Nodal Analysis

The aim of nodal analysis is to determine the voltage at each node relative to the reference node (or ground). Once you have done this you can easily work out anything else you need.

A node is all the points in a circuit

that are directly interconnected.

We assume the interconnections

have zero resistance so all points

within a node have the same

voltage. 

Five nodes: A, · · · ,E.

Ohm’s Law: VBD = IR5

KVL: VBD = VB − VD

KCL: Total current exiting any closed region is zero.

To find the voltage at each node, the first
step is to label each node with its voltage as follows

1. Pick any node as the voltage reference. Label its voltage as 0 V.

2. Assign node voltages to the other nodes.

3. Apply KCL to each node other than the reference node; express currents in terms of node voltages.

4. Solve the resulting system of linear equations for the nodal voltages.

Now, let us apply nodal analysis with voltage sources:

(Here, there are two cases we need to consider)

• Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one.

• Case 2: The voltage source is connected between two nonreferenced nodes: a generalized node (supernode) is formed.

Here’s the example of supernode:

A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.

The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the voltage source. 

You already know the steps on how to apply nodal analysis in a circuit, so we will apply the same steps on the circuit with inductor and capacitor.

Just remember these formulas: for the inductor, use the formula jwL, where w is the omega, and L is the inductance, and for the capacitor, use the formula 1/jwC, where C is the capacitance. Getting the equivalent impedance( resistance in ac circuit), helps you in solving problems as you encounter this kind of circuit in the sample problems .

Learnings:


Nodal Analysis  is based on Kirchoff's Current Law.

A set of independent equations are generated by applying KCL at each unknown node.
When a KCL equation is written at a node, the unknowns in the equation are node voltages.

All node voltages are expressed relative to a ground node, therefore one node is assigned
to be ground or zero volts. Also in this chapter we learn that computing in AC is just the same as in DC.

"Impedance Combinations"

 Now that we've seen how series and parallel AC circuit analysis is not fundamentally different than DC circuit analysis, it should come as no surprise that series-parallel analysis would be the same as well, just using complex numbers instead of scalar to represent voltage, current, and impedance.

Being a series-parallel combination circuit, we must reduce it to a total impedance in more than one step. The first step is to combine as a series combination of impedances, by adding their impedances together. Then, that impedance will be combined in parallel with the impedance of the resistor, to arrive at another combination of impedances. Finally, that quantity will be added to the impedance  to arrive at the total impedance.

Learnings:

In this chapter we learn about series-parallel combination circuit.  Ac Analysis are the same with Dc Analysis. It only involves complex numbers.

"Impedance & Admittance"

AC steady-state analysis using phasors
allows us to express the relationship
between current and voltage using a
formula that looks likes Ohm’s law:
V = I Z
 Z is called impedance.

In the preceding section, we obtained the voltage-current relations for the three passive elements as,

V = RI, V = jωLI, V =I/jωC

These equations may be written in terms of the ratio of the phasor voltage to the phasor current as,

V/I = R,  V/I = jωL,  V/I = 1/jωC

From these three expressions, we obtain Ohm’s law in phasor form for any type of element as

Z = V/I    or   V = ZI

The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms.

The admittance Y is the reciprocal of impedance, 

measured in siemens (S).

The admittance Y of an element (or a circuit) is the ratio of the phasor current through it to the phasor voltage across it, or

Y   =   1/Z   =   I/V

Some Thoughts on Impedance

• Impedance depends on the frequency ω.

• Impedance is (often) a complex number.

• Impedance allows us to use the same

solution techniques for AC steady state as

we use for DC steady state.

"Phasor in Circuit Elements"

Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor.How to represent a voltage or current in the phasor or frequency domain, one may legitimately ask how we apply this to circuits involving the passive elements R, L, and C. What we need to do is to transform the voltage-current relationship from the time domain to the frequency domain for each element. Again, we will assume the passive sign convention.

We begin with the resistor. If the current through a resistor 

R is i = Im cos(ωt + φ),

 the voltage across it is given by Ohm’s law as 

v = iR = RIm cos(ωt + φ)

Figure representation: 

Sinusoid & Phasor

Introduction

Any (non-pathological) function can be represented by an appropriate superposition of sine waves of

different frequencies, either as a Fourier series or as a Fourier integral. The response of a linear network toa composite sinusoidal signal can be determined for each sinusoidal term separately, and the overall

response then obtained by superposition. This is in part the basis of the fundamental importance of the

analysis of the response of a linear circuit to a single sinusoid of

 arbitrary frequency. Circuit behavior for

composite signals can be inferred from knowledge of this sinusoidal response. Hence what is considered

in this course primarily is the analysis of a linear circuit with an arbitrary

single-frequency sinusoidal

excitation.

