Assignment Task
Period
The Period goes from one peak to the next or from any point to the next matching point The period of the sine function is 2π, which means that the value of the function is the same every 2π units. The sine function, like cosine, tangent, cotangent, and many other trigonometric function, is a periodic function which means it repeats its values on regular intervals, or “periods.” In the case of the sine function, that interval is 2π.
Maximum value
Combining these figures yields the functions y = A + B sin x and also y = A + B cos x. These two functions have minimum and maximum values as defined by the following formulas. The maximum value of the function is M = A + |B|. This maximum value occurs whenever sin x = 1 or cos x = 1. The minimum value of the function is m = A ‐ |B|. This minimum occurs whenever sin x = −1 or cos x = −1.
Peak-to-peak voltage (Vpp) can be explained in terms of a sinusoidal-waveform voltage. Peak-to-peak voltage is a parameter measured between the maximum signal amplitude value and its minimum value (which can be negative, as in this case) over a single period (Fig. 1)
A similar parameter used in electronics is often mistaken for the peak-to-peak voltage, i.e., peak voltage denoted as Vp. Peak voltage (Vp) is measured from 0 to the maximum value (5V in the example). For a sinusoidal-waveform signal, Vpp will always be twice the Vp.
Instantaneous Valve
The instantaneous value of a sinusoidal waveform is given as the “Instantaneous value = Maximum value x sin θ ” and this is generalized by the formula. Where, Vmax is the maximum voltage induced in the coil and θ = WT, is the rotational angle of the coil with respect to time Next: The EMF induced in the coil at any instant of time depends upon the rate or speed at which the coil cuts the lines of magnetic flux between the poles and this is dependant upon the angle of rotation, Theta ( θ ) of the generating device. Because an AC waveform is constantly changing its value or amplitude, the waveform at any instant in time will have a different value from its next instant in time.
For example, the value at 1ms will be different to the value at 1.2ms and so on. These values are known generally as the Instantaneous Values, or Vi Then the instantaneous value of the waveform and also its direction will vary according to the position of the coil within the magnetic field as shown below.
Displacement of a Coil within a Magnetic Field
The instantaneous value of a sinusoidal waveform is given as the “Instantaneous value = Maximum value x sin θ ” and this is generalized by the formula. Where, Vmax is the maximum voltage induced in the coil and θ = ωt, is the rotational angle of the coil with respect to time. If we know the maximum or peak value of the waveform, by using the formula above the instantaneous values at various points along the waveform can be calculated. By plotting these values out onto graph paper, a sinusoidal waveform shape can be constructed. In order to keep things simple, we will plot the instantaneous values for the sinusoidal waveform at every 45o of rotation giving us 8 points to plot. Again, to keep it simple we will assume a maximum voltage, VMAX value of 100V. Plotting the instantaneous values at shorter intervals, for example at every 30o (12 points) or 10o (36 points) for example would result in a more accurate sinusoidal waveform construction. AVERAGE VALUE The process used to find the of Average value an alternating waveform is very similar to that for finding its RMS value, the difference this time is that the instantaneous values are not squared and we do not find the square root of the summed mean. The average voltage (or current) of a periodic waveform whether it is a sine wave, square wave or triangular waveform is the equivalent to the DC value of an alternating waveform. The average or mean value is defined as: “the quotient of the area under the waveform with respect to time“. In other words, the averaging of all the instantaneous values along time axis with time being one full period, (T). For a periodic waveform, the area above the horizontal axis is positive while the area below the horizontal axis is negative. The result is that the average or mean value of a symmetrical alternating quantity over the full 360o time period is therefore zero, (0). This is because the area above the horizontal axis (the positive half cycle) is the same as the area below the axis (the negative half cycle) and thus cancel each other out. In other words, when we do the maths of the two areas, the negative area cancels out the positive area producing a zero average value. Then the average or mean value of a symmetrical alternating
Quantity, such as a sine wave, is taken over the time period of only one half of a cycle, since as we have just stated, the average value over one complete cycle is zero regardless of the peak amplitude.
Average Voltage Summary Then to summarise. When dealing with alternating voltages (or currents), the term average valve is generally taken over one complete cycle, whereas the term mean value is used for one half of the periodic cycle. The average value of a whole sinusoidal waveform over one complete cycle is zero as the two halves cancel each other out, so the average value is taken over half a cycle. The average value of a sine wave of voltage or current is 0.637 times the peak value, (Vp or Ip. This mathematical relationship between the average values applies to both AC current and AC voltage.
Root-mean-square (RMS) value
The RMS value is the square root of the mean (average) value of the squared function of the instantaneous values. The symbols used for defining an RMS value are V rms or I rms,The term RMS, ONLY refers to time-varying sinusoidal voltages, currents or complex waveforms were the magnitude of the waveform changes over time and is not used in DC circuit analysis or calculations where the magnitude is always constant. When used to compare the equivalent RMS voltage value of an alternating sinusoidal waveform that supplies the same electrical power to a given load as an equivalent DC circuit, the RMS value is called the “effective value” and is generally presented as: V eff or I eff
In other words, the effective value is an equivalent DC value which tells you how many volts or amps of DC that a time-varying sinusoidal waveform is equal to in terms of its ability to produce the same power.
For example, the domestic mains supply in the Australia is 240Vac. This value is assumed to indicate an effective value of “240 Volts rms”. This means then that the sinusoidal rms voltage from the wall sockets of Australia home is capable of producing the same average
