FIRST ORDER CIRCUITS

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can, therefore, be described using only a first order differential equation. The two possible types of first-order circuits are:
     1. RC (resistor and capacitor) 
     2. RL (resistor and inductor)
RL and RC circuits is a term we will be using to describe a circuit that has either a) resistors and inductors (RL), or b) resistors and capacitors (RC).


RL Circuits

An RL Circuit has at least one resistor (R) and one inductor (L). These can be arranged in parallel, or in series. Inductors are best solved by considering the current flowing through the inductor. Therefore, we will combine the resistive element and the source into a Norton Source Circuit. The Inductor then, will be the external load to the circuit.

Example of RL circuit.

RC Circuits

An RC circuit is a circuit that has both a resistor (R) and a capacitor (C). Like the RL Circuit, we will combine the resistor and the source on one side of the circuit, and combine them into a thevenin source.

Example of RC circuit.

NORTON'S THEOREM

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. Just as with Thevenin's Theorem, the qualification of “linear” is identical to that found in the Superposition Theorem: all underlying equations must be linear (no exponents or roots). 

Remember that a current source is a component whose job is to provide a constant amount of current, outputting as much or as little voltage necessary to maintain that constant current. As with Thevenin's Theorem, everything in the original circuit except the load resistance has been reduced to an equivalent circuit that is simpler to analyze. 

THEVENIN'S THEOREM

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. The qualification of “linear” is identical to that found in the Superposition Theorem, where all the underlying equations must be linear (no exponents or roots).  
Thevenin's Theorem is especially useful in analyzing power systems and other circuits where one particular resistor in the circuit (called the “load” resistor) is subject to change, and re-calculation of the circuit is necessary with each trial value of load resistance, to determine voltage across it and current through it. 

SUPER POSITION

The superposition theorem for electrical circuits states that for a linear system the response (voltage or current) in any branch of a bilateral linear circuit having more than one independent source equals the algebraic sum of the responses caused by each independent source acting alone, where all the other independent sources are replaced by their internal impedances.

To ascertain the contribution of each individual source, all of the other sources first must be "turned off" (set to zero) by:
1.  Replacing all other independent voltage sources with a short circuit (thereby eliminating difference of potential i.e. V=0; internal impedance of ideal voltage source is zero (short circuit)).

2.  Replacing all other independent current sources with an open circuit (thereby eliminating current i.e. I=0; internal impedance of ideal current source is infinite (open circuit).
The superposition theorem is very important in circuit analysis. It is used in converting any circuit into its Norton equivalent or Thevenin equivalent.

The theorem is applicable to linear networks (time varying or time invariant) consisting of independent sources, linear dependent sources, linear passive elements (resistors, inductors, capacitors) and linear transformers.

MESH ANALYSIS

A mesh is a loop that does not contain other loop within it. It applies KVL to find the unknown currents and only applicable to a circuit that is a planar.

• Planar Circuit- is one that can be drawn in a with no branches crossing one another.

• Non Planar Circuit- can be handled using nodal analysis, but they will not be considered in this text.

Steps to determine mesh currents:

1. Assign mesh currents to the n meshes
2. Apply KVL to each of the n meshes. Use ohm's law to express the voltages in terms of the mesh currents.
3. Solve the resulting n simultaneous equations to get the mesh currents.

Mesh Analysis with Current Sources 

• Case 1: When a current source exists only in one mesh. Set the mesh current is equal to the the current source.

• Case 2: When a current source is exists between two meshes it will form a Super Mesh. Super mesh, created by excluding the current.

Properties of a Super Mesh:

1. Current source in the super mesh provides the constraint equation necessary to solve for the mesh currents.
2. A super mesh has no current of its own,
3. A super mesh requires the application of KVL and KCL.

NODAL ANALYSIS

Nodal Analysis with Voltage source

Two Cases to consider:

• Case 1: A voltage source is connected between a non-reference node and a reference node. The voltage of a non-reference is equal to the voltage source.

• Case 2: A voltage source is connected between two non-reference node will form the Super Node. Super node is formed by enclosing voltage source connected between two non-reference nodes and any elements connected in a parallel with it.

Properties of a Super Node:

1. The voltage source inside the super node provides a constant equation needed to solve for the node voltages.
2. It has no voltage of it's own.
3. It requires the application of both KCL and KVL. 

BASIC LAWS

These basic laws, known as Ohm's law and Kirchhoff's law, form the foundation upon which electric circuit analysis is built.

Ohm's Law
  Ohm's law states that  the voltage v across a resistor is directly proportional to the current i flowing through the resistor.

  The resistance R of an element denotes the ability to resist the flow of electric current; it is measured in ohms. It is represented by this equation

v = iR

Nodes, Branches and Loops 

Branch represents a single element such as a voltage source or a resistor.

Node is the point of connection between two or more branches.

Loop is any closed path in a circuit.

In this figure, an example of branches, node, and loop in an electrical circuit is shown.

Kirchhoff's Law

  There are two parts of Kirchhoff's law, the Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL).

• Kirchhoff's Current Law states that the algebraic sum currents entering a node is zero. In alternative form, the sum of current entering a node is equal to the sum of the current leaving the node.
• Kirchhoff's Voltage Law states that the algebraic sum of all voltages around a close path is zero.

The sum of voltage drop = The sum of the voltage rises.



Situations often arise in circuit analysis when the resistors are neither in  parallel nor in series. It can be simplified using three-terminal equivalent networks. These are the WYE or TEE  and DELTA and PI network, they can occur by themselves or as part of a large network.

In this figure, we can identify a Wye network if it is forming a letter Y and Tee if the form of the network is a letter T.

In identifying Delta network is when the form of the network is a triangle. In PI network by simply identifying it in pi form.

Delta to Wye Conversion

Wye to Delta Conversion

R_a = R₁R₂ + R₂R₃ + R₃R₁/ R₁

R_b = R₁R₂ + R₂R₃ + R₃R₁/ R₂

R_c = R₁R₂ + R₂R₃ + R₃R₁/ R₃