Series-Parallel Circuits

Most practical circuits have both series and parallel components. Components that are connected in series will share a common path. Components that are connected in parallel will be connected across the same two nodes. Identifying Series-Parallel Relationships

You can frequently simplify circuit analysis by combining series and parallel components. An important analysis method is to form an equivalent circuit. An equivalent circuit is one with characteristics that are electrically the same as another circuit, but is generally simpler. Combination Circuits

For example: There are no electrical measurements that can distinguish the boxes. Equivalent Circuits is equivalent to

Again, there are no electrical measurements that can distinguish the boxes. Equivalent Circuits is equivalent to Another example:

There are no electrical measurements that can distinguish between the three boxes. Equivalent Circuits is equivalent to

Kirchhoff’s voltage law and Kirchhoff’s current law can be applied to any circuit, including combination circuits. For example, applying KVL, the path shown will have a sum of 0 V. Kirchhoff’s Law So will this one!

Circuit Theorems 8

 Introduction  Linearity property  Superposition  Source transformations  Thevenin’s theorem  Norton’s theorem  Maximum power transfer Circuit Theorems9

Introduction 10 A large complex circuits A large complex circuits Simplify circuit analysis Simplify circuit analysis Circuit Theorems ‧ Thevenin’s theorem ‧ Norton theorem ‧ Circuit linearity ‧ Superposition ‧ source transformation ‧ max. power transfer ‧ Thevenin’s theorem ‧ Norton theorem ‧ Circuit linearity ‧ Superposition ‧ source transformation ‧ max. power transfer

Linearity Property Circuit Theorems11 Homogeneity property (Scaling) Additivity property

 A linear circuit is one whose output is linearly related (or directly proportional) to its input  Fig. 4.1 Circuit Theorems12 v V0V0 I0I0 i

 Linear circuit consist of ●linear elements ●linear dependent sources ●independent sources  Nonlinear: ●Exponential, square, logarithmic ●Example Circuit Theorems13

Example 1  For the circuit shown find I 0 when v s =12V and v s =24V. Circuit Theorems14

Example 1  KVL Eqs(1.1) and (1.3) we get Circuit Theorems15 (1.1) (1.2) (1.3)

Example 1 Eq(1.1), we get When Showing that when the source value is doubled, I 0 doubles. Circuit Theorems16

Example 2  Assume I 0 = 1 A and use linearity to find the actual value of I 0 in the circuit shown. Circuit Theorems17

Example 2 Circuit Theorems18

Superposition Superposition  The superposition principle states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltages across (or currents through) that element due to each independent source acting alone.  Turn off, killed, inactive source: ●independent voltage source: 0 V (short circuit) ●independent current source: 0 A (open circuit)  Dependent sources are left intact. Circuit Theorems19

 Steps to apply superposition principle: 1.Turn off all independent sources except one source. Find the output (voltage or current) due to that active source using nodal or mesh analysis. 2.Repeat step 1 for each of the other independent sources. 3.Find the total contribution by adding algebraically all the contributions due to the independent sources. Circuit Theorems20

How to turn off independent sources  Turn off voltages sources = short voltage sources; make it equal to zero voltage  Turn off current sources = open current sources; make it equal to zero current Circuit Theorems21

 Superposition involves more work but simpler circuits.  Superposition is not applicable to the effect on power. Circuit Theorems22

Example 3  Use the superposition theorem to find v in the circuit shown. Circuit Theorems23

Example 3 Since there are two sources, let Voltage division to get Current division, to get Hence And we find Circuit Theorems24

Example 4  Find I 0 in the circuit shown using superposition. Circuit Theorems25

Example 4 Circuit Theorems26 Fig. 4.10

Example 4 Circuit Theorems27

Source Transformation  A source transformation is the process of replacing a voltage source v s in series with a resistor R by a current source i s in parallel with a resistor R, or vice versa Circuit Theorems28

Circuit Theorems29

Equivalent Circuits Circuit Theorems30 i i vv v i vsvs -i s

 Arrow of the current source positive terminal of voltage source  Impossible source Transformation ●ideal voltage source (R = 0) ●ideal current source (R=  ) Circuit Theorems31

Example 6  Use source transformation to find v o in the circuit shown. Circuit Theorems32

Example 6 Circuit Theorems33

Example 6 we use current division in Fig. (c) to get and Circuit Theorems34

Example 7  Find v x in the next figure using source transformation Circuit Theorems35

Example 7 Applying KVL around the loop in Fig (b) gives (7.1) Appling KVL to the loop containing only the 3V voltage source, the resistor, and v x yields (7.2) Circuit Theorems36

Example 7 Substituting this into Eq.(7.1), we obtain Alternatively thus Circuit Theorems37

Thevenin’s Theorem Thevenin’s Theorem  Thevenin’s theorem states that a linear two- terminal circuit can be replaced by an equivalent circuit consisting of a voltage source V Th in series with a resistor R Th where V Th is the open circuit voltage at the terminals and R Th is the input or equivalent resistance at the terminals when the independent source are turn off. Circuit Theorems38

Property of Linear Circuits Circuit Theorems39 i v v i Any two-terminal Linear Circuits + - V th I sc Slope=1/R th

Circuit Theorems40

How to Find Thevenin’s Voltage  Equivalent circuit: same voltage-current relation at the terminals.  Open circuit voltage at a-b Circuit Theorems41

How to Find Thevenin’s Resistance  Circuit Theorems42

CASE 1  If the network has no dependent sources: ●Turn off all independent source. ●R TH : can be obtained via simplification of either parallel or series connection seen from a-b Circuit Theorems43

CASE 2  If the network has dependent sources ●Turn off all independent sources. ●Apply a voltage source v o at a-b ●Alternatively, apply a current source i o at a-b Circuit Theorems44

 The Thevenin’s resistance may be negative, indicating that the circuit has ability providing power Circuit Theorems45

Simplified circuit Voltage divider Circuit Theorems46

Example 8  Find the Thevenin’s equivalent circuit of the circuit shown, to the left of the terminals a-b. Then find the current through R L for R L = 6, 16, and 36 . Circuit Theorems47

Find R th Circuit Theorems48

Find V th Circuit Theorems49