Three - Phase Voltages

In electrical engineering, three-phase electric power systems have at least three conductors carrying alternating current voltages that are offset in time by one-third of the period. A three-phase system may be arranged in delta (∆) or star (Y) (also denoted as wye in some areas). A wye system allows the use of two different voltages from all three phases, such as a 230/400V system which provides 230V between the neutral (centre hub) and any one of the phases, and 400V across any two phases. A delta system arrangement only provides one voltage magnitude, however it has a greater redundancy as it may continue to operate normally with one of the three supply windings offline, albeit at 57.7% of total capacity. Harmonic currents in the neutral may become very large if non-linear loads are connected.


"Balanced Three-Phase Voltages"


Three-phase voltages are often produced with a three-phase ac generator or alternator whose cross-sectional view is shown below,

The voltage sources can be either wye-connected as shown in Fig.(a) or delta-connected as in Fig (b).

Balanced phase voltages are equal in magnitude and are out

of phase with each other by 120◦.

The phase sequence is the time order in which the voltages pass through their respective maximum values.

A balanced load is one in which the phase impedances

are equal in magnitude and in phase.

Types of Connections:

• Balanced Wye-Wye Connection

• Balanced Wye-Delta Connection

• Balanced Delta-Delta Connection

• Balanced Delta-Wye Connection

Power Factor and Complex Power

We see that if the voltage and current at the terminals of
a circuit are,

v(t) = Vm cos(ωt + θv)

and

i(t) = Im cos(ωt + θi)

The average power is a product of two terms. The product Vrms Irms is known as the apparent power S. The factor cos(θv − θi) is called the power factor (pf).

S = Vrms Irms

The apparent power (in VA) is the product ofthe rms values ofvoltage and current.

The power factor is dimensionless, since it is the ratio of the average power to the apparent power,

pf =P/S= cos(θv − θi)

The angle θv − θi is called the power factor angle, since it is the angle whose cosine is the power factor.

The power factor is the cosine ofthe phase difference between voltage and current. It is also the cosine ofthe angle ofthe load impedance.

"Complex Power"

Power engineers have coined the term complex power, which they use to find the total effect of parallel loads. Complex power is important in power analysis because it contains all the information pertaining to the power absorbed by a given load.


Complex power (in VA) is the product ofthe rms voltage phasor and the complex conjugate ofthe rms current phasor. As a complex quantity, its real part is real power P and its imaginary part is reactive power Q.


Introducing the complex power enables us to obtain the real and reactive powers directly from voltage and current phasors.

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.

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.

Applications of Maximum Power Transfer Theorem:

      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 for example to match an Amplifier with a Loudspeaker to yield maximum power to the speaker and thus produce maximum sound.

In some situations Transformer Coupling are also used to yield maximum power to the load when the matching of Load and Source impedance is not possible for example is the amplifier is of 1000 Ohms and the speaker if of 10 ohms.

      The application of Maximum Power Theorem is done only under the conditions when the maximum performance is desired over the overall efficiency of the circuit because as we discussed above the efficiency of a circuit under maximum power transfer condition is only 0.5. So, 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.

Thévenin's Theorem

Thévenin's Theorem for AC circuits with sinusoidal sources is very similar to the theorem we have learned for DC circuits. The only difference is that we must consider impedanceinstead of resistance. Concisely stated, Thévenin's Theorem for AC circuits says:

Any two terminal linear circuit can be replaced by an equivalent circuit consisting of a voltage source (V_Th) and a series impedance (Z_Th). 

In other words, Thévenin's Theorem allows one to replace a complicated circuit with a simple equivalent circuit containing only a voltage source and a series connected impedance. The theorem is very important from both theoretical and practical viewpoints.

It is important to note that the Thévenin equivalent circuit provides equivalence at the terminals only. Obviously, the internal structure of the original circuit and the Thévenin equivalent may be quite different. And for AC circuits, where impedance is frequency dependent, the equivalence is valid at one frequency only.

