THREE PHASE CIRCUITS

INTRODUCTION

A single- phase ac power system consists of a generator connected through a pair of wires (a transmission line) to a load.

It contains two identical sources (equal magnitude and the same phase) which are connected to two loads by two outer wires and the neutral. For example, the normal household system is a single-phase three-wire system because the terminal voltages have the same magnitude and the same phase. Such a system allows the connection of both 120-V and 240-V appliances.

BALANCED THREE-PHASE VOLTAGES

Three-phase voltages are often produced with a three-phase ac generator (or alternator)whose cross-sectional view

The generator basically consists of a rotating magnet(called the rotor)surrounded by a stationary winding (called the stator). Three separate windings or coils with terminals a-a, b-b, and c-c are physically placed 120◦ apart around the stator. Terminals a and a, for example, stand for one of the ends of coils going into and the other end coming out of the page. As the rotor rotates, its magnetic field “cuts” the flux from the three coils and induces voltages in the coils. Because the coils are placed 120◦ apart, the induced voltages in the coils are equal in magnitude but out of phase by 120◦ . Since each coil can be regarded as a single-phase generator by itself, the three-phase generator can supply power to both single-phase and three-phase loads.

A typical three-phase system consists of three voltage sources con- nected to loads by three or four wires (or transmission lines). (Three- phase current sources are very scarce.) A three-phase system is equiv- alent to three single-phase circuits. The voltage sources can be either wye-connected 

Balanced phase voltages are equal in magnitude and are out of phase with each other by 120◦.

Since the three-phase voltages are 120◦ out of phase with each other, there are two possible combinations. One possibility is and expressed mathematically as

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

The phase sequence is determined by the order in which the phasors pass through a fixed point in the phase diagram.

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A balanced load is one in which the phase impedances are equal in magnitude and in phase.

For a balanced wye-connected load,

Z1 = Z2 = Z3 = ZY

where ZY is the load impedance per phase. For a balanced delta-connected load, 

Za = Zb = Zc = Zdelta

where Zdelta is the load impedance per phase in this case.

LEARNING:

In this lesson I learned to determine the phase sequence of the set of voltages. It is appropriate to mention here that a balanced delta-connected load is more common than a balanced wye-connected load. I learned also that there are four possible connections in three phase source and three phase load. 

Y-Y connection (i.e., Y-connected source with a Y-connected load). 

Y-delta connection.

delta-delta connection.

delta-Y connection.

Summary

The phase sequence is the order in which the phase voltages of a three-phase generator occur with respect to time. In an abc sequence of balanced source voltages, Van leads Vbn by 120◦, which in turn leads Vcn by 120◦. In anacb sequence of balanced voltages, Van leads Vcn by 120◦, which in turn leads Vbn by 120◦. 

 A balanced wye- or delta-connected load is one in which the three- phase impedances are equal. 

 The easiest way to analyze a balanced three-phase circuit is to transform both the source and the load to a Y-Y system and then analyze the single-phase equivalent circuit. Table 12.1 presents a summary of the formulas for phase currents and voltages and line currents and voltages for the four possible configurations.

POWER FACTOR AND COMPLEX POWER

POWER FACTOR


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

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= Vrm Irms

The apparent power(inVA ) is the product of the rms values of voltage and current.

The apparent power is so called because it seems apparent that the power should be the voltage-current product, by analogy with dc resistive cir- cuits. It is measured in volt-amperes or VA to distinguish it from the average or real power, which is measured in watts. The power factor is dimensionless, since it is the ratio of the average power to the apparent power.

COMPLEX POWER

Considerable effort has been expended over the years to express power relations as simply as possible. 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(inVA) is the product of the rms voltage phasor and the complex conjugate of the rms current phasor. As a complex quantity,its real part is real power P and its imaginary part is reactive power Q.

 Given two of these items, the other two can easily be obtained from the triangle. , when S lies in the first quadrant, we have an inductive load and a lagging pf. When S lies in the fourth quadrant, the load is capacitive and the pf is leading. It is also possible for the complex power to lie in the second or third quadrant. This requires that the load impedance have a negative resistance, which is possible with active circuits.

LEARNING:

The apparent power(inVA )is the product of the rms values of voltage and current.

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

Apparent Power is simply:   P_apparent= V_RMS I_RMS

Therefore we can express Average Power as:    P_average= P_apparent × pf

We can also say:    pf = P_average/P_apparent

In other words, the power factor is the percentage of Apparent Power
that is actually being absorbed by the load.

AC Power Analysis

Instantaneous Power

The instantaneous power p(t) absorbed by an element
is the product of the instantaneous voltage v(t) across the element and the instantaneous current i(t) through it. Assuming the passive sign convention, 

p(t)= v(t)i(t)

 Let the voltage and current at the terminals of the circuit be 

v(t)= Vm cos(ωt +θv)

i(t)= Im cos(ωt +θi)

where Vm and Im are the amplitudes (or peak values), and θv and θi are the phase angles of the voltage and current, respectively. The instantaneous power absorbed by the circuit is 

p(t)= v(t)i(t)= VmIm cos(ωt +θv)cos(ωt +θi)

We apply the trigonometric identity 

cosAcosB =1/2[cos(A−B)+cos(A+B)]

and express Eq. (11.3) as

p(t)=1/2VmIm cos(θv −θi)+1/2VmIm cos(2ωt +θv +θi)

Average Power

The average power is the average of the instantaneous power over one period.

MAXIMUM AVERAGE POWER TRANSFER

maximum average power transfer, the load impedance ZL must be equal to the complex conjugate of the Thevenin impedance ZTh.

