SINUSOIDS AND PHASORS
Historical
Nikola Tesla (1856–1943) and George Westinghouse (1846–1914) helped establish alternating current as the primary mode of electricity transmission and distribution. Today it is obvious that ac generation is well established as the form of electric power that makes widespread distribution of electric power efficient and economical. However, at the end of the 19th century, which was the better—ac or dc—was hotly debated and had extremely out- spoken supporters on both sides. The dc side was led by Thomas Edison, who had earned a lot of respect for his many contributions. Power generation using ac really began to build after the successful contributions of Tesla. The real commercial success in ac came from George Westinghouse and the outstanding team, including Tesla, he assembled. In addition, two other big names were C. F. Scott and B. G. Lamme. The most significant contribution to the early success of ac was the patenting of the polyphase ac motor by Tesla in 1888. The induc- tion motor and polyphase generation and distribution systems doomed the use of dc as the prime energy source
Chapter 9.1
AC Circuit
A sinusoidal current is usually referred to as alternating current (ac). Such a current reverses at regular time intervals and has alternately positive and negative values. Circuits driven by sinusoidal current or voltage sources.
sinusoid is a signal that has the form of the sine or cosine function.
Chapter 9.2
SINUSOIDS
Consider the sinusoidal voltage;
Where
Vm = the amplitude of the sinusoid
ω = the angular frequency i radians
ωt = the argument of the sinusoid
The sinusoid is shown in Fig. 9.1(a) as a function of its argument and in Fig. 9.1(b) as a function of time. It is evident that the sinusoid repeats itself every T seconds; thus, T is called the period of the sinusoid. From the two plots in Fig. 9.1, we observe that ωt = 2π,
A periodic function is one that satisfies f(t) f(t nT), for all t and for all integers n.In Fig. 9.2. The starting point of in Fig. 9.2 occurs first in time. Therefore, we say that leads by Φ or that lags by Φ. If Φ is not equal to zero, we also say that and are out of phase. If Φ is n,ot equal to zero, then v1 and v2 are said to be in phase; they reach their minima and maxima at exactly the same time. We can compare v1 and v2in this manner because they operate at the same frequency; they do not need to have the same amplitude.
A sinusoid can be expressed in either sine or cosine form.
This is achieved by using the following trigonometric identities:
Chapter 9.3
PHASORS
A phasor is a complex number that represents the amplitude and phase of a sinusoid.
* The notion of solving ac circuits using phasors was first introduced by Charles Steinmetz in 1893. Before we completely define phasors and apply them to circuit analysis, we need to be thoroughly familiar with complex numbers.
A complex number z can be written in rectangular form as:
z = x + jy
The complex number z can also be written in polar or exponential form as:
The relationship between the rectangular form and the polar form is shown in Fig. 9.6, where the x axis represents the real part and the y axis represents the imaginary part of complex number. Given x and y, we can get r and Φ as
On the other hand, if we know r and Φ, we can obtain x and y as
Thus, z may be written as
* The following operations are important:
ADDITION:
Chapter 9.4
PHASORS RELATIONSHIP FOR CIRCUIT ELEMENTS
* Although it is equally correct to say that the inductor voltage leads the current by 90 degrees, convention gives the current phase relative to the voltage.
Chapter 9.5
IMPEDANCE AND ADMITTANCE
We obtain Ohm’s law in phasor form for any type of element as:
The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in ohms.
As a complex quantity, the impedance may be expressed in rectangular form as:
The impedance may also be expressed in polar form as:
The admittance Y is the reciprocal of impedance, measured in siemens (S).
Chapter 9.7
IMPEDANCE COMBINATIONS
Consider the N series - connected impedance shown in Fig. 9.18. The same current I flows through the impedance. Applying KVL around the loop gives.
The equivalent impedance at the input terminals is
If N = 2, as shown in Fig. 9.19, the current through the impedances is
"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.
It is a standard practice to represent S, P, and Q in the formof
a triangle, known as the power triangle, shown below,
Reactive Power:- Q = 0 for resistive loads (unity pf).
- Q < 0 for capacitive loads (leading pf).
- Q > 0 for inductive loads (lagging pf).
Videos:
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"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-Delta Connection
Three-phase Y and Delta
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 (windings) 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.
