TIME CONSTANTS BASIC DEFINITION AND TUTORIALS

When a capacitor is connected to a voltage source, it takes a certain length of time for the capacitor to become fully charged. If a high resistance is connected in series with the capacitor, the time for charging is increased.

For any given circuit containing capacitance and resistance only, the time in seconds required to charge the capacitor to 63.2 percent of its full charge is called the time constant for that circuit.

This same time constant applies when the capacitor is discharged through the same resistance and is the time required for the capacitor to lose 63.2 percent of its charge.

The charging and discharging of a capacitor in terms of time constants is illustrated in the graph of Fig. 3.8. It will be noted that it takes six time constants to charge the capacitor to 99.8 percent of full charge.

The discharge curve is the exact reverse of the charge curve. When the capacitor is short circuited, it will lose 63.2 percent of its charge in one time constant and almost 99.8 percent of its charge in six time constants.

To determine the length of a time constant in seconds for any particular capacitor-resistance circuit, it is necessary to multiply the capacitance in microfarads by the resistance in megohorns, that is,

T = CR

As an example of how the time constant may be used in determining the performance of a capacitance-resistance circuit, we shall assume that a 20-pf capacitor is connected in series with a 10,000-ohm resistor and that 110 volts is applied to the circuit at intervals off sec.

The time constant is equal to 20 x 0.01 or 0.2 sec. (Note that 10,000 ohms is equal to 0.01 megohm.)

The time interval is given as sec, hence, the number of time constants is 2.5.

If we examine a time constant chart or graph, we find that the voltage at 2.5 time constants will be approximately 92 percent of full voltage. Applying this to our problem, 92 percent of 110 volts is approximately 101 volts. Thus, we find that the capacitor in this problem will charge to approximately 10 1 volts.

FIXED RESISTORS DEFINITION BASIC AND TUTORIALS

The fixed resistors are those whose value cannot be varied after manufacture. Fixed resistors are classified into composition resistors, wire-wound resistors, and metal-film resistors. Table 1.2 outlines the characteristics of some typical fixed resistors.


TABLE 1.2 Characteristics of Typical Fixed Resistors
Operating
Resistor Types Resistance Range Watt Range Temp. Range a, ppm/°C
Wire-wound resistor
Precision 0.1 to 1.2 MW 1/8 to 1/4 –55 to 145 10
Power 0.1 to 180 kW 1 to 210 –55 to 275 260
Metal-film resistor
Precision 1 to 250 MW 1/20 to 1 –55 to 125 50–100
Power 5 to 100 kW 1 to 5 –55 to 155 20–100
Composition resistor
General purpose 2.7 to 100 MW 1/8 to 2 –55 to 130 1500



Wire-Wound Resistors. Wire-wound resistors are made by winding wire of nickel-chromium alloy on a ceramic tube covering with a vitreous coating. The spiral winding has inductive and capacitive characteristics that make it unsuitable for operation above 50 kHz. The frequency limit can be raised by noninductive winding so that the magnetic fields produced by the two parts of the winding cancel.

Composition Resistors. Composition resistors are composed of carbon particles mixed with a binder. This mixture is molded into a cylindrical shape and hardened by baking.

Leads are attached axially to each end, and the assembly is encapsulated in a protective encapsulation coating. Color bands on the outer surface indicate the resistance value and tolerance. Composition resistors are economical and exhibit low noise levels for resistances above 1 MW.

Composition resistors are usually rated for temperatures in the neighborhood of 70°C for power ranging from 1/8 to 2 W. Composition resistors have end-to-end shunted capacitance that may be noticed at frequencies in the neighborhood of 100 kHz, especially for resistance values above 0.3 MW.

Metal-Film Resistors. Metal-film resistors are commonly made of nichrome, tin-oxide, or tantalum nitride, either hermetically sealed or using molded-phenolic cases. Metal-film resistors are not as stable as the wire wound resistors.

Depending on the application, fixed resistors are manufactured as precision resistors, semiprecision resistors, standard general-purpose resistors, or power resistors. Precision resistors have low voltage and power coefficients, excellent temperature and time stabilities, low noise, and very low reactance.

These resistors are available in metal-film or wire constructions and are typically designed for circuits having very close resistance tolerances on values. Semiprecision resistors are smaller than precision resistors and are primarily used for current-limiting or voltage-dropping functions in circuit applications. Semiprecision resistors have long-term temperature stability.

General-purpose resistors are used in circuits that do not require tight resistance tolerances or long-term stability. For general-purpose resistors, initial resistance variation may be in the neighborhood of 5% and the variation in resistance under full-rated power may approach 20%.

Typically, general-purpose resistors have a high coefficient of resistance and high noise levels. Power resistors are used for power supplies, control circuits, and voltage dividers where operational stability of 5% is acceptable. Power resistors are available in wire-wound and film constructions. Film-type power resistors have the advantage of stability at high frequencies and have higher resistance values than wire-wound resistors for a given size.

WHAT IS CAPACITANCE? BASIC INFORMATION AND TUTORIALS

Capacitance (C) is the phenomenon whereby a circuit stores electrical energy. Whenever two conducting materials are separated by an insulating material, they have the ability of storing electrical energy.

Such an arrangement of materials (two conductors separated by an insulator) is called a capacitor or condenser. If a source of dc voltage is connected between the two conducting materials of a capacitor, a current will flow for a certain length of time.

The current initially will be relatively large but will rapidly diminish to zero. A certain amount of electrical energy will then be stored in the capacitor.

If the source of voltage is removed and the conductors of the capacitor are connected to the two ends of a resistor, a current will flow from the capacitor through the resistor for a certain length of time. The current initially will be relatively large but will rapidly diminish to zero.

The direction of the current will be opposite to the direction of the current when the capacitor was being charged by the dc source. When the current reaches zero, the capacitor will have dissipated the energy which was stored in it as heat energy in the resistor. The capacitor will then said to be discharged.

The two conducting materials, often called the plates of the capacitor, will be electrically charged when electrical energy is stored in the capacitor. One plate will have an excess of positive electricity and therefore will be positively charged with a certain number of coulombs of excess positive electricity.

The other plate will have an excess of negative electricity and therefore will be negatively charged with an equal number of coulombs of excess negative electricity. When in this state, the capacitor is said to be charged. When a capacitor is charged, a voltage is present between the two conductors, or plates, of the capacitor.

When a capacitor is in a discharged state, no electrical energy is stored in it, and there is no potential difference, no voltage, between its plates. Each plate contains just as much positive as negative electricity, and neither plate has any electric charge.

From the above discussion it is seen that a capacitor has a sustained current only as long as the voltage is changing. A capacitor connected to a dc supply will not have a sustained current. In an ac circuit, the voltage is continually changing from instant to instant.

Therefore, when a capacitor is connected to an ac supply, an alternating current continues to flow. The current is first in one direction, charging the capacitor, and then in the opposite direction, discharging the capacitor.

Farad (F) The unit of capacitance. It is designated by the symbol F. A circuit or capacitor will have a capacitance of 1 F if when the voltage across it is increased by 1 V, its stored electricity is increased by 1 C.

Another definition for a capacitance of 1 F, which results in the same effect, is given below. A circuit or capacitor will have a capacitance of 1 F when if the voltage impressed upon it is changed at the rate of 1 V/s, 1 A of charging current flows.

Capacitive reactance (Xc) is the name given to the opposition to the flow of alternating current due to capacitance. It is measured in ohms as resistance and inductive reactance are.

CIRCUIT ANALYSIS TYPE BASIC INFORMATION

Circuit Reduction Techniques.

When a circuit analyst wishes to find the current through or the voltage across one of the elements that make up a circuit, as opposed to a complete analysis, it is often desirable to systematically replace elements in a way that leaves the target elements unchanged, but simplifies the remainder in a variety of ways.

The most common techniques include series/parallel combinations, wye/delta (or tee/pi) combinations, and the Thevenin-Norton theorem.

Series Elements.

Two or more electrical elements that carry the same current are defined as being in series.

Parallel Elements.

Two or more electrical elements that are connected across the same voltage are defined as being in parallel.

Wye-Delta Connections.

A set of three elements may be connected either as a wye, shown in or a delta. These are also called tee and pi connections, respectively. The equations give equivalents, in terms of resistors, for converting between these connection forms.

