1. Passive components and circuits - CCP Lecture 3 Introduction

2. /24 2 Index Theorems for electric circuit analysis  Kirchhoff theorems  Superposition theorem  Thevenin theorem  Norton theorem

3. /24 3 Kirchhoff theorems http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kirchhoff.html http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Kirchhoff.html The theorems are applicable in circuit analysis for insulated circuits (the circuit is not exposed to external factors as electrical or magnetic fields). Kirchhoff’s voltage law : The algebraic sum of the voltages at any instant around any loop in a circuit is zero. Kirchhoff’s current law The algebraic sum of the currents at any instant at any node in a circuit is zero.

4. /24 4 Applying of Kirchhoff’s Theorem If a circuit has l branches and n nodes, then the complete description of its operation is obtained by writing KVL for l-n+1 loops and KCL for n-1 nodes. The loops must form an independent system. n Prior to the analysis of an electric circuit, the conventional directions of the currents in the circuit are not known. So, before writing the equations (Kirchhoff’s laws) for each loop, a positive arbitrary direction is selected for each branch of the circuit. n After performing the analysis of the circuit, if the value of the current is positive, the arbitrary and conventional directions of the current flow are identical. If the value of the current is negative, the conventional direction is opposite to the arbitrary selected direction.

5. /24 5 Applying of Kirchhoff’s Theorem n Step I – choosing the voltages and currents arbitrary directions n Step II – choosing the loop’s cover direction n Step III – writing the Kirkhhoff’s theorems

6. /24 6 Solving equation systems n In order to solve the equations, the Ohm’s Law is applied and the voltage across the resistors are substituted. n It is obtained a system with three equations and three variables, I R1, I R2 and I R3.

7. /24 7 The System Solutions The solutions are:  I R1  -6 mA  I R2  -13 mA  I R3  7 mA The voltages across the resistances:  V R1  -2 V  V R2  -2 V  V R3  7 V

8. /24 8 Linear and nonlinear circuits If the transmittances defined for a circuit are constant (are represented with linear segments in v-i, v-v or i-i planes), are called linear transmittances. A circuit or a component with only linear transmittances is called linear circuit or linear component. Important: generally, electronics devices and circuits made with them are nonlinear. The method used to approximate a nonlinear circuit operation with a linear circuit operation is called linearization.

9. /24 9 The Superposition Theorem The Superposition theorem states that the response in a linear circuit with multiple sources can be obtained by adding the individual responses caused by the separate independent sources acting alone. The source passivation  the sources are replaced by their internal resistance. By passivation, the ideal voltage source is replaced with a short-circuit, and the ideal current source is replaced with an open-circuit.

10. /24 10 The Superposition Theorem  +

11. /24 11 Thevenin’s Theorem Any two-terminal, linear network of sources and resistances can be replaced by a single voltage source in series with a resistance. The voltage source has a value equal to the open- circuit voltage appearing at the terminals of the network. The resistance value is the resistance that would be measured at the network’s terminals for passivated circuit. The source passivation= the sources are replaced by their internal resistance By passivation, the ideal voltage source is replaced with a short-circuit, and the ideal current source is replaced with an open-circuit.

12. /24 12 Thevenin’s Theorem Vo and Ro must be determined.

13. /24 13 Calculus of open-circuit voltage In order to calculate the open-circuit voltage, the Kirchhoff’s theorems can be applied. The superposition theorem will also be applied.

14. /24 14 The superposition theorem for calculus of open-circuit voltage

15. /24 15 The equivalent resistance calculus The circuit is passivated. A test voltage is applied (V TEST ) The current through the terminals is determinate (I TEST ) R O = V TEST / I TEST

16. /24 16 Conclusion n From the R3 resistance point of view, the equivalent circuit will have the same effect:

17. /24 17 Norton’s Theorem Any two-terminal, linear network of sources and resistances may be replaced by a single current source in parallel with a resistance. The value of the current source is the current flowing between the terminals of the network when they are short-circuited. The resistance value is the resistance that would be measured at the terminals of the network when all the sources have been replaced by their internal resistances.

18. /24 18 Norton’s Theorem Io and Ro must be determined.

19. /24 19 Calculus of short-circuit current In order to calculate the short-circuited current, the Kirchhoff’s theorems can be applied. The Superposition theorem!

20. /24 20 The superposition theorem for calculus of the short-circuit current

21. /24 21 Calculus of equivalent resistance

22. /24 22 Conclusion From the R3 resistance point of view, the equivalent circuit will have the same effect:

23. /24 23 Transfer from Thevenin to Norton equivalence Once having an equivalent circuit (Thevenin or Norton), the other one is obtained using the relation: For previous example:

24. /24 24 Recommendation for individual study For the following circuit determine the current through R resistor and the voltage across it, using:  Kirchhoff’s theorem  Thevenin and/or Norton equivalence (use the superposition theorem)