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AC circuits and Phasors
Introduction
A phasor or a vector is a quantity that can be described by two numbers which represents its
magnitude and direction. Thus it can be drawn as a directed line segment of arrow. The
difference between phasor and vector comes about as a result of direction. In a vector, the
direction is oriented in space for instance a force vector. A phasor direction represents a time
difference between two quantities in an AC circuit such as the voltage across a resistor and
voltage across an inductor. Thus vector and phasors are vectors significant in the study of AC
Circuits and use of the trigonometric concept.
In trigonometric function, there are three laws that will play a crucial role in this study and this
are the law of sines and the law of cosines which are useful in solving vector and phasor
problems. The extension of Kirchoffs voltage, ohms law and current laws to AC Circuits and
Phasors forms the foundation of AC Circuit theory.
AC Circuit
An alternating current is explained using phasor and time animations, phase relations,
impedance, RMS quantities and resonance. The AC electricity does supply power and also it is
significant in electronics and signal processing. In the life we are all using AC power in one way
or the other. The power that is supplied to our homes by electricity is always in AC. After the
power is supplied in AC, the devices we use like the television, computers among other devices
convert the AC into DC for usage.
Pure resistance
An AC voltage source is applied across a pure resistor R, where V is the r.m.s voltage and I=V/R
r.m.s current as shown in figure below.
Pure non inductive resistance
The waveform below is the current phase with the voltage
In phase waveforms
Phasors
The diagram below shows the phasor diagram if I is the reference phasor then V is in phase with
I.
Phasor diagram for a pure resistance
Pure inductance
An AC voltage source is applied across a pure inductance where V is the r.m.s voltage and I is
the r.m.s current. Taking I as the reference phasor, the voltage leads the current 90. As shown in
the figures above, the relevant waveform. The phasor diagram is shown below
Pure inductance
The waveform is
Out of phase waveform, i lags v
Notation of Phasors on rectangular coordinate axes
Consider a phasor V lying along OX-axis. If we multiply this phasor by -1, the phasor is reversed
in that it will rotate through 180° in the counter clockwise direction. With this then we can see
what factor the phasor will be multiplies so that it rotates through 90° in the counter clockwise
direction. Suppose the factor is j. as shown in the figure below,
Multiplying the phasor by j2 rotates the phasor through 180 in counter clockwise direction. This
means that multiplying the phasor by j2 is the same as multiplying by -1. Thus the final is that
With this then we can arrive at a very significant conclusion that when a phasor is multiplied by
, the phasor is rotated through 90 in counter clockwise direction. Each successive
multiplication by j rotates the phasor through an additional 90 in the counter clockwise direction.
It is easier to see the multiplying a phasor by
….90° counter clockwise from OX-axis
….180° counter clockwise from OX-axis
….270° counter clockwise rotation from OX-axis
….360° counter clockwise rotation from OX-axis
The significance of j
Just as a symbol of addition + indicates the operation of adding two or more numbers, similarly j
indicates an operation of rotating the phasor through 90° in counter clockwise directions. The
operation of j does not change the magnitude of the phasor. Consider a phasor V displaced 0
counter clockwise from OX- axis as shown in the figure below.
This phasor can be resolved into two rectangular components
the horizontal components a
along X-axis and the vertical components b along Y-axis. It can be seen that vertical components
is displaced 90° counter clockwise from OX-axis. Therefore, mathematically we can express the
component as jb meaning that components b is displaced 90° counter clockwise from
components a. therefore
Magnitude of V
Its angle with OX-axis,
It can be noted that a + jb is the mathematical form of the phasor V.
The quantity a + jb is called a complex number or complex quantity
The horizontal components is called the in-phase or also called active component while the
vertical component is called quadrature or also relative component.
The mathematical representation of phasors
There are three principles ways if representing a phasor in the mathematical form this includes

Rectangular form

Trigonometrical form

Polar form
Rectangular form
This method is also called a symbolic notation. In these methods, the phasor is resolved into
vertical and horizontal components and is expressed in the complex form. In discussion we have
to consider a voltage phasor V displaces 0° counter clockwise from the reference axis. The
horizontal or in-phase component of this phasor is a while the vertical or quadrature is b.
therefore, the phasor can be represented in the rectangular form as:
Magnitude of V
Its angle w.r.t OX-axis,
Phasor in one quadrant
Phasors in several quadrants
The phasors have been resolves into horizontal and vertical component. The various phasors can
be represented in the rectangular form as
The magnitude and phase angle of these phasors can be found out as explained above.
