================================ ELEC101: WWW Edition -1999 ================================
The main objective, however, is to cover the mathematics of sinusoidal currents and voltages. This is an essential first step to understand methods of solving ac circuits.
This presentation also highlights a complex number notation for representing sinusoids. This mathematical notation, that is called phasor, is a convenience even though it first seems to be complication and burden.
Main topics are covered in this presentation include sinusoidal signals, effective value of a sinusoid, phasors, complex impedance, impedance diagrams and total impedance of series and parallel combination of resistors, inductors and capacitors.
This presentation should be seen as a supporting learning material in ELEC101 and hence it does not replace the course notes covered in week 7.
It is strongly recommended you study your course notes before going through this presentation. This ensures a maximised learning outcome of this presentation module.
I hope you enjoy the presentation, and please let me know if you find this module useful in helping you for better understanding of the subject in any way.
Sinusoidal signals or waveforms are fundamental to the operation of ac circuits, and hence it is essential you become familiar with their properties.
It's worthwhile to note that engineers often prefer to use the terms sinusoidal waveform or sine wave rather than sinusoid signal. They also have a number of good reasons to pay extra attention to sinusoidal waveforms.
Many natural phenomena are sinusoidal (or very close to sinusoidal) in nature. Examples include - notes produce by musical instruments and - the projection of a satellite on the rotating earth.
Moreover, sinusoidal voltages are generated by electric power utilities throughout the world. The resulting ac power is used in domestic, commercial and industrial sites by electrical, electronic and telecommunication appliances and systems.
There is an attribute associated with sine waves that significantly facilitates the analysis of ac circuits.
Derivatives and integrals of sinusoids are themselves sinusoids. This means a sine voltage source produces a sine current in any linear circuit. Or, one may say, the shape of an ac sine waveform (say current) does not change as it passes through resistors, capacitors or inductors.
1- the frequency
2- the amplitude, and
3- the phase shift
Most features of a sinusoidal waveform are directly dependant on the choice of these parameters.
A sinusoidal waveform is a periodic function. This means it continually repeats itself after a fixed time interval. Here, the period is defined as the time between successive repetitions. The portion of a waveform contains one period is called a cycle, and the frequency is the number of cycles that occur in one second.
The amplitude is the maximum value of waveform as measured from its average. The amplitude determines how large the sinusoid is. Since the sine function oscillates between +1 and -1, the sine waveform oscillates between +E_m and -E_m, or positive maximum and negative maximum.
The phase shift, on the other hand, determines how far the sine waveform is from a reference sinusoidal waveform.
And finally, the magnitude of a waveform at any instant of time is referred to as instantaneous value. Soon you will be hearing about the instantaneous power that is the product of instantaneous current and instantaneous voltage.
Two sine waves may be "IN PHASE" or one LAGS or LEADS the other.
In figure shown in this slide both sine waves have the same frequency and hence the same period, and the second sine wave (v2) is leading the first sine wave (v1) by 30 degrees.
It is also true to say v1 is lagging v2 by 30 degrees.
It is shown that the effective (or rms) value of an ac sine wave is the most useful single value constant that could represent a sine wave.
Note should be taken that representation of a sine function by a constant single value is possible because sinusoidal functions are periodic.
Effective or RMS value of a sinusoidal waveform is well explained by comparing the ac quantity with its equivalent dc. For instance, a sine current has an effective value equivalent to the dc current if both generate the same amount of heat in a given resistor during a given time interval.
There are three steps involved in finding out the root mean square (rms) value of a periodic function.
First we need to Square the function, then the mean value of the resulting square should be found, and finally the square root of the resulting value is to be determined.
If the periodic function is a sinusoid, then its effective value becomes equal to its amplitude divided by square root 2.
It's worthwhile to note that when we say in Australia the mains voltage is 240 volts at 50 Hz, we mean the effective value of voltage is 240 volts and its frequency is 50 Hz. This also means our TV set, for example, sees an instantaneous voltage of 240 volts multiply by root 2 every 20 milliseconds.
These collisions convert electrical energy to heat and is called resistance of the material.
A purely resistive element opposes the flow of current and converts electrical energy to heat, but it does NOT introduce a phase shift between current and voltage. Hence, the sine voltage across the resistor and the sine current through the resistor are in phase.
Rather similar definitions can be put forward for inductive and capacitive reactances.
Inductive Reactance is the opposition to the flow of current by an inductor.
This opposition results in a continual interchange of energy between the source and the magnetic field of the inductor. This process involves no energy consumption whatsoever. Nevertheless, this process introduces a phase shift between sine current through the inductor and sine voltage across the inductor.
For a purely inductive element, the current through the element is lagging its voltage by 90 degrees.
Similarly, capacitive Reactance is the opposition to the flow of current by a capacitor.
This opposition results in a continual interchange of energy between the source and the electric field of the capacitor. This process, similar to inductor, involves no energy dissipation, but introduces a phase shift between sine current through the capacitor and sine voltage across the capacitor.
For a purely capacitive element, the current through the element leads the voltage by 90 degrees.
So we need to ask ourselves two fundamental questions:
In fact, the instantaneous power changes at a rate twice that of current and voltage. This means, the frequency of power delivered to your for instance washing machine is 100 Hz because its current and voltage are 50 Hz sine waves.
The average of instantaneous power is a useful fixed value that can be assigned to ac power in ac circuits.
The average power (or power in short) resulted from the application of a sine voltage across a resistor is the product of its rms current and its rms voltage. And, the average power associated with capacitor and inductors, as mentioned before, is ZERO.
How do we determine the algebraic sum of two or more voltages (or currents) that are varying sinusoidally?
We do have two options here.
A complex number represents a point in a two-dimensional plane with reference to two distinct axes. This point can also determine a radius vector drawn from the origin to the point of interest.
Manipulating sinusoids using complex notations turns trigonometric problems into simple arithmetic and algebra, and this is really handy.
If a complex number is plotted on rectangular co-ordinates with the real part on the X-axis and the imaginary part on the Y-axis, adding and subtracting complex numbers can be done using vector addition and subtraction.
At the same time the vectors representing the complex numbers could be represented by their magnitude and phase using polar co-ordinates.
In polar form, multiplication consists of multiplying the magnitudes and adding the phases while division consists of dividing the magnitudes and subtracting the phases.
So it is easier to add and subtract complex numbers if they are in rectangular form and it is easier to multiply and divide complex numbers if they are in polar form.
I trust you recall the two basic equation sets allowing the conversion of rectangular form to polar form and vice versa.
Some calculators have a built-in function to perform this task.
A complex number notation for representing sinusoids is called a phasor.
This mathematical notation is a convenience even though it first seems to be a burden for many students. The convenience comes from converting all trigonometric calculations into simple algebraic manipulations.
What is very important for you to remember is the fact that the term PHASOR is reserved for quantities such as currents and voltages, representing sinusoidal varying functions of time. This means if the function is NOT a sine function, then it can not be represented by PHASOR notation.
rule number 1-
The magnitude of a phasor is equal to the effective value
of the sine wave it represents.
similarly, the angle of a phasor is equal to the phase angle of the sine wave it represents.
rule number 2-
the phase angle of one waveform is specified with respect to the
reference waveform (the one passes through the origin), and
the reference phasor usually is selected on horizontal line.
rule number 3-
Counter clockwise rotation is used as positive direction for
angles; a leading angle is positive and a lagging angle
is negative
And finally An underline or boldface character usually is used to indicate a phasor, and also to distinguish it from the rms notation.
Information associated with time domain representation are:
In phasor domain, the associated information are:
Note: the only missing information in phasor domain is the frequency. This does not cause a problem as the response of a sine wave by R, L and C is also a sine wave with the same frequency.
Therefore, frequency domain calculations are to be conducted for a given single frequency shared by all currents and voltages within the circuit. The following example takes this point a bit further.
You need to extract information associated with each, and show under what condition these two voltages can be plotted in the same complex plane?
Information Associated with Time Domain Representation of v1 are:
Having established the required information, v1 can be represented in a phasor domain as
150 volts at an angle of 30 degrees
Information Associated with Phasor Representation of V2 are:
The two voltages can be added on the same complex plane if the frequency of v2 is 60Hz (ie. the same frequency as v1).
Phasor diagrams also provide a new insight into the voltage and current relations.
In ac circuit analysis, a phasor diagram shows at a glance the magnitude and phase relations among various quantities within the circuit.
A 220V/50Hz ac voltage supplying
The CURRENT drawn by the resistor in circuit1
Reminding you that resistors do not introduce phase angle between current and voltage.
Having the rms and the phase angle of current through the resistor and voltage across the resistor Phasor diagram1 can be drawn.
Similarly, the magnitude, frequency and phase angle of the CURRENT drawn by the capacitor and the inductor can be identified. This leads to Phasor Diagrams 2 and 3, as shown in this slide.
Note the phase angles of +90 and -90 degrees for circuits 2 and 3 respectively.
This also is a typical example involving the conversion of time domain quantities into phasor domain, and carrying out the calculations based on KVL and KCL in phasor domain.
It is much simpler to carry out calculations in phasor domain rather than in time domain.
I leave the time domain calculations for you to complete.
First of all, the impedance is a measure of how much a circuit containing R, L and C control or impede the level of current through an ac circuit.
If you recall the basic definition of resistance that was covered in dc circuits, you would find the same concept that is shared between the resistance and the impedance.
In simple terms, the impedance is the opposition of a circuit containing R, L, and C to the flow of ac current, and hence it is measured in "Ohms".
Impedance is a complex number, having both real and imaginary parts (in rectangular form) or magnitude and associated angle (in polar form).
In rectangular form, the impedance is represented by a point in a two-dimensional complex plane with reference to two distinct coordinates:
Please note that impedance is NOT a phasor, even though its format is very similar to the phasor notation.
Impedance of a circuit is a fixed value at a given frequency. A phasor represents a sinusoidal that is a time varying function at a given frequency.
To calculate a circuit impedance containing R, C and L, one should know the impedance of any one element when it appears in a circuit alone.
In this regard let us examine the following statements:
The points mentioned above will be used to construct an impedance diagram for an ac circuit containing series connection of R, C and L.
This total impedance has a magnitude of |Z_total| that can be calculated using the same approach as shown in finding equivalent resistance of series and parallel resistors in dc circuits. The total impedance also will have a phase angle that extends from -90 to +90 degrees.
recapping a few major points so far:
In fact, an impedance diagram reflects the individual and total impedance levels of a series ac circuit.
To construct the impedance diagram for a circuit containing series connection of R, L and C, the angles associated with R, XL and XC need special attention.
In placing the resistance, inductive reactance and capacitive reactance on a complex plane the following points should be taken into account:
Now, one may ask what if these impedances are connected in parallel?
Should we be able to use the impedance diagram for parallel connections of impedances? The answer to this question is NO. The following slide provides more information on finding total impedance of parallel connected impedances.
The only difference is the impedance is a complex number and the resistance is a real number.
The inverse of the total impedance (say viewed from the ac power source) of the circuit shown in this slide is equal to:
one over Z_1 plus one over Z_2, and so forth
This equation illustrates the following two important points:
Admittance Diagram is beyond the scope of this subject. Nevertheless, we define the complex number of admittance as being equal to the inverse of impedance.
This means that the admittance is a measure of how well an ac circuit will admit or allow current to flow through a circuit.
It is very important to remember that for any circuit configurations (series, parallel and combined series/parallel) the angle associated with the total impedance is equal to the angle by which the applied voltage leads or lags the source current.
This can be concluded from the Ohm's Law equation. The complex number on the left side of the equation can not be equal to the complex number appears on the right unless angles and magnitudes are equal.
The first one is an ac circuit with one energy source supplying a series and parallel combination of R, L and C.
In many instances, it is a good practice to redraw the circuit showing elements and/or sections of the circuit in BLOCK IMPEDANCES before attempting to calculate say the total impedance of the circuit.
In the second example, as shown in this slide, the energy source supplies a parallel combination of an inductor and series connection of a capacitor and a resistor.
Once again, the circuit is redrawn using BLOCK IMPEDANCES. And, the well known voltage divider rule is used to determine the voltage across the capacitor.
The complete solutions of examples covered in this presentation, including these two examples, are provided in your course notes, and I strongly suggest you take your time and go through solutions thoroughly.
Thank you for viewing and listening, and please send me your feedback and comments on the application of stream and synchronized multimedia educational modules in general and this presentation in particular.
All the best and Good luck with your final exam...
footnote
The presentation is the outcome of an UOW-ESDF project, awarded
in 1998. The project , aimed to set up a simple virtual
classroom environment to enhance student learning.