A very comprehensive introduction to circuit analysis which is very educational for all students

Circuit Theory

1. CIRCUIT THEORY CHAPTER 2 METHODS OF AC CIRCUIT ANALYSIS Electrical and Electronic Engineering

2. INDEPENDENT SOURCES An ideal independent source is an active element that provides a constant voltage or current that is completely independent of any other circuit element Figure 2.1 Independent sources: (a) constant or time-varying dc voltage, (b) constant dc voltage (c) ac voltage (d) dc or ac current 2

3. DEPENDENT SOURCES An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current. Dependent sources are useful in modeling elements such as transistors, operational amplifiers, and integrated circuits 3

4. DEPENDENT SOURCES An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current. Dependent sources are useful in modeling elements such as transistors, operational amplifiers, and integrated circuits 1. A voltage-controlled voltage source (VCVS) 2. A current-controlled voltage source (CCVS) 3. A voltage-controlled current source (VCCS) 4. A current-controlled current source (CCCS) 4

5. DEPENDENT SOURCES An ideal dependent (or controlled) source is an active element in which the source quantity is controlled by another voltage or current. Dependent sources are useful in modeling elements such as transistors, operational amplifiers, and integrated circuits Figure 2.2 Dependent sources: (a) VCVS (b) VCCS (c) CCVS (d) CCCS 5

6. DEPENDENT SOURCES EXAMPLE 1: In the circuit below, what is the voltage v across the 5 Ω resistor? 6

7. DEPENDENT SOURCES EXAMPLE 2: What is the output voltage, V out , for the circuit of below given that K a =130. 7

8. IMPEDANCES AND ADMITTANCES 8

9. MESH (LOOP) ANALYSIS The following steps provide a format which simplifies the process of using mesh analysis: Convert all sinusoidal expressions into equivalent phasor notation. Where necessary, convert current sources into equivalent voltage sources. Redraw the given circuit, simplifying the given impedances wherever possible and labelling the impedances (Z 1 , Z 2 , etc.). Arbitrarily assign clockwise loop currents to each interior closed loop within a circuit. Show the polarities of all impedances using the assumed current directions 9

10. MESH (LOOP) ANALYSIS Apply Kirchhoff’s voltage law to each closed loop in the circuit, writing each equation as follows: 10 If the current directions are originally assigned in a clockwise direction, then the resulting linear equations may be simplified to the following format: ƩZ x -summation of all impedances around loop x . ƩZ x -y is the summation of impedances common to loop x and loop y . ƩE x is the summation of voltage rises in the direction of the assumed current I x . If a voltage source has a polarity such that it appears as a voltage drop i n the assumed current direction, then the voltage is given a negative sign.

11. MESH (LOOP) ANALYSIS 11 Where Z jj is the summation of all impedances around loop j . The sign in front of all loop impedances will be positive. Z jk ( j ≠ k ) is the negative of the sum of impedances in the branch common to both loops j and k . All common impedance terms in the linear equations are given negative signs. Note that Z jk = Z kj E j is the summation of voltage rises in the direction of the assumed current I j . If a voltage source has a polarity such that it appears as a voltage drop in the assumed current direction, then the voltage is given a negative sign.

12. MESH (LOOP) ANALYSIS 12

13. MESH (LOOP) ANALYSIS 13 EXAMPLE 1 Solve for the loop equations in the circuit of the figure below

14. MESH (LOOP) ANALYSIS 14 EXAMPLE 1 Solve for the loop equations in the circuit of the figure below (left) SOLUTION 1.5 Step 1: The current source is first converted into an equivalent voltage source as shown in the figure above (middle). Steps 2 and 3: Next the circuit is redrawn as shown in the figure (right). The impedances have been simplified and the loop currents are drawn in a clockwise direction.

15. NODAL ANALYSIS Convert all sinusoidal expressions into equivalent phasor notation. If necessary, convert voltage sources into equivalent current sources. Redraw the given circuit, simplifying the given impedances wherever possible and relabeling the impedances as admittances (Y 1 , Y 2 , etc.). Select and label an appropriate reference node. Arbitrarily assign subscripted voltages (V 1 , V 2 , etc.) to each of the remaining n nodes within the circuit. Indicate assumed current directions through all admittances in the circuit. If an admittance is common to two nodes, it is considered in each of the two node equations. Apply Kirchhoff’s current law to each of the n nodes in the circuit. 15

16. NODAL ANALYSIS 16 Node 1: Node 2: Node n: ƩY x is the summation of all admittances connected to node x . The sign in front of all node admittances will be positive.   ƩY x – y is the summation of common admittances between node x and node y . If there are no common admittances between two nodes, this term is set to zero. All common admittance terms in the linear equations will have a negative sign.   ƩI x is the summation of current sources entering node x . If a current source leaves the node, the current is given a negative sign.

17. NODAL ANALYSIS 17 Where Y jj is the summation of all admittances connected to node j . The sign in front of all node admittances will be positive. Y jk (( j ≠ k )) is the summation of common admittances between node j and node k . If there are no common admittances between two nodes, this term is simply set to zero. All common admittance terms in the linear equations will have a negative sign. I j is the summation of current sources entering node j . If a current source leaves the node, the current is given a negative sign.

18. NODAL ANALYSIS 18 IN MATRIX FORM

19. NODAL ANALYSIS 19 EXAMPLE Given the circuit of the figure below , write the nodal equations and solve for the node voltages. SOLUTION

20. NODAL ANALYSIS 20 EXAMPLE Given the circuit of the figure below (left), write the nodal equations and solve for the node voltages. SOLUTION The circuit is redrawn in the figure on the right, showing the nodes and a simplified representation of the admittances. The nodal equations are written as: Node 1:   Node 2: Solving simultaneously gives: V 1 = 4.243 V  135º   V 2 = 6.324 V  –161.57º

21. NODAL ANALYSIS 21 Problem 1 W rite the nodal equations for the circuit below.

22. Example 22 Write the nodal equations for the circuit of the Figure below.

23. CIRCUIT THEORY CHAPTER 3 AC NETWORK THEOREMS Electrical and Electronic Engineering