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.

Capacitors

Capacitors are components designed to take advantage of this phenomenon by placing two conductive plates (usually metal) in close proximity with each other. There are many different styles of capacitor construction, each one suited for particular ratings and purposes. For very small capacitors, two circular plates sandwiching an insulating material will suffice. For larger capacitor values, the "plates" may be strips of metal foil, sandwiched around a flexible insulating medium and rolled up for compactness. The highest capacitance values are obtained by using a microscopic-thickness layer of insulating oxide separating two conductive surfaces. In any case, though, the general idea is the same: two conductors, separated by an insulator.

The schematic symbol for a capacitor is quite simple, being little more than two short, parallel lines (representing the plates) separated by a gap. Wires attach to the respective plates for connection to other components. An older, obsolete schematic symbol for capacitors showed interleaved plates, which is actually a more accurate way of representing the real construction of most capacitors:

When a voltage is applied across the two plates of a capacitor, a concentrated field flux is created between them, allowing a significant difference of free electrons (a charge) to develop between the two plates:

As the electric field is established by the applied voltage, extra free electrons are forced to collect on the negative conductor, while free electrons are "robbed" from the positive conductor. This differential charge equates to a storage of energy in the capacitor, representing the potential charge of the electrons between the two plates. The greater the difference of electrons on opposing plates of a capacitor, the greater the field flux, and the greater "charge" of energy the capacitor will store.

Just as Isaac Newton's first Law of Motion ("an object in motion tends to stay in motion; an object at rest tends to stay at rest") describes the tendency of a mass to oppose changes in velocity, we can state a capacitor's tendency to oppose changes in voltage as such: "A charged capacitor tends to stay charged; a discharged capacitor tends to stay discharged." Hypothetically, a capacitor left untouched will indefinitely maintain whatever state of voltage charge that its been left it. Only an outside source (or drain) of current can alter the voltage charge stored by a perfect capacitor:

Practically speaking, however, capacitors will eventually lose their stored voltage charges due to internal leakage paths for electrons to flow from one plate to the other. Depending on the specific type of capacitor, the time it takes for a stored voltage charge to self-dissipate can be a long time (several years with the capacitor sitting on a shelf!).

When the voltage across a capacitor is increased, it draws current from the rest of the circuit, acting as a power load. In this condition the capacitor is said to becharging, because there is an increasing amount of energy being stored in its electric field. Note the direction of electron current with regard to the voltage polarity:

Conversely, when the voltage across a capacitor is decreased, the capacitor supplies current to the rest of the circuit, acting as a power source. In this condition the capacitor is said to be discharging. Its store of energy -- held in the electric field -- is decreasing now as energy is released to the rest of the circuit. Note the direction of electron current with regard to the voltage polarity:

.

An obsolete name for a capacitor is condenser or condensor. These terms are not used in any new books or schematic diagrams (to my knowledge), but they might be encountered in older electronics literature. Perhaps the most well-known usage for the term "condenser" is in automotive engineering, where a small capacitor called by that name was used to mitigate excessive sparking across the switch contacts (called "points") in electromechanical ignition systems.

Maximum Power Transfer Theorem

Maximum power transfer theorem deals with the power transferred to the load on a circuit with a network of various sources or components on it. The maximum power transfer theorem defines the condition under which the maximum power is transferred to the load in a circuit.

Note: Here we are talking about maximum power transferred to the load only, not about the maximum power transferred to the load and internal components or resistance of the source combined, Under the condition of Maximum power transfer we only deal with the power transferred to the load and does not consider the power dissipated in internal circuits or resistance of the source so we are not talking about the maximum efficiency of power transfer but instead maximum possible power transfer from a source to a load.

The Maximum Power Transfer Theorem states that:

The power transferred from a source or circuit to a load is maximum when the resistance of the load is made equal or matched to the internal resistance of the source or circuit providing the power to the load.

Let,
 V= EMF supplied to the load.
RL = Load resistance.
Ri = Internal resistance of the source.

I = Current flowing through the load, internal resistance and the source of the circuit.
PL = Power transferred to the load.
Pi = Power dissipated at internal resistances.

Then,                                                    

Power transferred to the load = PL = I²RL 

or,
 =  = 

Now using the theorems of Differential calculus , If we keep the RL variable and want to calculate the maximum value of PL then we need to differentiate the PL with respect to RL and equate it with zero.
Thus,
Under Maximum power transfer to load condition:

or,

or,

Learning's:

The Maximum Power Transfer Theorem has a wide range of usage on real life situation. The theorem is used to maximize the power output to a load from any circuit. So they can be used to design circuits where the maximum output performance is desired. 

Maximum power transfer theorem is applied in radio electronics; for example: In Antenna Signal amplifier for radio and TV receivers; and various other fields where maximum performance is required but the maximum efficiency is not desired for example to match an Amplifier with a Loudspeaker to yield maximum power to the speaker and thus produce maximum sound.