When we measure voltage and current in three-phase systems, we need to be specific as to where we're measuring. Line voltage refers to the amount of voltage measured between any two line conductors in a balanced three-phase system. With the above circuit, the line voltage is roughly 208 volts. Phase voltage refers to the voltage measured across any one component (source winding or load impedance) in a balanced three-phase source or load. For the circuit shown above, the phase voltage is 120 volts. The terms line current and phase current follow the same logic: the former referring to current through any one line conductor, and the latter to current through any one component.
Y-connected sources and loads always have line voltages greater than phase voltages, and line currents equal to phase currents. If the Y-connected source or load is balanced, the line voltage will be equal to the phase voltage times the square root of 3:
However, the “Y” configuration is not the only valid one for connecting three-phase voltage source or load elements together. Another configuration is known as the “Delta,” for its geometric resemblance to the Greek letter of the same name (Δ). Take close notice of the polarity for each winding in Figure below.
Three-phase, three-wire Δ connection has no common.
At first glance it seems as though three voltage sources like this would create a short-circuit, electrons flowing around the triangle with nothing but the internal impedance of the windings to hold them back. Due to the phase angles of these three voltage sources, however, this is not the case.
One quick check of this is to use Kirchhoff's Voltage Law to see if the three voltages around the loop add up to zero. If they do, then there will be no voltage available to push current around and around that loop, and consequently there will be no circulating current. Starting with the top winding and progressing counter-clockwise, our KVL expression looks something like this:
Indeed, if we add these three vector quantities together, they do add up to zero. Another way to verify the fact that these three voltage sources can be connected together in a loop without resulting in circulating currents is to open up the loop at one junction point and calculate voltage across the break: (Figure below)
Voltage across open Δ should be zero.
Starting with the right winding (120 V ∠ 120o) and progressing counter-clockwise, our KVL equation looks like this:
Sure enough, there will be zero voltage across the break, telling us that no current will circulate within the triangular loop of windings when that connection is made complete.
Having established that a Δ-connected three-phase voltage source will not burn itself to a crisp due to circulating currents, we turn to its practical use as a source of power in three-phase circuits. Because each pair of line conductors is connected directly across a single winding in a Δ circuit, the line voltage will be equal to the phase voltage. Conversely, because each line conductor attaches at a node between two windings, the line current will be the vector sum of the two joining phase currents. Not surprisingly, the resulting equations for a Δ configuration are as follows:
Let's see how this works in an example circuit: (Figure below)
The load on the Δ source is wired in a Δ.
With each load resistance receiving 120 volts from its respective phase winding at the source, the current in each phase of this circuit will be 83.33 amps:
So each line current in this three-phase power system is equal to 144.34 amps, which is substantially more than the line currents in the Y-connected system we looked at earlier. One might wonder if we've lost all the advantages of three-phase power here, given the fact that we have such greater conductor currents, necessitating thicker, more costly wire. The answer is no. Although this circuit would require three number 1 gage copper conductors (at 1000 feet of distance between source and load this equates to a little over 750 pounds of copper for the whole system), it is still less than the 1000+ pounds of copper required for a single-phase system delivering the same power (30 kW) at the same voltage (120 volts conductor-to-conductor).
One distinct advantage of a Δ-connected system is its lack of a neutral wire. With a Y-connected system, a neutral wire was needed in case one of the phase loads were to fail open (or be turned off), in order to keep the phase voltages at the load from changing. This is not necessary (or even possible!) in a Δ-connected circuit. With each load phase element directly connected across a respective source phase winding, the phase voltage will be constant regardless of open failures in the load elements.
Perhaps the greatest advantage of the Δ-connected source is its fault tolerance. It is possible for one of the windings in a Δ-connected three-phase source to fail open (Figure below) without affecting load voltage or current!
Even with a source winding failure, the line voltage is still 120 V, and load phase voltage is still 120 V. The only difference is extra current in the remaining functional source windings.
The only consequence of a source winding failing open for a Δ-connected source is increased phase current in the remaining windings. Compare this fault tolerance with a Y-connected system suffering an open source winding in Figure below.
Open “Y” source winding halves the voltage on two loads of a Δ connected load.
With a Δ-connected load, two of the resistances suffer reduced voltage while one remains at the original line voltage, 208. A Y-connected load suffers an even worse fate (Figure below) with the same winding failure in a Y-connected source
Open source winding of a "Y-Y" system halves the voltage on two loads, and looses one load entirely.
In this case, two load resistances suffer reduced voltage while the third loses supply voltage completely! For this reason, Δ-connected sources are preferred for reliability. However, if dual voltages are needed (e.g. 120/208) or preferred for lower line currents, Y-connected systems are the configuration of choice.
