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Sunday, March 18, 2012

AC Power Analysis

Instantaneous and Average Power

The instantaneous electrical power P delivered to a component is given by

$P(t) = I(t) \cdot V(t) \,$

where

P(t) is the instantaneous power, measured in watts (joules per second)

V(t) is the potential difference (or voltage drop) across the component, measured in volts

I(t) is the current through it, measured in amperes

If the component is a resistor with time-invariant voltage to current ratio, then:

$P=I \cdot V = I^2 \cdot R = \frac{V^2}{R} \,$

where

$R = V/I \,$

is the resistance, measured in ohms.

Average Power:

Maximum Average Power Transfer

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.

R_L = Re{Z_Th} and X_L= - Im{Z_Th}

The maximum power in this case:

P_max =

Where V²_Th and I²_N represent the square of the sinusoidal peak values.

Effective RMS Value

The RMS value of a set of values (or a continuous-time waveform) is the square root of the arithmetic mean (average) of the squares of the original values (or the square of the function that defines the continuous waveform).

In the case of a set of $n$ values $\{x_1,x_2,\dots,x_n\}$ , the RMS value is given by:

$x_{\mathrm{rms}} = \sqrt {\frac{1}{n}\left({{x_1}^2 + {x_2}^2 + \cdots + {x_n}^2}\right)}$

The corresponding formula for a continuous function (or waveform) $f(t)$ defined over the interval $T_1 \le t \le T_2$ is

$f_{\mathrm{rms}} = \sqrt {{1 \over {T_2-T_1}} {\int_{T_1}^{T_2} {[f(t)]}^2\, dt}},$

and the RMS for a function over all time is

$f_\mathrm{rms} = \lim_{T\rightarrow \infty} \sqrt {{1 \over {2T}} {\int_{-T}^{T} {[f(t)]}^2\, dt}}.$

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a series of equally spaced samples.

Apparent Power and Power Factor

Apparent power is the product of the root-mean-square of voltage and current.

Engineers care about apparent power, because even though the current associated with reactive power does no work at the load, it heats the wires, wasting energy. Conductors, transformers and generators must be sized to carry the total current, not just the current that does useful work.

Another consequence is that adding the apparent power for two loads will not accurately give the total apparent power unless they have the same displacement between current and voltage (the samepower factor).

$P = S = V_\mathrm{RMS} I_\mathrm{RMS} = I_\mathrm{RMS}^2 R = \frac{V_\mathrm{RMS}^2} {R}\,\!$

The ratio between real power and apparent power in a circuit is called the power factor. It's a practical measure of the efficiency of a power distribution system. For two systems transmitting the same amount of real power, the system with the lower power factor will have higher circulating currents due to energy that returns to the source from energy storage in the load. These higher currents produce higher losses and reduce overall transmission efficiency. A lower power factor circuit will have a higher apparent power and higher losses for the same amount of real power.

The power factor is one when the voltage and current are in phase. It is zero when the current leads or lags the voltage by 90 degrees. Power factors are usually stated as "leading" or "lagging" to show the sign of the phase angle of current with respect to voltage.

Purely capacitive circuits cause reactive power with the current waveform leading the voltage wave by 90 degrees, while purely inductive circuits cause reactive power with the current waveform lagging the voltage waveform by 90 degrees. The result of this is that capacitive and inductive circuit elements tend to cancel each other out.

Where the waveforms are purely sinusoidal, the power factor is the cosine of the phase angle (φ) between the current and voltage sinusoid waveforms. Equipment data sheets and nameplates often will abbreviate power factor as " $\cos \phi$ " for this reason.

$pf = {Pa + Pb + Pc \over |Sa| + |Sb| + |Sc|}$

Impedance and Admittance

Impedance

impedance, is the measure of the opposition that a circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current (AC) circuit. Impedance extends the concept of resistance to AC circuits, and possesses both magnitude and phase, unlike resistance which has only magnitude. When a circuit is driven with direct current (DC), there is no distinction between impedance and resistance; the latter can be thought of as impedance with zero phase angle.

It is necessary to introduce the concept of impedance in AC circuits because there are other mechanisms impeding the flow of current besides the normal resistance of DC circuits. There are an additional two impeding mechanisms to be taken into account in AC circuits: the induction of voltages in conductors self-induced by the magnetic fields of currents (inductance), and the electrostatic storage of charge induced by voltages between conductors (capacitance). The impedance caused by these two effects is collectively referred to as reactance and forms the imaginary part of complex impedance whereas resistance forms the real part.

The symbol for impedance is usually $\scriptstyle Z$ and it may be represented by writing its magnitude and phase in the form $\scriptstyle |Z| \angle \theta$ . However, complex number representation is often more powerful for circuit analysis purposes. The term impedancewas coined by Oliver Heaviside in July 1886. Arthur Kennelly was the first to represent impedance with complex numbers in 1893.

Impedance is defined as the frequency domain ratio of the voltage to the current. In other words, it is the voltage–current ratio for a single complex exponential at a particular frequency ω. In general, impedance will be a complex number, with the same units as resistance, for which the SI unit is the ohm (Ω). For a sinusoidal current or voltage input, the polar form of the complex impedance relates the amplitude and phase of the voltage and current. In particular,

The magnitude of the complex impedance is the ratio of the voltage amplitude to the current amplitude.
The phase of the complex impedance is the phase shift by which the current is ahead of the voltage.