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).

Contrasting our original example circuit against the Norton equivalent: it looks something like this:

. . . after Norton conversion . . .

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. Also similar to Thevenin's Theorem are the steps used in Norton's Theorem to calculate the Norton source current (INorton) and Norton resistance (RNorton).

As before, the first step is to identify the load resistance and remove it from the original circuit:

Then, to find the Norton current (for the current source in the Norton equivalent circuit), place a direct wire (short) connection between the load points and determine the resultant current. Note that this step is exactly opposite the respective step in Thevenin's Theorem, where we replaced the load resistor with a break (open circuit):

With zero voltage dropped between the load resistor connection points, the current through R1 is strictly a function of B1's voltage and R1's resistance: 7 amps (I=E/R). Likewise, the current through R3 is now strictly a function of B2's voltage and R3's resistance: 7 amps (I=E/R). The total current through the short between the load connection points is the sum of these two currents: 7 amps + 7 amps = 14 amps. This figure of 14 amps becomes the Norton source current (INorton) in our equivalent circuit:

Remember, the arrow notation for a current source points in the direction opposite that of electron flow. Again, apologies for the confusion. For better or for worse, this is standard electronic symbol notation. Blame Mr. Franklin again!

To calculate the Norton resistance (RNorton), we do the exact same thing as we did for calculating Thevenin resistance (RThevenin): take the original circuit (with the load resistor still removed), remove the power sources (in the same style as we did with the Superposition Theorem: voltage sources replaced with wires and current sources replaced with breaks), and figure total resistance from one load connection point to the other:

Now our Norton equivalent circuit looks like this:

Learning's:

Thevenin’s theorem and Norton’s theorem are two important theorems used in fields such as electrical engineering, electronic engineering, physics, circuit analysis and circuit modeling. These two theorems are used to reduce large circuits to simple voltage sources, current sources and resistors. These theories are very useful in calculating and simulating changes for large scale circuits. Norton’s theorem uses a current source, whereas  Thevenin’s theorem uses a voltage source and also the Norton’s equivalent circuit and Thevenin’s equivalent circuit can be easily interchanged.

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). If we're dealing with passive components (such as resistors, and later, inductors and capacitors), this is true. However, there are some components (especially certain gas-discharge and semiconductor components) which are nonlinear: that is, their opposition to current changes with voltage and/or current. As such, we would call circuits containing these types of components, nonlinear circuits.

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. Let's take another look at our example circuit:

Let's suppose that we decide to designate R2 as the “load” resistor in this circuit. We already have four methods of analysis at our disposal (Branch Current, Mesh Current, Millman's Theorem, and Superposition Theorem) to use in determining voltage across R2 and current through R2, but each of these methods are time-consuming. Imagine repeating any of these methods over and over again to find what would happen if the load resistance changed (changing load resistance is verycommon in power systems, as multiple loads get switched on and off as needed. the total resistance of their parallel connections changing depending on how many are connected at a time). This could potentially involve a lot of work!

Thevenin's Theorem makes this easy by temporarily removing the load resistance from the original circuit and reducing what's left to an equivalent circuit composed of a single voltage source and series resistance. The load resistance can then be re-connected to this “Thevenin equivalent circuit” and calculations carried out as if the whole network were nothing but a simple series circuit:

. . . after Thevenin conversion . . .

The “Thevenin Equivalent Circuit” is the electrical equivalent of B1, R1, R3, and B2 as seen from the two points where our load resistor (R2) connects.

The Thevenin equivalent circuit, if correctly derived, will behave exactly the same as the original circuit formed by B1, R1, R3, and B2. In other words, the load resistor (R2) voltage and current should be exactly the same for the same value of load resistance in the two circuits. The load resistor R2 cannot “tell the difference” between the original network of B1, R1, R3, and B2, and the Thevenin equivalent circuit of EThevenin, and RThevenin, provided that the values for EThevenin and RThevenin have been calculated correctly.

The advantage in performing the “Thevenin conversion” to the simpler circuit, of course, is that it makes load voltage and load current so much easier to solve than in the original network. Calculating the equivalent Thevenin source voltage and series resistance is actually quite easy. First, the chosen load resistor is removed from the original circuit, replaced with a break (open circuit):

Next, the voltage between the two points where the load resistor used to be attached is determined. Use whatever analysis methods are at your disposal to do this. In this case, the original circuit with the load resistor removed is nothing more than a simple series circuit with opposing batteries, and so we can determine the voltage across the open load terminals by applying the rules of series circuits, Ohm's Law, and Kirchhoff's Voltage Law:

The voltage between the two load connection points can be figured from the one of the battery's voltage and one of the resistor's voltage drops, and comes out to 11.2 volts. This is our “Thevenin voltage” (EThevenin) in the equivalent circuit: