ADVANCED ELECTRICAL CIRCUIT: 02/28/10

Sunday, February 28, 2010

WEEK 1: Generation of AC Voltage

Alternating Current vs. Direct Current

Schematic diagram of a basic DC circuit.

The figure to the right shows the schematic diagram of a very basic DC circuit. It consists of nothing more than a source (a producer of electrical energy) and a load (whatever is to be powered by that electrical energy). The source can be any electrical source: a chemical battery, an electronic power supply, a mechanical generator, or any other possible continuous source of electrical energy. For simplicity, we represent the source in this figure as a battery.

At the same time, the load can be any electrical load: a light bulb, electronic clock or watch, electronic instrument, or anything else that must be driven by a continuous source of electricity. The figure here represents the load as a simple resistor.

Regardless of the specific source and load in this circuit, electrons leave the negative terminal of the source, travel through the circuit in the direction shown by the arrows, and eventually return to the positive terminal of the source. This action continues for as long as a complete electrical circuit exists.

Schematic diagram of a basic AC circuit.

Now consider the same circuit with a single change, as shown in the second figure to the right. This time, the energy source is constantly changing. It begins by building up a voltage which is positive on top and negative on the bottom, and therefore pushes electrons through the circuit in the direction shown by the solid arrows. However, then the source voltage starts to fall off, and eventually reverse polarity. Now current will still flow through the circuit, but this time in the direction shown by the dotted arrows. This cycle repeats itself endlessly, and as a result the current through the circuit reverses direction repeatedly. This is known as an alternating current.

This kind of reversal makes no difference to some kinds of loads. For example, the light bulbs in your home don't care which way current flows through them. When you close the circuit by turning on the light switch, the light turns on without regard for the direction of current flow.

Of course, there are some kinds of loads that require current to flow in only one direction. In such cases, we often need to convert alternating current such as the power provided at your wall socket to direct current for use by the load. There are several ways to accomplish this, and we will explore some of them in later pages in this section.

Properties of Alternating Current

A DC power source, such as a battery, outputs a constant voltage over time, as depicted in the top figure to the right. Of course, once the chemicals in the battery have completed their reaction, the battery will be exhausted and cannot develop any output voltage. But until that happens, the output voltage will remain essentially constant. The same is true for any other source of DC electricity: the output voltage remains constant over time.

By contrast, an AC source of electrical power changes constantly in amplitude and regularly changes polarity, as shown in the second figure to the right. The changes are smooth and regular, endlessly repeating in a succession of identical cycles, and form a sine wave as depicted here.

Because the changes are so regular, alternating voltage and current have a number of properties associated with any such waveform. These basic properties include the following list:

Frequency. One of the most important properties of any regular waveform identifies the number of complete cycles it goes through in a fixed period of time. For standard measurements, the period of time is one second, so the frequency of the wave is commonly measured in cycles per second (cycles/sec) and, in normal usage, is expressed in units of Hertz (Hz). It is represented in mathematical equations by the letter 'f.' In North America (primarily the US and Canada), the AC power system operates at a frequency of 60 Hz. In Europe, including the UK, Ireland, and Scotland, the power system operates at a frequency of 50 Hz.
Period. Sometimes we need to know the amount of time required to complete one cycle of the waveform, rather than the number of cycles per second of time. This is logically the reciprocal of frequency. Thus, period is the time duration of one cycle of the waveform, and is measured in seconds/cycle. AC power at 50 Hz will have a period of 1/50 = 0.02 seconds/cycle. A 60 Hz power system has a period of 1/60 = 0.016667 seconds/cycle. These are often expressed as 20 ms/cycle or 16.6667 ms/cycle, where 1 ms is 1 millisecond = 0.001 second (1/1000 of a second).

Measuring the wavelength of a sine wave.