The reciprocal of impedance is admittance

Admittance

admittance (Y) is a measure of how easily a circuit or device will allow a current to flow. It is defined as the inverse of the impedance (Z). The SI unit of admittance is the siemens (symbol S). Oliver Heaviside coined the term in December 1887.

$Y = Z^{-1} = 1/Z \,$

where

Y is the admittance, measured in siemens

Z is the impedance, measured in ohms

Note that the synonymous unit mho, and the symbol ℧ (an upside-down uppercase omega Ω), are also in common use.

Resistance is a measure of the opposition of a circuit to the flow of a steady current, while impedance takes into account not only the resistance but also dynamic effects (known as reactance). Likewise, admittance is not only a measure of the ease with which a steady current can flow, but also the dynamic effects of the material's susceptance to polarization:

$Y = G + j B \,$

where

$Y$ is the admittance, measured in siemens (a.k.a. mho, the inverse of ohm).
$G$ is the conductance, measured in siemens.
$B$ is the susceptance, measured in siemens.
$j = \sqrt{-1}$

Sinusoids and Phasors

Sinusoids

a signal that has the form of the sine and cosine functions.

is a mathematical function that describes a smooth repetitive oscillation. It occurs often in pure and applied mathematics, as well as physics,engineering, signal processing and many other fields. Its most basic form as a function of time (t) is:

$y(t) = A \cdot \sin(\omega t + \phi)$

where:

A, the amplitude, is the peak deviation of the function from its center position.
ω, the angular frequency, specifies how many oscillations occur in a unit time interval, in radians per second
φ, the phase, specifies where in its cycle the oscillation begins at t = 0.
- When the phase is non-zero, the entire waveform appears to be shifted in time by the amountφ/ω seconds. A negative value represents a delay, and a positive value represents an advance.

The sine wave is important in physics because it retains its waveshape when added to another sine wave of the same frequency and arbitrary phase. It is the only periodic waveform that has this property. This property leads to its importance in Fourier analysis and makes it acoustically unique.

The graphs of the sine and cosine functions are sinusoids of different phases.

Phasors

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

is a representation of a sine wavewhose amplitude (A) and angular frequency(ω) are time-invariant. It is a subset of a more general concept called analytic representation. Phasors decompose the behavior of a sinusoid into three independent factors that relay amplitude, frequency and phase information. This can be particularly useful because the frequency factor (which includes the time-dependence of the sine wave) is often common to all the components of a linear combination of sine waves. In these situations, phasors allow this common feature to be factored out, leaving just the time-independent amplitude and phase information (the latter simply defining the phase at t=0 as θ), which can be combined algebraically rather than trigonometrically. Similarly, linear differential equations can be reduced to algebraic ones. The term phasortherefore often refers to just those two factors. In older texts, a phasor is also referred to as a sinor.

An example of series RLC circuit and respective phasor diagram for a specific ω

Sunday, March 11, 2012

Balanced Wye-Wye Connection

The balanced three-phase wye-wye connection is shown below. Note that the line impedance for each of the individual phases in included in the circuit. The line impedance are assumed to be equal for all three phases.

The line currents (IaA, IbB and IcC) are designated according to the source/load node naming convention. The source current, line current, and load current are all one in the same current for a given phase in a wye-wye

connection.

*This is an example of a balanced Wye-Wye connection:

The magnitude of the Line Voltages VL is √3 time the magnitude of the phase voltages Vp, or

VL = √3 Vp

Three-Phase Voltages

We are about to discuss about the Balance Three-Phase Voltages.

3 phase voltages are often produced with a three phase AC generator. The Generator basically consists of a rotating magnet (called the rotor) surrounded by a stationary winding (called the stator). Three separate coils with terminals a-a', b-b' and c-c' are physically placed 120° apart around the stator. The induced voltages in the coils are equal in magnitude but out of phase by 120° . The 3 phase generator can supply power to both single phase and three phase loads. The three-phase system was introduced and patented by NIKOLA TESLA in 1887 and 1888 a Croatian-American engineer.

Three phase system is consist of three voltages sources connected to load by three or four wires (transmission lines). It has two types of voltage sources a wye-connected and a delta-connected sources.

a.) b.)

a.) wye-connected source

b.) delta connected source

V_1, V_2, and V_{3 is the same with}V_an, V_bn, V_bn.

V_an, V_bn, V_{bn = 0 This implies that}| V_a | = |V_bn | = |V_bn|

We all know that voltages in a 3 phase system is the same in magnitude but they are 120° out of phase with each other.

V_{an =}Vp <0°

V_{bn =}Vp<-120°

Vcn = Vp<120°

where Vp is the effective or rms value of the phase voltages. This is also known as the abc sequence or positive sequence. Where V_{an leads}Vbn and leads Vcn.

Like also in the voltage source, the load also has a 3 phase system. It is also connected in wye and delta.

But a load is said to be unbalanced if the phase impedance of the system is not equal in magnitude or phase. A balanced load is one in which the phase impedance are equal in magnitude and in phase.

Z₁ = Z₂= Z₃ = Z₄ ; Balanced wye - connected load,

Z_a = Z_b= Z_c = Z_{∆ ;} Balanced delta - connected load,

Now we discussed about the 3 phase voltage and loads. Both have 2 connection the wye and the delta. Therefore we could have 4 connections the

wye-wye connection

wye-delta connection

delta - delta connection

delta - wye connection