Phasors
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Waveforms
Ø DC waveforms
Ø AC waveforms
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Phase Difference and the Phase Shift Equation
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Phase Relationship of a Sinusoidal Waveform
Ø In-Phase
Ø Out-of-Phase
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Concept of Leading and Lagging Phase Difference
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Phasor Diagram of a Sinusoidal waveform
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The cosine waveform
Sine and cosine relationship
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Ways to express sinusoidal waves in form of phasors
Ø Converting rectangular form in form of phasors
Ø Converting polar form in form of phasors
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Phasor Mathematical Relationships
Ø Addition & Subtraction
Ø Multiplication & Division
Ø Reciprocal Property
Ø Complex Conjugation
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Basic Terms
Ø Impedance
Ø Reactance
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Phasor Relationship to basic passive circuit element
Ø Phasor relationship of A.C Resistance
· Sinusoidal waveforms for A.C Resistor
· Phasor diagram for A.C Resistance
Ø Phasor relationship of A.C Inductance
· Sinusoidal waveforms for A.C Inductance
· Phase diagram for A.C Inductance
Ø Phase relationship of A.C Capacitance
· Sinusoidal waveforms for A.C Capacitance
· Phasor diagram for A.C Capacitance
Conclusion for three basic passive elements
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Importance of Phasors
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Phasors
· Any physical variable, be it a sinusoidal signal—voltage or current—can be represented by a phasor, which encodes the amplitude and phase information of the signal.
· Phasor is a rotating vector; it is a scaled line, whose length represents an AC quantity, that has both magnitude (i.e. peak amplitude) and direction (i.e. phase), which is frozen at some point in time.
· The one difference between the vectors and the phasors lies in it that a vector’s magnitude is the “peak-value” of the sinusoid, while a phasor’s magnitude is the “rms” of the sinusoid. In both cases, the phase angle and the direction remains the same.
Note: Phasors in linear circuits are related if all of the signals in the circuit are at the same frequency.
Therefore, the phase of an alternating quantity at any instant in time can be represented by a phasor diagram, which means they can be thought of as “functions of time”.
A complete sine wave can be represented by a single vector, rotating at an angular velocity of ω = 2πƒ, where ƒ is the frequency of the waveform. Consequently, we can say that Phasor is a quantity that has both "Magnitude" and "Direction".
-Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/s.
Waveforms:
Basically, waveforms are visual representations of the alteration suffered by voltage or current plotted to a base of time.
Waveforms can be divided into two:
1. DC waveforms
2. AC waveforms
DC WAVEFORMS:
· DC current/voltage is unidirectional.
AC WAVEFORMS (‘BI-DIRECTIONAL’ WAVEFORM):
· When the instantaneous values of either voltage or current are plotted against time to obtain the shape, it is called an AC waveform.
· An AC waveform constantly changes its polarity. Every half cycle, it alternates between a positive maximum value and a negative maximum value with respect to time.
Example: the domestic mains supply used in our homes.
· It is a time-variant waveform.
Example: a sinusoidal waveform, also known as sine waves.
AC waveforms can have a variety of shapes. Examples include complex waves, square waves, and triangular waves.
Phase Difference and the Phase Shift Equation:
· When a sine wave does not pass through zero at time=0, then it has a phase shift. This is also known as phase difference.
· The phase difference or phase shift is the angle Φ (Greek letter Phi), in degrees or radians. It indicates that the waveform has shifted from a certain reference point along the horizontal zero axis.
The phase difference, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle, and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ = 0 to 360o depending upon the angular units used.
Then the equation for the instantaneous value of a sinusoidal voltage or current waveform becomes.
Also, known as phase difference equation, where:
Am – is the amplitude of the waveform.
ωt – is the angular frequency of the waveform in radian/sec.
Φ (phi) – is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.
If the positive slope of the sinusoidal waveform passes through the horizontal axis "before" t =0, then the waveform has shifted to the left, so Φ >0, and the phase angle will be positive in nature.
Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal axis "after" t = 0, then the waveform has shifted to the right, so Φ <0, and the phase angle will be negative in nature.
The following figure aptly demonstrates the phenomenon of phase shift:
Phase Relationship of a Sinusoidal Waveform:
Two Sinusoidal Waveforms - "in-phase":
When the frequency of the voltage(v) and the current(i) are same, then must both reach their max positive, negative, and zero values during a single cycle at the same time(their amplitudes maybe different). In this case, the two alternating quantities, v and i, are said to be “in-phase”.
