AC Instantaneous Value and RMS Value:
AC Instantaneous Value and RMS Value
Instantaneous Value:
The Instantaneous value of an alternating voltage or current is the value of voltage or current at one particular instant.
The value may be zero if the particular instant is the time in the cycle at which the polarity of the voltage is changing.
It may also be the same as the peak value, if the selected instant is the time in the cycle at which the voltage or current stops increasing and starts decreasing.
There are actually an infinite number of instantaneous values between zero and the peak value.
RMS Value:
The average value of an AC waveform is NOT the same value as that for a DC waveforms average value.
This is because the AC waveform is constantly changing with time and the heating effect given by the formula ( P = I 2.R ), will also be changing producing a positive power consumption.
The equivalent average value for an alternating current system that provides the same power to the load as a DC equivalent circuit is called the “effective value”.
This effective power in an alternating current system is therefore equal to: ( I2.R. Average).
As power is proportional to current squared, the effective current, I will be equal to √ I 2 Ave.
Therefore, the effective current in an AC system is called the Root Mean Squared or RMS.
Pure Resistive circuit:
Resistors are “passive” devices that are they do not produce or consume any electrical energy, but convert electrical energy into heat.
In DC circuits the linear ratio of voltage to current in a resistor is called its resistance.
However, in AC circuits this ratio of voltage to current depends upon the frequency and phase difference or phase angle ( φ ) of the supply.
So when using resistors in AC circuits the term Impedance, symbol Z is the generally used and we can say that DC resistance = AC impedance, R = Z.
It is important to note, that when used in AC circuits, a resistor will always have the same resistive value no matter what the supply frequency from DC to very high frequencies, unlike capacitor and inductors.
For resistors in AC circuits the direction of the current flowing through them has no effect on the behaviour of the resistor so will rise and fall as the voltage rises and falls.
The current and voltage reach maximum, fall through zero and reach minimum at exactly the same time.
i.e, they rise and fall simultaneously and are said to be “in-phase” as shown below.
We can see that at any point along the horizontal axis that the instantaneous voltage and current are in-phase because the current and the voltage reach their maximum values at the same time, that is their phase angle θ is 0o.
Then these instantaneous values of voltage and current can be compared to give the ohmic value of the resistance simply by using ohms law.
Consider below the circuit consisting of an AC source and a resistor.
The instantaneous voltage across the resistor, VR is equal to the supply voltage, Vt and is given as:
VR = Vmax sinωt
The instantaneous current flowing in the resistor will therefore be:
IR = VR / R
= Vmax sinωt / R
= I max sinωt
In purely resistive series AC circuits, all the voltage drops across the resistors can be added together to find the total circuit voltage as all the voltages are in-phase with each other.
Likewise, in a purely resistive parallel AC circuit, all the individual branch currents can be added together to find the total circuit current because all the branch currents are in-phase with each other.
Since for resistors in AC circuits the phase angle φ between the voltage and the current is zero, then the power factor of the circuit is given as cos 0o = 1.0.
The power in the circuit at any instant in time can be found by multiplying the voltage and current at that instant.
Then the power (P), consumed by the circuit is given as P = Vrms Ι cos Φ in watt’s. But since cos Φ = 1 in a purely resistive circuit, the power consumed is simply given as, P = Vrms Ι the same as for Ohm’s Law.
This then gives us the “Power” waveform and which is shown below as a series of positive pulses because when the voltage and current are both in their positive half of the cycle the resultant power is positive.
When the voltage and current are both negative, the product of the two negative values gives a positive power pulse.
Then the power dissipated in a purely resistive load fed from an AC rms supply is the same as that for a resistor connected to a DC supply and is given as:
P = V rms * I rms
= I 2 rms * R
= V 2 rms / R
Pure Inductive circuits:
This simple circuit above consists of a pure inductance of L Henries ( H ), connected across a sinusoidal voltage given by the expression: V(t) = Vmax sin ωt.
When the switch is closed this sinusoidal voltage will cause a current to flow and rise from zero to its maximum value.
This rise or change in the current will induce a magnetic field within the coil which in turn will oppose or restrict this change in the current.
But before the current has had time to reach its maximum value as it would in a DC circuit, the voltage changes polarity causing the current to change direction.
This change in the other direction once again being delayed by the self-induced back emf in the coil, and in a circuit containing a pure inductance only, the current is delayed by 90o.
The applied voltage reaches its maximum positive value a quarter ( 1/4ƒ ) of a cycle earlier than the current reaches its maximum positive value, in other words, a voltage applied to a purely inductive circuit “LEADS” the current by a quarter of a cycle or 90o as shown below.
The instantaneous voltage across the resistor, VR is equal to the supply voltage, Vt and is given as:
VL = Vmax sin (ωt + 90)
IL = V / XL
XL = 2πfL
Pure Capacitive circuits:
When the switch is closed in the circuit above, a high current will start to flow into the capacitor as there is no charge on the plates at t = 0.
The sinusoidal supply voltage, V is increasing in a positive direction at its maximum rate as it crosses the zero reference axis at an instant in time given as 0o.
Since the rate of change of the potential difference across the plates is now at its maximum value, the flow of current into the capacitor will also be at its maximum rate as the maximum amount of electrons are moving from one plate to the other.
As the sinusoidal supply voltage reaches its 90o point on the waveform it begins to slow down and for a very brief instant in time the potential difference across the plates is neither increasing nor decreasing therefore the current decreases to zero as there is no rate of voltage change.
At this 90opoint the potential difference across the capacitor is at its maximum ( Vmax ), no current flows into the capacitor as the capacitor is now fully charged and its plates saturated with electrons.
At the end of this instant in time the supply voltage begins to decrease in a negative direction down towards the zero reference line at 180o.
Although the supply voltage is still positive in nature the capacitor starts to discharge some of its excess electrons on its plates in an effort to maintain a constant voltage.
These results in the capacitor current flowing in the opposite or negative direction.
When the supply voltage waveform crosses the zero reference axis point at instant 180o, the rate of change or slope of the sinusoidal supply voltage is at its maximum but in a negative direction, consequently the current flowing into the capacitor is also at its maximum rate at that instant.
Also at this 180o point the potential difference across the plates is zero as the amount of charge is equally distributed between the two plates.
Then during this first half cycle 0o to 180o, the applied voltage reaches its maximum positive value a quarter (1/4ƒ) of a cycle after the current reaches its maximum positive value, in other words, a voltage applied to a purely capacitive circuit “LAGS” the current by a quarter of a cycle or 90o as shown below.
IC = Imax sin (ωt + 90)
IL = V / XC
XC = 1 / 2πfC
For more details about AC Instantaneous Value and RMS Value click here
To see other topics in Basic Electrical and Electronics Engineering click here
Click Here to Download the pdf of this topic AC Instantaneous Value and RMS Value.
Other links
AC Circuits and Kirchhoff’s law
Steady State Solution of DC Circuits and Problems based on ohm’s law
RL Series Circuit And LR Series Circuit
Electrical Instruments and Classification of instruments
Permanent Magnet Moving Coil Instruments (PMMC)
Operating Principles of Moving Iron Instruments Ammeters and Voltmeters