Wavelength. Because an AC wave moves physically as well as changing in time, sometimes we need to know how far it moves in one cycle of the wave, rather than how long that cycle takes to complete. This of course depends on how fast the wave is moving as well. Electrical signals travel through their wires at nearly the speed of light, which is very nearly 3 × 10⁸ meters/second, and is represented mathematically by the letter 'c.' Since we already know the frequency of the wave in Hz, or cycles/second, we can perform the division of c/f to obtain a result in units of meters/cycle, which is what we want. The Greek letter (lambda) is used to represent wavelength in mathematical expressions. Thus, = c/f. As shown in the figure to the right, wavelength can be measured from any part of one cycle to the equivalent point in the next cycle. Wavelength is very similar to period as discussed above, except that wavelength is measured in distance per cycle where period is measured in time per cycle.
Amplitude. Another thing we have to know is just how positive or negative the voltage is, with respect to some selected neutral reference. With DC, this is easy; the voltage is constant at some measurable value. But AC is constantly changing, and yet it still powers a load. Mathematically, the amplitude of a sine wave is the value of that sine wave at its peak. This is the maximum value, positive or negative, that it can attain. However, when we speak of an AC power system, it is more useful to refer to the effective voltage or current. This is the rating that would cause the same amount of work to be done (the same effect) as the same value of DC voltage or current would cause. We won't cover the mathematical derivations here; for the present, we'll simply note that for a sine wave, the effective voltage of the AC power system is 0.707 times the peak voltage. Thus, when we say that the AC line voltage in the US is 120 volts, we are referring to the voltage amplitude, but we are describing the effective voltage, not the peak voltage of nearly 170 volts. The effective voltage is also known as the rms voltage.

When we deal with AC power, the most important of these properties are frequency and amplitude, since some types of electrically powered equipment must be designed to match the frequency and voltage of the power lines. Period is sometimes a consideration, as we'll discover when we explore electronic power supplies. Wavelength is not generally important in this context, but becomes much more important when we start dealing with signals at considerably higher frequencies.

WEEK 2: Phasor concept in AC Analysis

A phase vector ("phasor") is a representation of a sine wave whose amplitude (A), phase (θ), and frequency (ω) are time-invariant. It is a subset of a more general concept called analytic representation. Phasors reduce the dependencies on these parameters to three independent factors, thereby simplifying certain kinds of calculations. In particular the frequency factor, which also includes the time-dependence of the sine wave, is often common to all the components of a linear combination of sine waves. Using phasors, it can be factored out, leaving just the static amplitude and phase information to be combined algebraically (rather than trigonometrically). Similarly, linear differential equations can be reduced to algebraicones. The term phasor therefore often refers to just those two factors. In older texts, a phasor is also referred to as a sinor.

WEEK 3: Impedance and Admittance

In the study of DC circuits, the student of electricity comes across a term meaning the opposite of resistance: conductance. It is a useful term when exploring the mathematical formula for parallel resistances: R_parallel = 1 / (1/R₁ + 1/R₂ + . . . 1/R_n). Unlike resistance, which diminishes as more parallel components are included in the circuit, conductance simply adds. Mathematically, conductance is the reciprocal of resistance, and each 1/R term in the “parallel resistance formula” is actually a conductance.

Whereas the term “resistance” denotes the amount of opposition to flowing electrons in a circuit, “conductance” represents the ease of which electrons may flow. Resistance is the measure of how much a circuit resists current, while conductance is the measure of how much a circuitconducts current. Conductance used to be measured in the unit of mhos, or “ohms” spelled backward. Now, the proper unit of measurement isSiemens. When symbolized in a mathematical formula, the proper letter to use for conductance is “G”.

Reactive components such as inductors and capacitors oppose the flow of electrons with respect to time, rather than with a constant, unchanging friction as resistors do. We call this time-based opposition, reactance, and like resistance we also measure it in the unit of ohms.

As conductance is the complement of resistance, there is also a complementary expression of reactance, called susceptance. Mathematically, it is equal to 1/X, the reciprocal of reactance. Like conductance, it used to be measured in the unit of mhos, but now is measured in Siemens. Its mathematical symbol is “B”, unfortunately the same symbol used to represent magnetic flux density.

The terms “reactance” and “susceptance” have a certain linguistic logic to them, just like resistance and conductance. While reactance is the measure of how much a circuit reacts against change in current over time, susceptance is the measure of how much a circuit is susceptible to conducting a changing current.

If one were tasked with determining the total effect of several parallel-connected, pure reactances, one could convert each reactance (X) to a susceptance (B), then add susceptances rather than diminish reactances: X_parallel = 1/(1/X₁ + 1/X₂ + . . . 1/X_n). Like conductances (G), susceptances (B) add in parallel and diminish in series. Also like conductance, susceptance is a scalar quantity.

