Three-Phase Circuits

Three-Phase Circuits =)

       Initially we explored the idea of three-phase power systems by connecting three voltage sources together in what is commonly known as the “Y” (or “star”) configuration. This configuration of voltage sources is characterized by a common connection point joining one side of each source. (Figure below)

Three-phase “Y” connection has three voltage sources connected to a common point.

If we draw a circuit showing each voltage source to be a coil of wire (alternator or transformer winding) and do some slight rearranging, the “Y” configuration becomes more obvious in Figure below.

Three-phase, four-wire “Y” connection uses a "common" fourth wire.

The three conductors leading away from the voltage sources (winding's) toward a load are typically called lines, while the windings themselves are typically called phases. In a Y-connected system, there may or may not (Figure below) be a neutral wire attached at the junction point in the middle, although it certainly helps alleviate potential problems should one element of a three-phase load fail open, as discussed earlier.

Three-phase, three-wire “Y” connection does not use the neutral wire.

Overview and Insights

-The conductors connected to the three points of a three-phase source or load are called lines.

-The three components comprising a three-phase source or load are called phases.

-Line voltage is the voltage measured between any two lines in a three-phase circuit.

-Phase voltage is the voltage measured across a single component in a three-phase source or load.

-Line current is the current through any one line between a three-phase source and load.

-Phase current is the current through any one component comprising a three-phase source or load.

-In balanced “Y” circuits, line voltage is equal to phase voltage times the square root of 3, while line current is equal to phase current.

 

-In balanced Δ circuits, line voltage is equal to phase voltage, while line current is equal to phase current times the square root of 3.

-Δ-connected three-phase voltage sources give greater reliability in the event of winding failure than Y-connected sources. However, Y-connected sources can deliver the same amount of power with less line current than Δ-connected sources.

Power Factor & Complex Power

Power Factor & Complex Power =)

Power Factor 

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

        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.

It is a standard practice to represent S, P, and Q in the form of

a triangle, known as the power triangle, shown below,

Overview and Insights

- Real Power (P) in measured in W, Reactive Power (Q) in VAR, and Apparent Power (S) in VA.

- For Power Factor (PF), when theta increases PF decreases, and when theta decreases PF increases.

Reactive Power:

Q = 0 for resistive loads (unity pf).

Q < 0 for capacitive loads (leading pf).

Q > 0 for inductive loads (lagging pf).

AC Power Analysis

AC Power Analysis =)

Instantaneous and Average Power

The instantaneously power, p(t) 

Instantaneous power is the power of a any object at an instant. if you differentiate work done w.r.t time it will be the instantaneous power. If the given velocity is instantaneous the power=F*v

Average Power

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

Maximum Average Power Transfer

We have already seen that an AC circuit can (at one frequency) be replaced by a Thévenin or Norton equivalent circuit. Based on this technique, and with the Maximum Power Transfer Theorem for DC circuits, we can determine the conditions for an AC load to absorb maximum power in an AC circuit. For an AC circuit, both the Thévenin impedance and the load can have a reactive component. Although these reactances do not absorb any average power, they will limit the circuit current unless the load reactance cancels the reactance of the Thévenin impedance. Consequently, for maximum power transfer, the Thévenin and load reactances must be equal in magnitude but opposite in sign; furthermore, the resistive parts -according to the DC maximum power theorem- must be equal. In another words the load impedance must be the conjugate of the equivalent Thévenin impedance. The same rule applies for the load and Norton admittances.

RL= Re{ZTh} and XL = - Im{ZTh}

The maximum power in this case:

Where V2Th and I2N represent the square of the sinusoidal peak values.

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

Overview and Insights

              Power analysis is another chapter and view of understanding in our class since it has lesser circuit analysis. Power is the most important quantity in electric utilities, electronic, and communication systems, because such systems involve transmission of power from one point to another.

Thevenins Theorem

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:

Overview and Insights

• Thevenin's Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load.
• Steps to follow for Thevenin's Theorem:
• (1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be.
• (2) Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.
• (3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit.
• (4) Analyze voltage and current for the load resistor following the rules for series circuits.

Source Transformation

Source Transformation :)

Source transformation is simplifying a circuit solution, especially with mixed sources, by transforming a voltage into a current source, and vice versa. Finding a solution to a circuit can be difficult without using methods such as this to make the circuit appear simpler. Source transformation is an application of Thévenin's theorem and Norton's theorem.