Using Thévenin's Theorem is especially advantageous when:

· we want to concentrate on a specific portion of a circuit. The rest of the circuit can be replaced by a simple Thévenin equivalent.

· we have to study the circuit with different load values at the terminals. Using the Thévenin equivalent we can avoid having to analyze the complex original circuit each time. 

We can calculate the Thévenin equivalent circuit in two steps:

1. Calculate Z_Th. Set all sources to zero (replace voltage sources by short circuits and current sources by open circuits) and then find the total impedance between the two terminals.

2. Calculate V_Th. Find the open circuit voltage between the terminals. 

Superposition Theorem

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:

• 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)).

• 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)).

This procedure is followed for each source in turn, then the resultant responses are added to determine the true operation of the circuit. The resultant circuit operation is the superposition of the various voltage and current sources.

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.

Another point that should be considered is that superposition only works for voltage and current but not power. In other words the sum of the powers of each source with the other sources turned off is not the real consumed power. To calculate power we should first use superposition to find both current and voltage of each linear element and then calculate the sum of the multiplied voltages and currents.

Source Transformation

Source transformation is the process of replacing a voltage source vs in series with a resistor R by a current source is in parallel with a resistor R, or vice versa.

Source transformations are easy to perform as long as there is a familiarity with Ohm's law. If there is a voltage source in series with an impedance, it is possible to find the value of the equivalent current source in parallel with the impedance by dividing the value of the voltage source by the value of the impedance. The converse also applies here: if a current source in parallel with an impedance is present, multiplying the value of the current source with the value of the impedance will result in the equivalent voltage source in series with the impedance.

Learnings 
Using Source transformations are easy to perform as long as you have knowledge with Ohm's Law. Just transform a voltage source in series with an impedance to a current source in parallel with the impedance, in AC source transformation is also the same in the DC source transformation. 

Mesh Analysis

Mesh analysis (or the mesh current method) is a method that is used to solve planar circuits for the currents (and indirectly the voltages) at any place in the circuit. Planar circuits are circuits that can be drawn on a plane surface with no wires crossing each other. A more general technique, called loop analysis (with the corresponding network variables called loop currents) can be applied to any circuit, planar or not. Mesh analysis and loop analysis both make use of Kirchhoff’s voltage law to arrive at a set of equations guaranteed to be solvable if the circuit has a solution. Mesh analysis is usually easier to use when the circuit is planar, compared to loop analysis.

Determine the currents in the example using mesh analysis.

We first define mesh currents I1 and I2 matching those currents defined in the figure.

Wring the KVL equation for the two meshes gives

mesh1 = 12 - 2I1 - (2+j2)(I1-I2) = 0

mesh2 = -(2+j2)(I1-I2) - (-j4+4+j6)I2 = 0

The equation in matrix form are 

Solving for the mesh currents gives

The third phasor current I3 is related to the mesh currents by

Use these steps to sytematically apply Mesh Analysis when solving a circuit problem:

1. Find all meshes and label them. Show mesh currents going clockwise.
2. Write down everything you know by inspection. Current sources give you information about
   mesh currents. This means that you will not have to generate as many KVL equations.
3. If there are dependent sources in the problem, express their values in terms of mesh currents.
   (There may be enough information to express the dependent source as a constant, which is even better.)
4. Write a KVL equation in each mesh where the mesh current is unknown.
   You may have to temporarily remove current sources to visualize where the loops are.
5. After step 4 you should have N equations if you started with N unknowns. Using Algebra
   or Linear Algebra you can now solve for all the mesh currents.
6. If you know all the mesh currents for the circuit, then anything else can be found (e.g. voltages,
   power, current through an element, etc.)

Learnings:

Mesh Analysis is a step-by-step approach to solving circuits.
It is based on Kirchoff's Voltage Law.

With Mesh Analysis the mesh currents of the circuit are the unknowns in the equations.Solving Mesh analysis in AC is just the same as in DC.

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