When resistive and reactive components are interconnected, their combined effects can no longer be analyzed with scalar quantities of resistance (R) and reactance (X). Likewise, figures of conductance (G) and susceptance (B) are most useful in circuits where the two types of opposition are not mixed, i.e. either a purely resistive (conductive) circuit, or a purely reactive (susceptive) circuit. In order to express andquantify the effects of mixed resistive and reactive components, we had to have a new term: impedance, measured in ohms and symbolized by the letter “Z”.

To be consistent, we need a complementary measure representing the reciprocal of impedance. The name for this measure is admittance.Admittance is measured in (guess what?) the unit of Siemens, and its symbol is “Y”. Like impedance, admittance is a complex quantity rather than scalar. Again, we see a certain logic to the naming of this new term: while impedance is a measure of how much alternating current is impeded in a circuit, admittance is a measure of how much current is admitted.

Given a scientific calculator capable of handling complex number arithmetic in both polar and rectangular forms, you may never have to work with figures of susceptance (B) or admittance (Y). Be aware, though, of their existence and their meanings.

WEEK 4: Frequency Domain Analysis of Simple RLC Circuits

The series RLC can be analyzed in the frequency domain using complex impedance relations. If the voltage source above produces a complex exponential waveform with complex amplitude V(s) and angular frequency $s = σ + i ω$ , KVL can be applied:

$V(s) = I(s) \left ( R + Ls + \frac{1}{Cs} \right )$

where I(s) is the complex current through all components. Solving for I(s):

$I(s) = \frac{1}{ R + Ls + \frac{1}{Cs} } V(s)$

And rearranging, we have at

$I(s) = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) } V(s)$

Complex admittance

Next, we solve for the complex admittance Y(s):

$Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + {R \over L}s + \frac{1}{LC} \right ) }$

Finally, we simplify using parameters α and ω_o

$Y(s) = { I(s) \over V(s) } = \frac{s}{ L \left ( s^2 + 2 \alpha s + {\omega_0}^2 \right ) }$

Notice that this expression for Y(s) is the same as the one we found for the Zero State Response.

Poles and zeros

The zeros of Y(s) are those values of s such that $Y (s) = 0$ :

$s = 0 \,$ and $|s| \rightarrow \infty$

The poles of Y(s) are those values of s such that $Y(s) \rightarrow \infty$ . By the quadratic formula, we find

$s = - \alpha \pm \sqrt{\alpha^2 - {\omega_0}^2}$

Notice that the poles of Y(s) are identical to the roots $λ 1$ and $λ 2$ of the characteristic polynomial.

Sinusoidal steady state

If we now let $s = i ω$ ....

Considering the magnitude of the above equation:

$| Y(s=i \omega) | = \frac{1}{\sqrt{ R^2 + \left ( \omega L - \frac{1}{\omega C} \right )^2 }}.$

Next, we find the magnitude of current as a function of ω

$| I( i \omega ) | = | Y(i \omega) | | V(i \omega) |.\,$

If we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt, then the graph of magnitude of the current i (in amperes) as a function of ω (in radians per second) is:

Sinusoidal steady-state analysis

Note that there is a peak at $i m a g (ω) = 1$ . This is known as the resonance frequency. Solving for this value, we find:

$\omega_0 = \frac{1}{\sqrt{L C}}.$

Parallel RLC circuit

Parallel RLC Circuit notations:

V - the voltage of the power source (measured in volts V)

I - the current in the circuit (measured in amperes A)

R - the resistance of the resistor (measured in ohms = V/A);

L - the inductance of the inductor (measured in henrys = H = V·s/A)

C - the capacitance of the capacitor (measured in farads = F = C/V = A·s/V)

The complex admittance of this circuit is given by adding up the admittances of the components:

${1\over Z}={1\over Z_L}+{1\over Z_C}+{1\over Z_R}={1\over{j\omega L}}+{j\omega C}+{1\over R}$

The change from a series arrangement to a parallel arrangement has some very real consequences for the behaviour. This can be seen by plotting the magnitude of the current $I={V\over Z}$ . For comparison with the earlier graph we choose values where R = 1 ohm, C = 1 farad, L = 1 henry, and V = 1.0 volt and ω in radians per second: