 2+G 22 V 2 +G 23 V 3 =I 22 G 31 V 1+G 32 V 2+G 33 V 3=I 33 500 V 3 2 500 • Node 3: V 2 3 1 k 500 I 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis What if there are dependent sources? Example: 5 m. A Node ①： V 1 Ib 1 1 k V 2 50 100 Ib 2 1 k + Vo - Node ② ： Matrix is not symmetric due to the dependent source.

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis What if there are voltage sources? 1 V 1 Node 2: Node 3: + - R 1 Ib V 2 1 k 2 0. 7 V + - V 3 3 50 100 Ib 3 k V 4 R 3 R 2 + 4 R 4 Vo 1 k - Difficulty: We do not know Ib – the current through the voltage source? Node 4: Independent Voltage Source: Equations: KCL at node 2, node 3, node 4, and Unknowns: Ib, V 2, V 3, V 4

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis What if there are voltage sources? CURRENT CONTROLLED VOLTAGE SOURCE KCL AT SUPERNODE Io=?

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Nodal Analysis Advantages of Nodal Analysis • • • Solves directly for node voltages. Current sources are easy. Voltage sources are either very easy or somewhat difficult. Works best for circuits with few nodes. Works for any circuit.

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh(Loop) Analysis 1) Identifying the Meshes 1 k V 1 1 k + - + Mesh 1 Mesh 2 1 k - V 2 Mesh: A special kind of loop that doesn’t contain any loops within it.

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh(Loop) Analysis 2) Assigning Mesh Currents 1 k V 1 1 k + - 1 k I 1 + I 2 - V 2 3) Apply KVL around each loop to get an equation in terms of the loop currents. I 1 ( 1 k + 1 k ) - I 2 1 k = V 1 For Mesh 1: -V 1 + I 1 1 k + (I 1 - I 2) 1 k = 0 For Mesh 2: (I 2 - I 1) 1 k + I 2 1 k + V 2 = 0 - I 1 1 k + I 2 ( 1 k + 1 k ) = -V 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh(Loop) Analysis 3) Apply KVL around each loop to get an equation in terms of the loop currents. 1 k I 1 ( 1 k + 1 k ) - I 2 1 k = V 1 - I 1 1 k + I 2 ( 1 k + 1 k ) = -V 2 V 1 + - 1 k 1 k I 1 4) Solve the resulting system of linear equations. I 2 + - V 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis I 1 Im 1 I 2 I 3 Mesh 1: Mesh 2: Mesh 3: Mesh 2 Im 2 Mesh 1 I 6 I 5 Mesh 3 Im 3 I 4

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis P 2. 4(P 2. 2 ) Im 1 Mesh 1: Im 2 Mesh 2:

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis What if there are current sources? P 2. 5 • The current sources in this circuit will have whatever voltage is necessary to make the current correct. • We can’t use KVL around the loop because we don’t know the voltage. P 2. 6 Mesh 1 I=? I m 2 + Im 1 Im 2 V Im 1 - Super Mesh: Mesh 2 Im 3 Super Mesh 2: Mesh 1:

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis What if there are current sources? 2 k The Supermesh surrounds this source! 12 V 2 m. A I 3 + - 2 k I 2 I 1 I 0 1 k The Supermesh does not include this source! 4 m. A

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis What if there are current sources? 2 k 2 m. A 12 V + - I 1 I 3 2 k I 0 1 k I 2 • The 4 m. A current source sets I 2: I 2 = -4 m. A • The 2 m. A current source sets a constraint on I 1 and I 3: 4 m. A I 1 - I 3 = 2 m. A • We have two equations and three unknowns. Where is the third equation?

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis What if there are current sources? P 2. 6 Im 2 + Mesh 1: Mesh 2: V Im 1 - Node 3: Im 3 Node 3 Mesh 2

Ch 2 Basic Analysis Methods to Circuits 2. 2 Useful Circuit Analysis Techniques Mesh(Loop) Analysis Dependent current source. Current sources not shared by meshes. We treat the dependent source as a conventional source. Equations for meshes with current sources We are asked for Vo. We only need to solve for I 3. Replace and rearrange Then KVL on the remaining loop(s) And express the controlling variable, Vx, in terms of loop currents

Ch 2 Basic Analysis Methods to Circuits 2. 2 Basic Nodal and Mesh Analysis Mesh (Loop) Analysis Advantages of Loop Analysis • • Solves directly for some currents. Voltage sources are easy. Current sources are either very easy or somewhat difficult. Works best for circuits with few loops. Disadvantages of Loop Analysis • Some currents must be computed from loop currents. • Choosing the supermesh may be difficult.