Thevenin-Norton Theorem.

The Thevenin theorem and its dual, the Norton theorem, provide the engineer with a convenient way of characterizing a network at a terminal pair. The method is most useful when one is considering various loads connected to a pair of output terminals. The equivalent can be determined analytically, and in some cases, experimentally.

Thevenin Theorem.

This theorem states that at a terminal pair, any linear network can be replaced by a voltage source in series with a resistance (or impedance). It is possible to show that the voltage is equal to the voltage at the terminal pair when the external load is removed (open circuited), and that the resistance is equal to the resistance calculated or measured at the terminal pair with all independent sources de-energized. 

De-energization of an independent source means that the source voltage or current is set to zero but that the source resistance (impedance) is unchanged. Controlled (or dependent) sources are not changed or de-energized.

Norton Theorem.

This theorem states that at a terminal pair, any linear network can be replaced by a current source in parallel with a resistance (or impedance). It is possible to show that the current is equal to the current that flows through the short-circuited, terminal pair when the external load is short circuited, and that the resistance is equal to the resistance calculated or measured at the terminal pair with all independent sources de-energized. 

De-energization of an independent source means that the source voltage or current is set to zero but that the source resistance (impedance) is unchanged. Controlled (or dependent) sources are not changed or de-energized.

KIRCHHOFF'S CURRENT LAW BASIC INFORMATION AND TUTORIALS

Kirchhoff’s current law (KCL) states that the currents entering and leaving any branch point or node in the circuit must add up to zero.

This follows directly from the conservation property: electric charge is neither created nor destroyed, nor is it “stored” (in appreciable quantity) within our wires, so that all the charge that flows into any junction must also flow out.

Thus, if three wires connect at one point, and we know the current in two of them, they determine the current in the third.

Again, the analogy of flowing water helps make this more obvious. At a point where three pipes are connected, the amount of water flowing in must equal the amount flowing out (unless there is a leak). 

For the purpose of computation, we assign positive or negative signs to currents flowing in and out of the node, respectively.

It does not matter which way we call positive, as long as we remain consistent in our definition. Then, the sum of currents into (or out of) the node is zero.

This is illustrated with the simple example in Figure 2.5, where KCL applied to the branch point proves that the current through the battery equals the sum of currents through the individual resistors.

Despite their simple and intuitive nature, the fundamental importance of Kirchhoff’s laws cannot be overemphasized. They lie at the heart of the interdependence of the different parts and branches of power systems: whenever two points are electrically connected, their voltages and the currents through them must obey KVL and KCL, whether this is operationally and economically desirable or not.

For example, managing transmission constraints in power markets is complicated by the fact that the flow on any one line cannot be changed independently of others. Thus the engineer’s response to the economist’s lamentation of how hard it is to manage power transmission: “Blame Kirchhoff.”

KIRCHHOFF'S VOLTAGE LAW BASIC INFORMATION AND TUTORIALS

Kirchhoff’s voltage law (often abbreviated KVL) states that the sum of voltages around any closed loop in a circuit must be zero. In essence, this law expresses the basic properties that are inherent in the definition of the term “voltage” or “electric potential.”

Specifically, it means that we can definitively associate a potential with a particular point that does not depend on the path by which a charge might get there. 

This also implies that if there are three points (A, B, and C) and we know the potential differences between two pairings (between A and B and between B and C), this determines the third relationship (between A and C).

Without thinking in such abstract and general terms, we apply this principle when we move from one point to another along a circuit by adding the potential differences or voltages along the way, so as to express the cumulative voltage between the initial and final point.

Finally, when we go all the way around a closed loop, the initial and final point are the same, and therefore must be at the same potential: a zero difference in all.

The analogy of flowing water comes in handy. Here, the voltage at any given point corresponds to the elevation. A closed loop of an electric circuit corresponds to a closed system like a water fountain. The voltage “rise” is a power source—say, a battery—that corresponds to the pump.

From the top of the fountain, the water then flows down, maybe from one ledge to another, losing elevation along the way and ending up again at the bottom. Analogously, the electric current flows “down” in voltage, maybe across several distinct steps or resistors, to finish at the “bottom” end of the battery.