We now begin the analysis of circuits in which the source voltage or current is time-varying. In this chapter, we are particularly interested in sinusoidally time-varying excitation, or simply, excitation by a sinusoid. 

  

A sinusoid is a signal that has a form of the sine or cosine function.

v(t) = Vm sinωt

Vm=the amplitude of the sinusoid

ω=the angular frequency in radians/s

ωt=the argument of the sinusoid

A periodic function is one that satisfies v(t) = v(t+nT), for all 

t and for all integers n.

While...

A phasor is a complex number that represents the amplitude and phase of a sinusoid.

Arithmetic With Complex Numbers

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. We remember the equation for the inductor:

If we apply KCL on the node that forms the positive terminal of the voltage source, we can solve to get the following differential equation:

An RL parallel 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. Then if we apply KVL around the resulting loop, we get the following equation:

A parallel RC Circuit

Inductors

An inductor, also called a coil or reactor, is a passive two-terminal electrical component which resists changes in electric current passing through it. It consists of a conductor such as a wire, usually wound into a coil. When a current flows through it, energy is stored temporarily in a magnetic field in the coil. When the current flowing through an inductor changes, the time-varying magnetic field induces a voltage in the conductor, according to Faraday’s law of electromagnetic induction, which opposes the change in current that created it.

An inductor is characterized by its inductance, the ratio of the voltage to the rate of change of current, which has units of henries (H). Along with capacitors and resistors, inductors are one of the three passive linear circuit elements that make up electric circuits. Inductors are widely used inalternating current (AC) electronic equipment, particularly in radio equipment. They are used to block the flow of AC current while allowing DC to pass; inductors designed for this purpose are called chokes. They are also used in electronic filters to separate signals of different frequencies, and in combination with capacitors to make tuned circuits, used to tune radio and TV receivers.

Inductance (L) results from the magnetic field around a current-carrying conductor; the electric current through the conductor creates a magnetic flux. Mathematically speaking, inductance is determined by how muchmagnetic flux φ through the circuit is created by a given current i.

 (1)

For materials that have constant permeability with magnetic flux (which does not include ferrous materials) L is constant and (1) simplifies to

Any wire or other conductor will generate a magnetic field when current flows through it, so every conductor has some inductance. The inductance of a circuit depends on the geometry of the current path as well as the magnetic permeability of nearby materials. In inductors, the wire or other conductor is shaped to increase the magnetic field. Winding the wire into a coil increases the number of times the magnetic flux lines link the circuit, increasing the field and thus the inductance. The more turns, the higher the inductance. The inductance also depends on the shape of the coil, separation of the turns, and many other factors. By winding the coil on a "magnetic core" made of a ferromagnetic material like iron, the magnetizing field from the coil will induce magnetization in the material, increasing the magnetic flux. The high permeability of a ferromagnetic core can increase the inductance of a coil by a factor of several thousand over what it would be without it.

Any change in the current through an inductor creates a changing flux, inducing a voltage across the inductor. By Faraday's law of induction, the voltage induced by any change in magnetic flux through the circuit is[4]

From (1) above

     (2)

So inductance is also a measure of the amount of electromotive force (voltage) generated for a given rate of change of current. For example, an inductor with an inductance of 1 henry produces an EMF of 1 volt when the current through the inductor changes at the rate of 1 ampere per second. This is usually taken to be the constitutive relation (defining equation) of the inductor.

The dual of the inductor is the capacitor, which stores energy in an electric field rather than a magnetic field. Its current-voltage relation is obtained by exchanging current and voltage in the inductor equations and replacing L with the capacitance C.

Learning's:

An capacitor and inductor are similar in the way that a capacitor resists a change of a voltage and an inductor resists a change in current. Capacitors are widely used to clean up a power supply line. Inductors are used in switching power supplies where a relatively constant current is passed through an inductor. A switching power supply works in that a switch is opened and closed very quickly. When the switch is closed, the inductor is 'charged'. When the switch is open, the energy is drawn from the inductor into the load. Usually such a power supply is being decoupled with a capacitor to create a stable power supply line.In capacitors, energy is stored in their electric field while in inductors, energy is stored in their magnetic field.