Remember phase angle are to be measured form the reference axis for instance OX-axis. If the
angle is measures counter clockwise from this reference axis, is assigned a positive sing. If the
angle is measure clockwise from the reference axis, it is assigned a negative sign.
Trigonometrical form
This one is similar form except that in-phase and quadrature components of the phasor are
expressed in the trigonometric form. Thus referring to the figure above, a=V cosø and b=V sinø
where V is the magnitude of the phasor V. hence the phasor V can be expressed in the
Trigonometrical form as
Addition and subtraction of phasors
The rectangular form is best suited for addition or subtraction of phasors. If the phasor are given
in polar, they should be first converted to rectangular form and then addition or subtraction be
carried out.
Its angle w.r.t OX-axis will be
Polar form: to multiply the phasors that is in polar form multiply their magnitude and add the
angles algebraically.
It can be seen that multiplication of phasors becomes easier when they are expressed in polar
form.
AC Circuit in relation to resistance, inductance and impedance
An AC circuit consist of circuit elements and power source that provides an alternating voltage
change in V. this time varying voltage form the source is described by
Where
is the maximum output of the source or the voltage amplitude. There are various
possible for AC source including generators and electrical oscillators. In our homes, each
electrical outlet serves as an AC source. Because the output voltage of an AC source varies
sinusoidal with time, the voltage is positive during one half of the cycle and negative during the
other half has shown in the figure below.
The voltage supplied by an AC source is sinusoidal with period t
Likewise, the current in any circuit driven by an AC source is an alternating current that also
varies sinusoidally with time. Thus the angular frequency of the AC voltage is
Where f us the frequency of the source and T is he period. The source determines the frequency
of the current in any circuit connected to it. Commercial electric power plants in a country like
United States use frequency of 60Hz, which corresponds to an angular frequency of 377rad/s.
Resistance in AC circuit
Consider a simple AC circuit consisting of a resistor and an AC source as shown below. At any
instant, the algebraic sum of the voltages around a closed loop in a circuit must be zero according
to Kirchhoff’s loop rule. Therefore,
If we arrange this expression and substitute
resistor is
Where
is the maximum current
The instantaneous voltage across the resistor is
thus the equation given is
for ∆V, the instantaneous current in the
A plot of voltage and current versus time for this circuit is shown below. At point a, the current
has a maximum value in one direction arbitrarily called the positive direction. Between pints a
and b the current is decreasing in magnitude but is still in the positive direction. At point b, the
current is momentarily zero, it them begins to increase in the negative direction between points b
and c. at pint c, the current has reached its maximum value in the negative direction.
This is the plot of the instantaneous current
and instantaneous voltage
across a resistor as
functions of time. The current is in phase with the voltage, which means that the current is zero
when the voltage is zero, maximum when the voltage is maximum and minimum when the
voltage is minimum. At time t=T, one cycle of the time varying and current has been completed.
This is the phasor diagram for the resistive circuit showing that the current is in phase with the
voltage.
The current and voltage are in step with each other because they vary identically with time
because
and
both vary as sin wt reach their maximum values at the same time, hence they
are said to be in phase, similar to the way that two waves can be in phase. Therefore, the
sinusoidal applied voltage, the current in a resistor is always in phase with the voltage across the
resistor. For resistors in AC circuits, there are two concepts to learn. Resistors behave essentially
the same way in both DC and AC circuits but that is not the case for the capacitors and inductors.
To simplify, the analysis of circuits containing two o more elements, we use a graphical
representation called a phasor diagram. A phasor diagram is a vector whose length is
proportional to the maximum value of the variable it represents
for voltage and
for
current). The phasor rotates counter clockwise at an angular speed equal to the angular frequency
associated with the variable the projection of the phasor onto the vertical axis represents the
instantaneous value of the quantity it represents.
Inductance in AC circuit
Let us consider an AC circuit consisting only of an inductor connected to the terminals of an AC
source as shown in the figure below.
If
the self induced instantaneous voltage across the inductor. Kirchhoff’s loop rule applied to this
circuit give ∆V + ∆VL=0 or
Substituting
for
Solving the equation for
and rearranging give
gives
Integrating this expression gives the instantaneous current
in the inductor as a function of time
Using the trigonometric identity
we can express the equation above as
Comparing this result with equation for
inductor and the instantaneous voltage
When the current
shows that the instantaneous current
in the
across the inductor are out of phase by
in the inductor is a maximum, it is momentarily not changing so that voltage
across the inductor is zero. At points such as a and e, the current is zero and the rate of change of
current is at a maximum. Therefore the voltage across the inductor is also at a maximum. Notice
that the voltage reaches its maximum value on quarter of a period before the current reaches its
maximum values. Therefore, for a sinusoidal applied voltage, the current is an inductor always
lags behind the voltage across the inductor by 90°.