This notion is illustrated by in the simple circuit in Figure 2.4 that includes one battery and two resistors. Note that it is irrelevant which point we choose to label as the “zero” potential: no matter what the starting point,adding all the potential gains and drops encountered throughout the complete loop will give a zero net gain.

INDUCTANCE - BASIC ELECTRICAL PARAMETERS INFORMATION AND TUTORIALS

What Is Inductance?


The basic inductive device is a coil of wire, called an inductor or a solenoid. Its functioning is based on the physical fact that an electric current produces a magnetic field around it.

This magnetic field describes a circular pattern around a current-carrying wire; the direction of the field can be specified with a “right-hand rule.” When a wire is coiled up as shown in figure below, it effectively amplifies this magnetic field, because the contributions from the individual loops add together.

The sum of these contributions is especially great in the center, pointing along the central axis of the coil. The resulting field can be further amplified by inserting a material of high magnetic permeability (such as iron) into the coil; this is how an electromagnet is made.


When such a coil is placed in an a.c. circuit, a second physical fact comes into play, namely, that a changing magnetic field in the vicinity of a conducting wire induces an electric current to flow through this wire. If the current through the coil oscillates back and forth, then so does the magnetic field in its center.

Because this magnetic field is continuously changing, it induces another current in the coil. This induced current is proportional to the rate of change of the magnetic field. The direction of the induced current will be such as to oppose the change in the current responsible for producing the magnetic field.

In other words, the inductor exerts an inhibitive effect on a change in current flow. This inhibitive effect results in a delay or phase shift of the alternating current with respect to the alternating voltage. Specifically, an ideal inductor (with no resistance at all) will cause the current to lag behind the voltage by a quarter cycle, or 90 degrees.

This result is difficult to explain intuitively. We will not attempt to detail the specific changes in the current and magnetic field over the course of a cycle. One thing that can readily be seen from the graph, though, is that the current has its maximum at the instant that the magnetic field changes most rapidly.

As the magnetic field increases and decreases during different parts of the cycle, it stores and releases energy. This energy is not being dissipated, only repeatedly exchanged between the magnetic field and the rest of the circuit. This exchange process becomes very important in the context of power transfer. Because the induced current in an inductor is related to the change in the field per unit time, the frequency of the applied alternating current is important.

The higher the frequency, the more rapidly the magnetic field is changing and reversing, and thus the greater the induced current with its impeding effect is. The lower the frequency, the easier it is for the current to pass through the inductor.

A direct current corresponds to the extreme case of zero frequency. When a steady d.c. voltage is applied to an inductor, it essentially behaves like an ordinary piece of wire. After a brief initial period, during which the field is established, the magnetic field remains constant along with the current.

An unchanging magnetic field exerts no further influence on an electric current, so the flow of a steady direct current through a coil of wire is unaffected by the inductive property. Overall, the effect of an inductor on an a.c. circuit is expressed by its reactance, denoted by X (to specify inductive reactance, the subscript L is sometimes added).

The inductive reactance is the product of the angular a.c. frequency7 and the inductance, denoted by L, which depends on the physical shape of the inductor and is measured in units of henrys (H). In equation form, XL = wL

Thus, unlike resistance, the reactance is not solely determined by the intrinsic characteristics of a device. In the context of power systems, however, because the frequency is always the same, reactance is treated as if it were a constant property.

When describing the behavior of electrical devices in the context of circuit analysis, we are generally interested in writing down a mathematical relationship between the current passing through and the voltage drop across the device. For a resistor, this is simply Ohm’s law, V = IR, where the resistance R is the proportionality constant between voltage and current.

It turns out that the inductance L also works as a proportionality constant between current and voltage across an inductor, but in this case the equation involves the rate of change of current, rather than simply the value of current at any given time. Readers familiar with calculus will recognize the notation dI/dt, which represents the time derivative or rate of change of current with respect to time.

Thus, we write V = L dI/dt meaning that the voltage drop V across an inductor is the product of its inductance L and the rate of change of the current I through it. This equation is used in circuit analysis in a manner analogous to Ohm’s law to establish relationships between current and voltage at different points in the circuit, except that it is more cumbersome to manipulate owing to the time derivative.