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Key Words: Words Linearity Superposition Thevenin’s and Norton’s theorems

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Linearity • Linearity is a mathematical property of circuits that makes very powerful analysis techniques possible. • Linearity leads to many useful properties of circuits: – Superposition: the effect of each source can be considered separately. – Equivalent circuits: Any linear network can be represented by an equivalent source and resistance (Thevenin’s and Norton’s theorems)

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Linearity • Linearity leads to simple solutions: – Nodal analysis for linear circuits results in systems of linear equations that can be solved by matrices V 1 500 1 I 1 V 2 500 V 3 3 1 k 500 I 2

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Linearity • The relationship between current and voltage for a linear elements satisfies two properties: – Homogeneity – Additivity *Real circuit elements are not linear, but can be approximated as linear

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Linearity • Homogeneity: – Let v(t) be the voltage across an element with current i(t) flowing through it. – In an element satisfying homogeneity, if the current is increased by a factor of K, the voltage increases by a factor of K. • Additivity – Let v 1(t) be the voltage across an element with current i 1(t) flowing through it, and let v 2(t) be the voltage across an element with current i 2 (t) flowing through it – In an element satisfying additivity, if the current is the sum of i 1 (t) and i 2 (t), then the voltage is the sum of v 1 (t) and v 2 (t). Example: Resistor: V = R I – If current is KI, then voltage is R KI = KV – If current is I 1 + I 2, then voltage is R(I 1 + I 2) = RI 1 + RI 2 = V 1 + V 2

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition I 2 I • Superposition is a direct consequence of linearity • It states that “in any linear circuit containing multiple independent sources, the current or voltage at any point in the circuit may be calculated as the algebraic sum of the individual contributions of each source acting alone. ”

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition How to Apply Superposition? • To find the contribution due to an individual independent source, zero out the other independent sources in the circuit. – Voltage source short circuit. – Current source open circuit. • Solve the resulting circuit using your favorite techniques. – Nodal analysis – Loop analysis

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition For the above case: Zero out Vs, we have : Zero out E 2, we have : I I 2’’ R 1 R 3 I 2’ R 2 E 2 R 1 + Vs_ R 3 R 2 E 2

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition P 2. 7 2 k 4 m. A 12 V - 2 m. A 1 k I 0 + 2 k

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition P 2. 7 2 k 2 m A KVL for mesh 2: I 1 1 k I’o Mesh 2 I 2 2 k

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition P 2. 7 2 k 4 m. A I 1 1 k I’’ 0 Mesh 2 KVL for mesh 2: I 2 2 k

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition P 2. 7 12 V 2 k 1 k I’’’ 0 Mesh 2 + I 2 KVL for mesh 2: 2 k

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Superposition P 2. 7 2 k 4 m. A 12 V - 2 m. A 1 k + 2 k I 0 = I’ 0 +I’’ 0+ I’’’ 0 = -16/3 m. A

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Thevenin’s theorem • Any circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single voltage source and a single resistor • Thevenin’s theorem implies that we can replace arbitrarily complicated networks with simple networks for purposes of analysis

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Thevenin’s theorem Independent Sources RTh Voc Circuit with independent sources + - Thevenin equivalent circuit

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Thevenin’s theorem No Independent Sources RTh Circuit without independent sources Thevenin equivalent circuit

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Norton’s theorem • Very similar to Thevenin’s theorem • It simply states that any circuit with sources (dependent and/or independent) and resistors can be replaced by an equivalent circuit containing a single current source and a single resistor

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Norton’s theorem Norton Equivalent: Independent Sources Isc Circuit with one or more independent sources RTh Norton equivalent circuit

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Norton’s theorem Norton Equivalent: No Independent Sources RTh Circuit without independent sources Norton equivalent circuit

Ch 2 Basic Analysis Methods to Circuits • Motivation of applying the Thevenin’s theorem and Norton’s theorem: – Sometimes, in a complex circuit, we are only interested in working out the voltage /current or power being consumed by a single load (resistor); – We can then treat the rest of the circuit (excluding the interested load) as a voltage (current) source concatenated with a source resistor; – Simplify our analysis. • Thevenin’s theorem: – Given a linear circuit, rearrange it in the form of two networks of A and B connected by two wires. Define Voc as the open-circuit voltage which appears across the terminals of A when B is disconnected. Then all currents and voltage in B will remain unchanged, if we replace all the independent current or voltage source in A by an independent voltage source which is in series with a resistor (RTh). • The Norton’s Theorem: – Given a linear circuit, rearrange it in the form of two networks of A and B connected by two wires. Define isc as the short-circuit current which appears across the terminals of A when B is disconnected. Then all currents and voltage in B will remain unchanged, if we replace all the independent current or voltage source in A by an independent current source isc which is in parallel with a resistor (RN).