This result is known as the maximum average power transfer theorem for the sinusoidal steady state. Setting RL = RTh and XL =−XTh

Pmax = |VTh|^2/8RTh

EFFECTIVE OR RMS VALUE

The effective value of a periodic current is the dc current that delivers the same average power to a resistor as the periodic current.

The idea of effective value arises from the need to measure the effectiveness of a voltage or current source in delivering power to a resistive load.

The effective value of a periodic signal is its root mean square(rms)value.

LEARNINGS:

In this topic I learned that the instanstanues power is valid for signals of any waveform. The unit of Instantaneous power is VA. Complex power is the product of the complex effective voltage and the complex effective conjugate current. The unit of complex power is VA. 

Thevenins Theorem on AC Analysis

           Solving on Thevenins theorem in ac anaylsis, is just the same process in dc analysis, which we've  done to tackle. The difference is in ac, there's a presence of impedance which involve in solving series-parallel combination.

Thevenins Theorem states that:

            Any combination of sinusoidal AC sources and impedances with two terminals can be replaced by a single voltage source  e and a single series impedance z. The value of e is the open circuit voltage at the terminals, and the value of z is e divided by the current with the terminals short circuited. In this case, that impedance evaluation involves a series-parallel combination.

Summary of Thevenin's Theorem

    Remember that the Thevenin equivalent circuit is always a voltage source in series with a resistance regardless of the original circuit that it replaces. The significance of Thevenin's theorem is that the equivalent circuit can replace the original circuit as far as any external load is concerned. Any load connected between the terminals of a Thevenin equivalent circuit experiences the same current and voltage as if it were connected to the terminals of the original circuit.

A summary of steps for applying Thevenin's theorem follows.


Step 1. Open the two terminals between which you want to find the Thevenin circuit, This is done by removing the component from which the circuit is to be viewed.


Step 2. Determine the voltage across the two open terminals.

Step 3. Determine the impedance viewed from the two open terminals with ideal voltage sources replaced with shorts and ideal current sources replaced with opens (zeroed).

Step 4. Connect Vr7, andZ,Tinseies to produce the complete Thevenin equivalent circuit.

Here are some example of solving thevenins theorem(note that they are the same process on solving the thevenins theorem through ac analysis in a dc analysis):

Insight Learnings:

       Remember that the Thevenin equivalent circuit is always a voltage source in series with a resistance regardless of the original circuit that it replaces. The significance of Thevenin's theorem is that the equivalent circuit can replace the original circuit as far as any external load is concerned. Any load connected between the terminals of a Thevenin equivalent circuit experiences the same current and voltage as if it were connected to the terminals of the original circuit.

Source Transformation on AC Analysis

The source transformation in ac analysis involves transforming a voltage source in series with an impedance to a current source in parallel with an impedance.


where:

Vs = Zs Is       <--->      Is = Vs / Zs


as a recall to the dc analysis :

 DC 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.

but in AC Source transformation A voltage source with impedance Z in series is the same as a current source with an impedance Z in parallel.

Here are some examples of source transformation solving in dc analysis( note that they are the same on solving in ac analysis):

Insight Learnings:

In dc analysis:

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.

In ac analysis:

A voltage source with impedance Z in series is the same as a current source with an impedance Z in parallel.

Transform a voltage source in series with an impedance to a current source in     parallel with an impedance for simplification or vice versa.

Superposition Theorem On AC Analysis

   
       The superposition theorem in ac circuit is just the same on dc circuits. What is add on ac is just there's a presence of Impedance which are the conductor and capacitor. They had a variable and value. But the process is just the same, which all sources (except dependent sources) other than the one being considered are eliminated and then replace current sources with opens, replace voltage sources with shorts. Based on the book, The superposition theorem can be stated as follows:

        The current in any given branch of a multiple-source circuit can be found by determining the currents in that particular branch produced by each source acting alone, with all other sources replaced by their internal impedances. The total current in the given branch is the phasor sum of the individual source currents in that branch.

The procedure for the application of the superposition theorem is as follows:

Step 1. Leave one of the sources in the circuit, and replace all others with their internal impedance. For ideal voltage sources, the internal impedance is zero. For ideal current sources, the intemal impedance is infinite. We will call this procedure zeroing the source.

Step 2. Find the cunent in the branch of interest produced by the one remaining source.

Step 3. Repeat Steps 1 and 2 for each source in turn. When complete, you will have a number of current values equal to the number of sources in the circuit.

Step 4. Add the individual current values as phasor quantities.


Here are an example of Superpostion Theorem, but these videos are just in dc circuit, as like what I've said earlier in ac circuit there's a presence of impedances and solving these are just the same in dc analysis.

Insight Learning:

To calculate the contribution of each source independently, all the other sources must be removed and replaced without affecting the final result.

When removing a voltage source, its voltage must be set to zero, which is equivalent to replacing the voltage source with a short circuit.

When removing a current source, its current must be set to zero, which is equivalent to replacing the current source with an open circuit. 

Mesh Analysis on AC Circuit

Mesh Analysis on AC Circuit


Since KVL is valid for phasors, we can analyze AC circuits by  MESH analysis

Example :

Determine current Io in the circuit of Fig. 10.7 using mesh analysis.

Solution:
Applying KVL to mesh 1, we obtain

 (8+j10−j2)I1 −(−j2)I2 −j10I3 = 0     (10.3.1)

For mesh 2, (4−j2−j2)I2 −(−j2)I1 −(−j2)I3 +20 90◦ = 0     (10.3.2) 

For mesh 3, I3 = 5.

 Substituting this in Eqs. (10.3.1) and (10.3.2), we get

 (8+j8)I1 +j2I2 = j50 (10.3.3) j2I1 +(4−j4)I2 =− j20−j10    (10.3.4)

Equations (10.3.3) and (10.3.4) can be put in matrix form as 

Example 2: