What is the Signal?
There are different measurable quantities in the world surrounding us. Some quantities are constant like acceleration due to gravity, speed of light, velocity of sound in air. Some are time-varying like AC voltage, Pressure, Temperature. It means they change their value as time passes on. Signal simply means the value of any quantity taken over a period of time. Signals are usually time varying in nature. Generally a graph is plotted between values at different time instants. This is called graphical representation of signal.
What is Sine Wave or Sinusoidal Wave Signal?
Sine Wave or Sinusoidal Wave Signal is a special type of signal. It is given by the function
When Sine wave starts from zero and covers positive values, reaches zero; and again covers negative values, reaches zero, it is said to have completed one cycle or single cycle.
The upper part of sine wave is called positive cycle and the lower part is called negative cycle in a single cycle.
For different values of time, the Signal gives the values of quantity at that time. Therefore Signal is a function of time. It is therefore written asf (t). The Maximum value of the Sinusoidal Signal is also called its amplitude (A). Here ω is called Angular Frequency of Signal and f is the Frequency of Signal. ∅ Is called Phase difference.
Frequency is measured in Hertz (Hz). It shows number of cycles of signal that took place in a second. Large ω or large f value indicates that the signal completes more oscillations (i.e., going from positive values to negative values) in less amount of time. Hence the Signal is more Oscillatory in nature.
Sinusoidal signal need not start at zero. It may start after certain duration of time. This is time after which Sinusoidal Signal starts is indicated with the help of phase difference (∅). It is measured in Radians.
Periodic signals are those which repeat their pattern after certain amount of time. This time after which pattern is repeated is called time period (T) of Periodic Signal. It is inverse of frequency of Signal.
A sinusoidal signal is a periodic signal, because the pattern keeps on repeating after one Wavelength as shown in the Figure above.
All the power signals in our home, office and industries are AC sinusoidal signals. The frequency (f) in India and British countries is 50 Hz and in American countries it is 60 Hz.
Why is Sinusoidal Wave Signal so Important?
Sinusoidal signals are important in both electrical and electronic engineering domains.
According to Fourier Series Theory, any signal (Periodic Signal) can be written in terms of only sine and cosine Signals of different frequencies. Therefore a complex signal can be broken-down into simple sine and cosine signals and mathematical analysis becomes easy. Hence it is widely used in electrical and electronic analysis.
Also, in transformers the output voltage is time derivative of magnetic flux. Magnetic flux is itself time derivative of input voltage. But we want same voltage signal both at input and output. The only functions that satisfy this condition are sine and cosine functions. As sine signal starts from zero value, it is preferred. Therefore majority of power systems in the world today are using sinusoidal AC voltage. All the household equipment also work on Sinusoidal AC voltage.
Power in AC Circuit
AC circuits are usually three-phase for electrical distribution and electrical transmission purposes. Single phase circuits are commonly used in our domestic supply system. The total power of a three-phase AC circuit is equal to three times the single phase power. So if the power in a single phase of a three-phase system is ‘P’, then the total power of the three-phase system would by 3P (provided the three-phase system is perfectly balanced). But if the three-phase system is not exactly balanced, then the total power of the system would be the sum of the power of individual phases. Suppose, in a three phase system, the power at R phase is PR , at Y phase is PY and at B phase is PB, then total power of the system would be
This is simple scalar sum, since power is a scalar quantity. This is the season, if we consider only single phase during calculating and analyzing of three phase power, it is enough.
Let us consider, network A is electrically connected with network B as shown in the figure below:
Let us consider the expression of the voltage waveform of a single phase system is:
Where V is the amplitude of the waveform, ω is the angular velocity of propagation of the wave.Now, consider the current of the system is i(t) and this current has a phase difference from the voltage by an angle φ. That means current wave propagates with φ radiant lag in respect of the voltage. The voltage and current waveform can be represented graphically as shown below:
The current waveform in this case can be represented as:
Now, the expression of the instantaneous power,
[where Vrms and Irms is the root mean square value of voltage and current waveform]Now, let us plot the term P versus time,
It is seen from the graph that, the term P does not have any negative value. So, it will have a nonzero average value. It is sinusoidal with a frequency twice of system frequency. Let us now plot second term of the power equation, i.e. Q.
This is purely sinusoidal and has a zero average value. So from of these two graphs, it is clear that P is the component of power in an AC circuit, which actually transported from network A to network B. This power is consumed in network B as electric power.
Q on the other hand does not really flow from network A to network B. Rather it oscillate between network A and B. This is also component of power, actually flowing into and out of the inductor, capacitor like energy storage elements of the network.
Here, P is known as the real or active part of the power and Q is known as imaginary or reactive part of the power
Hence, P is called real power or active power, and Q is called imaginary or active power. The unit of active power is Watt, whereas the unit of reactive power is Voltage Ampere Reactive or VAR.
We have already considered,
where, S is the product of root mean value of voltage and current i.e.
This product of RMS value of voltage and current of a system is referred as apparent power is Voltage Ampere or VA. So,
This can be represented in complex form as
Again, the expression of the real power is
where ɸ is the angle between voltage and current phasor. So,
So, here in the expression P, cos ɸ is the factor which determines the real power component of an apparent power S.
This is why the term cos ɸ in the expression of real power is called power factor. For both positive and negative value of ɸ, cos ɸ is always positive.
This implies, regardless of the sign of ɸ (which is dependent on whether the current is lagging or leading the voltage) real power is always positive.
That means it flows from the sending end (Network A) to the receiving end (Network B). We have also shown the same earlier when looking at the waveform for real power.
Now if the current is leading the voltage, then the angle between voltage and current phasor is negative, taking the voltage phasor as reference:
In that case, the reactive component of the power is negative,
Power Triangle
The relation between apparent power to active power and reactive power can be represented in trigonometric form as shown below.
Now, if the current is lagging the voltage the angle between voltage and current phasor is positive, taking the voltage phasor as reference.
In this case, the reactive component of power is positive. Since,
The power triangle is represented as shown below.
If the impedance of the network is capacitive, the current leads the voltage and in case of inductive network the current lags voltage. So we can conclude, the reactive power is negative in the case of capacitive reactance and it is positive and in the case of inductive reactance.
If the network is purely resistive, there would not be any angular difference between current and voltage. Hence,
So the reactive power, in this case, would be,
Thus, there is no reactive power generated or consumed in the network.
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