Performing a source transformation consists of using Ohm's law to take an existing voltage source in series with a resistance, and replace it with a current source in parallel with the same resistance. Remember that Ohm's law states that a voltage on a material is equal to the material's resistance times the amount of current through it (V=IR). Since source transformations are bilateral, one can be derived from the other. ^[2] Source transformations are not limited to resistive circuits however. They can be performed on a circuit involving capacitors and inductors, as long as the circuit is first put into the frequency domain. In general, the concept of source transformation is an application of Thévenin's theorem to a current source, or Norton's theorem to avoltage source.

Specifically, source transformations are used to exploit the equivalence of a real current source and a real voltage source, such as a battery. Application of Thévenin's theorem and Norton's theorem gives the quantities associated with the equivalence. Specifically, suppose we have a real current source I, which is an ideal current source in parallel with an impedance. If the ideal current source is rated at I amperes, and the parallel resistor has an impedance Z, then applying a source transformation gives an equivalent real voltage source, which is ideal, and in series with the impedance. This new voltage source V, has a value equal to the ideal current source's value times the resistance contained in the real current source. The impedance component of the real voltage source retains its real current source value.

In general, source transformations can be summarized by keeping two things in mind:

·         Ohm's Law

·         Impedance's remain the same

Overview and Insights

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.

Superposition Theorem

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:

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)).

This procedure is followed for each source in turn, and 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.

Overview and Insights

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. 

AC Circuits

AC Circuits w/ Complex Numbers

Direct current (DC) circuits involve current flowing in one direction. In alternating current (AC) circuits, instead of a constant voltage supplied by a battery, the voltage oscillates in a sine wave pattern, varying with time as:

In a household circuit, the frequency is 60 Hz. The angular frequency is related to the frequency, f, by:

Vo represents the maximum voltage, which in a household circuit in North America is about 170 volts. We talk of a household voltage of 120 volts, though; this number is a kind of average value of the voltage. The particular averaging method used is something called root mean square (square the voltage to make everything positive, find the average, take the square root), or rms. Voltages and currents for AC circuits are generally expressed as rms values. For a sine wave, the relationship between the peak and the rms average is:

rms value = 0.707 peak value

Resistance in an AC circuit

The relationship V = IR applies for resistors in an AC circuit, so

In AC circuits we'll talk a lot about the phase of the current relative to the voltage. In a circuit which only involves resistors, the current and voltage are in phase with each other, which means that the peak voltage is reached at the same instant as peak current. In circuits which have capacitors and inductors (coils) the phase relationships will be quite different.

Capacitance in an AC circuit

Consider now a circuit which has only a capacitor and an AC power source (such as a wall outlet). A capacitor is a device for storing charging. It turns out that there is a 90° phase difference between the current and voltage, with the current reaching its peak 90° (1/4 cycle) before the voltage reaches its peak. Put another way, the current leads the voltage by 90° in a purely capacitive circuit.

To understand why this is, we should review some of the relevant equations, including:

relationship between voltage and charge for a capacitor: CV = Q

RLC Circuits

Consider what happens when resistors, capacitors, and inductors are combined in one circuit. If all three components are present, the circuit is known as an RLC circuit (or LRC). If only two components are present, it's either an RC circuit, an RL circuit, or an LC circuit.

The overall resistance to the flow of current in an RLC circuit is known as the impedance, symbolized by Z. The impedance is found by combining the resistance, the capacitive reactance, and the inductive reactance. Unlike a simple series circuit with resistors, however, where the resistances are directly added, in an RLC circuit the resistance and reactances are added as vectors.

This is because of the phase relationships. In a circuit with just a resistor, voltage and current are in phase. With only a capacitor, current is 90° ahead of the voltage, and with just an inductor the reverse is true, the voltage leads the current by 90°. When all three components are combined into one circuit, there has to be some compromise.

To figure out the overall effective resistance, as well as to determine the phase between the voltage and current, the impedance is calculated like this. The resistance R is drawn along the +x-axis of an x-y coordinate system. The inductive reactance is at 90° to this, and is drawn along the +y-axis. The capacitive reactance is also at 90° to the resistance, and is 180° different from the inductive reactance, so it's drawn along the -y-axis. The impedance, Z, is the sum of these vectors, and is given by:

The current and voltage in an RLC circuit are related by V = IZ. The phase relationship between the current and voltage can be found from the vector diagram: its the angle between the impedance, Z, and the resistance, R. The angle can be found from:

If the angle is positive, the voltage leads the current by that angle. If the angle is negative, the voltage lags the currents.

The power dissipated in an RLC circuit is given by:

Overview and Insights

I learned that all of this power is lost in the resistor; the capacitor and inductor alternately store energy in electric and magnetic fields and then give that energy back to the circuit. I am also having difficulty in solving Complex Numbers because the values are not accurate when i solve.