As with the relationship between current and voltage, for a resistor, we can represent this
relationship for an inductor with a phasor diagram. The phasor are at 90 to each other,
representing the 90° phase difference between current and voltage.
This plot is of instantaneous current
and instantaneous voltage
across a capacitor as
functions of time. The voltage lags behind the current by 90
This is a phasor diagram for the capacitive circuit showing that the current leads the voltage by
90°.
Using trigonometric identity
We can express the equation in another form
Comparing the expression <v =
sin wt shows that the current is pie/2rad =90° out of
phase with the voltage across the capacitor. A plot of current and voltage versus tome shows that
the current reaches its maximum value of a cycle sooner than the voltage reaches its maximum
value.
Consider a point such as b where the current is zero at this instant. That occurs when the
capacitor reaches its maximum charge so that the voltage across the capacitor is a maximum. At
points such as a and e, the current reaches zero and the capacitor begins to recharge with the
opposite polarity. When the charge is zero, the voltage across the capacitor is zero. Therefore the
current and voltage are out of phase.
As with inductor, we can represent the current and voltage for a capacitor on a phasor diagram.
The phasor diagram shows that for a sinusoidal applied voltage, the current always leads the
voltage the voltage across a capacitor by 90°.
The equation below shows that the current in the circuit reaches its maximum value when
As in this case we give the denominator
the symbol
frequency we define it as the capacitive reactance
We can now write the equation as
and because the function varies with
Impedance
The RLC series circuit
The figure below shown a circuit that contains a resistor an inductor and a capacitor connected in
series across an alternating voltage source. If the applied voltage varies sinusoidally with time,
the instantaneous applied voltage
While the current varies as
Where ø is some phase angle between the current and the applied voltage. Based the discussion
of phase, the expected is that the current will generally no be in phase with voltage in an RL
circuit. The aim is to determine the ø and
. The figure below shows the voltage versus time
across each element in the circuit and their phase relationship.
First, because the elements are in series, the current everywhere in the circuit must be the same at
any instant. That is, the current at all points in a series AC circuit has the same amplitude and
phase. Based in the preceding sections, it is known that the voltage across each element has a
different amplitude and phase. In particular, the voltage across the resistor is in phase with the
current, the voltage across the inductor leads the current by 90°. Using the phase relationship we
can express the instantaneous voltage across the three circuit element as
The sum of these voltages must equal the voltage from the AC source, but it is important to
recognize that because the three voltages have different phase relationship with the current, they
cannot be assed directly. The figure below represents the phasor at an instant at which the current
in all three elements in momentarily zero.
The zero current is represented by the current phasor along the horizontal axis in each part of the
figure. Next the voltage phasor is drawn at the appropriate phase angle to the current for each
element.
Because phasor are rotating vectors the voltage phasors in the figure above can be combined
using vector addition.
This is a phasor diagram for the series RLC circuit. The phasor
phasor
phasor
leads
b y90 and the phasor
lags
is in phase with the current
by 90.
This is the inductance and capacitance phasors which are added together and then added
vertically to the resistance phasor. The total voltage
makes an angle ø with
.
The voltage phasors
and
construct the different phasor
are in opposite directions along the same line, so we can
-
which is perpendicular to the phasor
shows that the vector sum of the voltage amplitude
length is the maximum applied voltage
phasor
,
and
. This diagram
equals a phasor whose
and which makes an angle ø with the current
. Form the inductance and capacitance phasors we can see that
Therefore we can express the maximum current as
Once again, this expression has the denominator of the fraction which plays a role of resistance
and is called the impedance Z of the circuit
Where impedance also has units of ohms. Therefore the equation can also be written as
It is not that the impedance and therefore the current in an AC circuit depend on the resistance,
the inductance and capacitance and the frequency this is because the reactances are frequency
dependent. Form the phasor diagram above, the phase angle ø between the current and voltage is
found as follows
Where
>
which occurs in high frequency, the phase angle is positive, signifying that the
current lags the applied voltage. We can describe the situation by saying that the circuit is more
inductive than capacitive. Where
<
the phase angle in negative signifying that the current
leads the applied voltage and the circuit is more capacitive than inductive. When
phase angle is zero and the circuit is purely resistive.
=
the