RESISTANCE IN PARALLEL BASIC INFORMATION AND TUTORIALS

What Happens To Resistance When Connected In Parallel?

When resistors are combined in parallel, the effect is perhaps less obvious than for the series case: rather than adding resistance, we are in fact decreasing the overall resistance of the combination by providing alternative paths for the current.

This is so because in the parallel case the individual charge is not required to travel through every element, only one branch, so that the presence of the parallel elements “alleviates” the current flow through each branch, and thereby makes it easier for the charge to traverse.

It is convenient here to consider resistors in terms of the inverse property, conductance. Thus, we think of the resistor added in parallel not as posing a further obstacle, but rather as providing an additional conducting option: after all, as far as the current is concerned, any resistor is still better than no path at all. Accordingly, the total resistance of a parallel combination will always be less than any of the individual resistances.


Using conductance (G = 1/R), the algebraic rule for combining any number of resistive elements in parallel is simply that the conductance of the parallel combination equals the sum of the individual conductances.

For example, suppose a 10-ohm and a 2.5-ohm resistor are connected in parallel, as in figure below.

We know already that their combined (parallel) resistance must be less than 2.5 ohm. To do the math, it is convenient to first write each in terms of conductance: 0.1 mho and 0.4 mho. The combined conductance is then simply the sum of the two, 0.5 mho. Expressed in terms of resistance, this result equals 2 ohm. In equation form, we would write for resistors in parallel:

1/Rt = 1/R1 + 1/R2 + ....


Note that the voltage drop across any number of elements in parallel is the same. This can easily be seen because all the elements share the same terminals: the points where they connect to the rest of the circuit are, in electrical terms, the same.

While elements connected in parallel thus have a common voltage drop across them, the current flowing through the various elements or branches will typically differ. Intuitively, we might guess that more current will flow through a branch with a lower resistance, and less current through one with a higher resistance.

This can be shown rigorously by applying Ohm’s law for each of the parallel resistances: If V is the voltage drop common to all the parallel resistances, and R1 is the individual resistance of one branch, then the current I1 through this branch is given by V/R1. Thus, the amount of current through each branch is inversely proportional to its resistance.

To summarize, there is a tidy correspondence between the series and parallel cases: In a series connection, the current through the various elements is the same, but the voltage drops across them vary (proportional to their resistance); in a parallel connection, the voltage drop across the various elements is the same, but the currents through them vary (inversely proportional to their resistance).

RESISTANCE IN SERIES BASIC INFORMATION AND TUTORIALS

What Happens To Resistance In Series?


The simplest kind of combination of multiple circuit elements has resistors connected in series (figure below).

The rule is easy: to find the resistance of a series combination of resistors, add their individual resistances.

For example, if a 10-ohm resistor is connected in series with a 20-ohm resistor, their combined resistance is 30 ohm. This means that we could replace the two resistors with a single resistor of 30 ohm and make no difference whatsoever to the rest of the circuit.

In fact, if the series resistors were enclosed in a box with only the terminal ends sticking out, there would be no way for us to tell by electrical testing on the terminals whether the box contained a single 30-ohm resistor or any series combination of two or more resistors whose resistances added up to 30 ohm.

Thus, an arbitrary number of resistances can be added in series, and their order does not matter.

Intuitively, the addition rule makes sense because if we think of a resistor as posing an “obstacle” to the current, and note that the same current must travel through each element in the series, each obstacle adds to the previous ones.

This notion can be formalized in terms of voltage drop. Across each resistor in a series combination, there will be a voltage drop proportional to its resistance. It is always true that, regardless of the nature of the elements (whether they are resistors or something else), the voltage drop across a set of elements connected in series equals the sum of voltage drops across the individual elements.

This notion reappears in the context of Kirchhoff’s Voltage Law.

HOW ALTERNATING CURRENT WORKS - THE BASICS OF ALTERNATING CURRENT

The modern electric power system is an alternating current, three phase system. Electricity is generated by synchronous generators which are machines which convert the rotational energy of a shaft into electrical energy.

The energy conversion is based on a phenomenon associated with magnetism and electricity called induction. If a stationary wire loop is placed in the field of a rotating magnet, an electric current will be induced in the wire.

