SUPERPOSITION THEOREM

Definition: A linear network which contains two or more independent source can be analyzed to obtain the various voltages and branch currents by allowing the source to act one at a time, then superposition one at the time.

The principles apply of the linear relationship between current and voltages are:

1.      The Superposition Theorem states that a circuit can be analyzed with only one source of power at a time, the corresponding component voltages and currents algebraically added to find out what they'll do with all power sources in effect.

2.      To negate all but one power source for analysis, replace any source of voltage (batteries) with a wire; replace any current source with an open (break).

NORTON THEOREM

Definition: Norton's Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single current source and parallel resistance connected to a load.

Norton equivalent circuit

Four steps to follow for Norton’s Theorem:

1.    Find the Norton source current by removing the load resistor from the original circuit.

2.    Find the Norton current (I_N) through a short wire jumping across the open connection points where the load resistor used to be.

3.    Find the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.

4.    Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance. The load resistor re-attaches between the two open points of the equivalent circuit.

 

MAXIMUM POWER TRANSFER

Definition: The Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power.

For example, Thevenin equivalent circuit show the Maximum Power Transfer Theorem tells us that the load resistance (R_L) resulting in greatest power dissipation is equal in value to the Thevenin resistance (R_TH) , in this case, 0.8 Ω:

The formula of the Power maximum;

THEVENIN THEOREM

Definition:  Thevenin’s Theorem states that it is possible to simplify any linear circuit, no matter how complex, to an equivalent circuit with just a single voltage source and series resistance connected to a load.

Four steps to follow for Thevenin’s Theorem:

1.    Find the Thevenin source by removing the load resistor in the original circuit.

2.    Find the Thevenin resistor (R_TH) by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open points of the equivalent circuit.

3.    Find Thevenin voltage (V_TH) across the open connection points where the load resistor used to be.

4.    Draw the Thevenin equivalent circuit, with the Thevenin Voltage (V_TH) source in series with the Thevenin resistance (R_TH). The load resistor (R_L) re-attaches between the two open points of the equivalent circuit.

KIRCHOFF LAW

KIRCHOFF CURRENT LAW (KCL)

A node with four connected branches

 I1 + I2 = I3 + I4

I1 + I2 - I3 - I4 = 0

Total current = 0

Total Current Out = Total Current In

KIRCHOFF VOLTAGE LAW (KVL)

Definition: Kirchhoff’s Current Law states that the algebraic sum of currents entering a node is  zero. The sum of the currents entering a node is equal to the sum of the currents leaving a node.

NODE AND MESH

• NODE VOLTAGE METHOD

In the node voltage method, one of the principle nodes is selected as the reference and equations base on Kirchoff Current Law (KCL) are written at the other principles nodes.

FIGURE 1

The network shown in figure 1 contains five nodes, where 11 and 22 are simple node and 1, 2, and E₀ are principal node.
Node E0 selected as the reference for voltage V1 and V2. KCL requires that the total out of node 1 be zero.

Total current out of node 2 must be zero.

• MESH CURRENT METHOD

The Mesh Current Method, also known as the Loop Current Method, is quite similar to the Branch Current method in that it uses simultaneous equations, Kirchhoff's Voltage Law, and Ohm's Law to determine unknown currents in a network. Steps to follow for the “Mesh Current” method of analysis: 

1. Draws mesh currents in loops of circuit, enough to account for all components.
2.  Label resistor voltage drop polarities based on assumed directions of mesh currents.
3. Write KVL equations for each loop of the circuit, substituting the product IR for E in each resistor term of the equation. Where two mesh currents intersect through a component, express the current as the algebraic sum of those two mesh currents (i.e. I₁ + I₂) if the currents go in the same direction through that component. If not, express the current as the difference (i.e. I₁ - I₂).
4. Solve for unknown mesh currents (simultaneous equations).
5.  If any solution is negative, then the assumed current direction is wrong.
6. Algebraically add mesh currents to find current in components sharing multiple mesh currents.
7.  Solve for voltage drops across all resistors (E=IR).

DC EQUIVALENT CIRCUIT AND NETWORK THEOREMS

INTRODUCTION

Network analysis is any structured technique used to mathematically analyze a circuit (a “network” of interconnected components). Quite often the technician or engineer will encounter circuits containing multiple sources of power or component configurations which defy simplification by series/parallel analysis techniques. In those cases, he or she will be forced to use other means. This chapter presents a few techniques useful in analyzing such complex circuits. There are six types of analysis methods for resistive circuits which are:

1. Node and Mesh
2. Kirchoff Laws
3. Thevenin's Law
4. Maximum Power Transfer
5. Norton's Theorem
6. Superposition Theorem