Ch 2 Basic Analysis Methods to Circuits • Equivalent transform between a Thevenin equivalent circuit and a Norton equivalent circuit Thevenin equivalent If Norton equivalent and =

Ch 2 Basic Analysis Methods to Circuits • Therefore, for a source transform: Thevenin Norton: Thevenin :

Ch 2 Basic Analysis Methods to Circuits • Example 45 (P 88) Find a Thevenin and Norton equivalent circuit for the following circuit excluding A B

Ch 2 Basic Analysis Methods to Circuits Thevenin equivalent Norton equivalent

Ch 2 Basic Analysis Methods to Circuits • Application of Thevenin’s theorem when there are only independent sources. step : 1) Determine of two connection points between network A and network B. 2) Determine of two connection points by replacing the voltage source by a shortcircuit or the current source by a open-circuit. • Similarly, for Norton’s Theorem steps : 1) Determine between the two connection points between network A and B. 2) Determine by two connection points by replacing the voltage source by a shortcircuit or the current source by an open circuit.

Ch 2 Basic Analysis Methods to Circuits • Practice 4. 6 (P 90)

Ch 2 Basic Analysis Methods to Circuits Thevenin equivalent :

Ch 2 Basic Analysis Methods to Circuits • When there are multiple independent source, we shall use “superposition”. Example 4. 7 (P 91) Find the Thevenin and Norton equivalent for the network excluding the resistor. To determine , when only 4 v voltage source is functioning.

Ch 2 Basic Analysis Methods to Circuits When only 2 m. A current source is functioning : Therefore,

Ch 2 Basic Analysis Methods to Circuits To determine Thevenin equivalent:

Ch 2 Basic Analysis Methods to Circuits Norton equivalent Try to look into this problem from the Norton approach. (Figure out the Norton equivalent circuit first)

Ch 2 Basic Analysis Methods to Circuits When there are both independent source and dependent source. —Dependent source cannot be “zero out” as far as its control variable is not zero. —Similar as before, —But we cannot determine directly, however, we can use Example 4. 8 (P 92) — Determine the Thevenin equivalent of the following circuit To determine since , applying KVL to the supermesh:

Ch 2 Basic Analysis Methods to Circuits To determine Therefore its Thevenin equivalent is:

Ch 2 Basic Analysis Methods to Circuits • When there are only dependent sources: – VOC = 0 – RTH can be determined by implying a test (imaginary) voltage across the two terminals. 3Ω i Example 4. 9 (p 93) 1. 5 i Open circuit : To determine RTh, applying an independent circuit source x. A. As : Apply KCL: Giving : i 1. 5 i 3Ω 2Ω v x. A 0. 6Ω 2Ω

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Thevenin’s theorem • Circuits with independent sources – Compute the open circuit voltage, this is Voc – Compute the Thevenin resistance (set the sources to zero – short circuit the voltage sources, open circuit the current sources), and find the equivalent resistance, this is RTh • Circuits with independent and dependent sources: – Compute the open circuit voltage – Compute the short circuit current – The ratio of the two is RTh • Circuits with dependent sources only* – Voc is simply 0 – RTh is found by applying an independent voltage source (V volts) to the terminals and finding voltage/current ratio * Not required by this course.

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Norton’s theorem • Circuits with independent sources, w/o dependent sources – Compute the short circuit current, this is Isc – Compute the Thevenin resistance (set the sources to zero – short circuit the voltage sources, open circuit the current sources), and find the equivalent resistance, this is RN • Circuits with both independent and dependent sources – Find Voc and Isc – Compute RN= Voc/ Isc • Circuits w/o independent sources* – Apply a test voltage (current) source – Find resulting current (voltage) – Compute RN

Ch 2 Basic Analysis Methods to Circuits 2. 3 Useful Circuit Analysis Techniques Maximum power transfer For every choice of RL we have a different power. How do we find the maximum value? Consider PL as a function of RL and find the maximum of such function The maximum power transfer theorem The load that maximizes the power transfer for a circuit is equal to the Thevenin equivalent resistance of the circuit. The value of the maximum power that can be transferred is

Ch 2 Basic Analysis Methods to Circuits Analysis methods Review KVL, KCL, I — V Combination rules Node method Any circuits Mesh method Superposition Thévenin Norton linear circuits