Explain the expression for instantaneous current i and its phase relationshp to applied voltage
Answers
We have seen in the previous tutorials that in an AC circuit containing sinusoidal waveforms, voltage and current phasors along with complex numbers can be used to represent a complex quantity.
We also saw that sinusoidal waveforms and functions that were previously drawn in the time-domain transform can be converted into the spatial or phasor-domain so that phasor diagrams can be constructed to find this phasor voltage-current relationship.
Now that we know how to represent a voltage or current as a phasor we can look at this relationship when applied to basic passive circuit elements such as an AC Resistance when connected to a single phase AC supply.
Any ideal basic circuit element such as a resistor can be described mathematically in terms of its voltage and current, and in the tutorial about resistors, we saw that the voltage across a pure ohmic resistor is linearly proportional to the current flowing through it as defined by Ohm’s Law. Consider the circuit below.
AC Resistance with a Sinusoidal Supply
AC Resistance
When the switch is closed, an AC voltage, V will be applied to resistor, R. This voltage will cause a current to flow which in turn will rise and fall as the applied voltage rises and falls sinusoidally. As the load is a resistance, the current and voltage will both reach their maximum or peak values and fall through zero at exactly the same time, i.e. they rise and fall simultaneously and are therefore said to be “in-phase ”.
Then the electrical current that flows through an AC resistance varies sinusoidally with time and is represented by the expression, I(t) = Im x sin(ωt + θ), where Im is the maximum amplitude of the current and θ is its phase angle. In addition we can also say that for any given current, i flowing through the resistor the maximum or peak voltage across the terminals of R will be given by Ohm’s Law as:
voltage across a resistance
and the instantaneous value of the current, i will be:
current through a resistance
So for a purely resistive circuit the alternating current flowing through the resistor varies in proportion to the applied voltage across it following the same sinusoidal pattern. As the supply frequency is common to both the voltage and current, their phasors will also be common resulting in the current being “in-phase” with the voltage, ( θ = 0 ).
In other words, there is no phase difference between the current and the voltage when using an AC resistance as the current will achieve its maximum, minimum and zero values whenever the voltage reaches its maximum, minimum and zero values as shown below.
Sinusoidal Waveforms for AC Resistance
AC resistance waveforms
This “in-phase” effect can also be represented by a phasor diagram. In the complex domain, resistance is a real number only meaning that there is no “j” or imaginary component. Therefore, as the voltage and current are both in-phase with each other, there will be no phase difference ( θ = 0 ) between them, so the vectors of each quantity are drawn super-imposed upon one another along the same reference axis. The transformation from the sinusoidal time-domain into the phasor-domain is given as.
Phasor Diagram for AC Resistance
ac resistance phasor diagram
As a phasor represents the RMS values of the voltage and current quantities unlike a vector which represents the peak or maximum values, dividing the peak value of the time-domain expressions above by √2 the corresponding voltage-current phasor relationship is given as.
RMS Relationship
voltage and current magnitude
Phase Relationship
phase relationship