The rotor of an electric generator is made to look like a magnet by energizing conductors embedded in it with a source of direct current.The system that provides direct current to the rotor windings is called the excitation system.

The energized windings on the rotor are conventionally called the field or field circuit. In modern generators the direct current excitation is derived from an alternating current source that has been rectified to provide dc.

The direct current excitation establishes a magnetic field in the metal of the rotor which extends across the air gap between the rotor and the stationary part of the generator (stator or armature). Electricity is induced in coils which are placed in slots in the stator.

The voltage induced in any one coil reflects the time varying characteristic of the magnetic field, as viewed by a stationary observer, caused by the rotation of the rotor. The magnitude of the induced voltage can be adjusted up or down by changing the magnitude of the direct current flowing in the rotor.

This is done by a voltage regulator in the excitation system which monitors the voltage at the terminal of the electric generator and adjusts the field voltage up or down as required to maintain the desired generator terminal voltage.

The voltage and current have a sinusoidal shape, that is, in each cycle of 360 degrees, it starts at a zero value at zero degrees, rises to a positive maximum at 90 degrees, falls to zero at 180 degrees, continues to fall to a negative maximum at 270 degrees and returns to zero at 360 degrees, where the process repeats as shown in figure below.

This sinusoidal shape reflects the rotating pattern of the magnetic field produced on the rotor. If the stator coil is connected to an external load, current will flow. The current will also be oscillatory in nature, hence the name alternating current. The number of full cycles that occur in a set time defines the frequency of the electricity.

In the United States and many other areas of the world, the frequency is 60 hertz or cycles per second. In other areas a frequency of 50 cycles is used. The frequency is set by the number of magnetic circuits that are established on the rotor.

The frequency of the electricity produced by a particular generator is defined as: where n is the speed in revolutions per minute (rpm) and p is the number of pairs of magnetic poles. Steam turbines rotate at high speeds.

For example, if one magnetic circuit is established, that is, there two magnetic poles established (a single pair consisting of a north and a south pole), a speed of 3,600 rpm will result in a frequency of 60 hertz.

Alternately, if two magnetic circuits are established using two pairs of poles, a speed of only 1,800 rpm is needed to produce a frequency of 60 hertz. Hydraulic turbines rotate at relatively low speeds and will have many poles to produce the required frequency.

Because of the oscillatory nature of the voltage and current, an “effective” voltage and current value is defined. These effective values are considered to be equivalent to the direct current voltage and currents that would produce the same power dissipation (as heat) in a resistance.

The effective value for a sine wave is equal to 0.707 x the peak value. In the United States, for example, the oft quoted household voltage of 120 volts is an effective value and corresponds to a peak value of 169.7 volts.

NODAL ANALYSIS OF A DC NETWORK BASIC AND TUTORIALS

This is an example of circuit analysis using the nodal method. Solve the circuit below:

Calculation Procedure
1. Label the Circuit
Label all nodes (Fig. 1.12). One of the nodes (node A) is chosen as the reference node. It can be thought of as a circuit ground, which is at zero voltage or ground potential.

Nodes B and D are already known to be at the potential of the source voltages. The voltage
at node C (VC) is unknown.

Assume that VC VB and VC VD. Draw all three currents I1, I2, and I3 away from node C, that is, toward the reference node.

2. Write KCL at Node C
I1 + I2 + I3= 0.

3. Express Currents in Terms of Circuit Voltages Using Ohm’s Law
Refer to Fig. 1.12: I1= V1/R1= (VC - 8)/2, I2= V2 /R2= (VC - 24)/1, and I3= V3 /R3= VC/4.

4. Substitute in KCL Equation of Step 2
Substituting the current equations obtained in Step 3 into KCL of Step 2, we find I1 + I2 + I3= 0 or (VC - 8)/2 + (VC - 24)/1 + VC/4= 0.

Because the only unknown is VC , this simple equation can be solved to obtain VC= 16 V.


5. Solve for All Currents
I1= (VC - 8)/2= (16 - 8)/2= 4 A (true direction) and I2= (VC - 24)/1= (16 -24)/1 = -8 A. The negative sign indicates that I2 flows toward node C instead of in the assumed direction (away from node C). I3= VC/4= 16/4= 4 A (true direction).