Two Sinusoidal Waveforms - "out of-phase":
If aphase difference exists between the two waveforms as the current crosses the horizontal reference axis, reaching its max peak and zero values after the voltage waveform. Due to this phase shift, the two waveforms are said to be “out-of-phase”.
Concept of leading and Lagging Phase difference:
Note that they have a Lagging Phase Difference, where, i lags v by angle Φ. i.e. the equations for the wave forms are
If current, i has a positive value and crosses the reference axis reaching its maximum peak and zero value at some time before the voltage, v then the current waveform will be "leading" the voltage by some phase angle. Then the two waveforms are said to have a Leading Phase Difference, where, i leads v by angle Φ. The expression for both the voltage and the current will be
Phasor Diagram of a Sinusoidal Waveform:
The phasor diagram on the right is drawn corresponding to time zero (t = 0) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage (V) and the current (I) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle Φ, as the two phasors rotate in an anticlockwise direction.
· The Cosine Waveform
If the waveform crosses the horizontal zero axis with a positive going slope 90o or π/2 radians before the reference waveform, the waveform is called a Cosine Waveform. It is a sinusoidal function (same shape as sine waves), but is shifted by +90o or one full quarter of a period ahead of it.
· Sine and Cosine Wave Relationships
A sine wave is a cosine wave that has been shifted in the other direction by -90o. Either way when dealing with sine waves or cosine waves with an angle, the following rules will always apply.
§ When comparing two sinusoidal waveforms it more common to express their relationship as either a sine or cosine with positive going amplitudes and this is achieved using the following mathematical identities.
§ By using these relationships above we can convert any sinusoidal waveform with or without an angular or phase difference from either a sine wave into a cosine wave or vice versa.
§ Adding or subtracting 180 degrees from the argument will change the negative amplitude to positive.
NOTE: For calculations use cosine waveform.
Ways to express sinusoidal waves in form of phasors:
RECTANGULAR FORM: Z = x + jy
POLAR FORM: Z = A ∠Φ
EXPONENTIAL FORM: Z = A ejΦ
Phasors have two components, the magnitude (M) and the phase angle (φ). Phasors are related to sinusoids through our cosine convention:
RECTANGULAR AND POLAR FORM TOO, CAN BE INTERCHANGED:
Converting Rectangular Form into Polar Form, ( R→P )
For expressing a phasor in polar form, the following example can be taken into consideration:
Example: Convert the complex number 5 + 2 i to polar form.
Solution: We have a = 5 and b = 2. We compute
Solution: We have a = 5 and b = 2. We compute
So, the complex number in polar form must be 5.39 ∠ 21.8°.
Converting Polar Form, ( P→R ) into Rectangular Form
A is the part of the phasor along the real axis
B is the part of the phasor along the imaginary axis
Example: Convert the complex number 5∠53° to rectangular form.
Solution: We have R = 5 and Phase angle = 53°. We compute
A = 5 cos (53°) = 3 and b = R sin (53°) = 4
So, the complex number in rectangular form must be 3 + 4 i.
PHASOR MATHEMATICAL RELATIONSHIPS:
Phasor Addition/ Subtraction
Addition and subtraction of phasors can occur in many ways. One of the most widely used methods is:
.
Rectangular Form: In the rectangular form, the phasor is divided up into a real part, x and an imaginary part, y forming the generalized expression Z = x ± jy.
EXAMPLE:
PHASOR ADDITION IN RECTANGULAR FORM:Q. (6+j 12) + (7+j2) = (6+ 7) +j (12+2) = (13+j 14) Ans.
PHASOR SUBTRACTION IN RECTANGULAR FORM:Q.(6+j12) - (7+j2) = (6-7) + (12-j2) = (-1+j10)Ans.
Complex Numbers using the Rectangular Form
· A/B - is the Complex Number representing the Vector
· x - is the Real part or the Active component Z= x+jy
· y - is the Imaginary part or the Reactive component
· j - is defined by √-1
· In the rectangular form, a complex number can be represented as a point on a two-dimensional plane called the complex or s-plane.
Complex Numbers (The j-operator)
· In engineering, the letter “j”, commonly known as the j-operator, is used to distinguish an imaginary number from the real number. The letter j is used in front of a number to signify its imaginary number operation. Examples: j3, j12, j100, etc.
· The value of j-operator is equal to √-1, so successive multiplication of "j", (j x j) will
Polar Form Multiplication and Division
For adding and subtracting complex numbers, rectangular form is the most efficient, but polar form is often better for multiplying and dividing. To multiply together two vectors in polar form, we must first multiply together the two modulus or magnitudes and then add together their angles.
Multiplication of phasors in Polar Form
Two phasors can be easily multiplied according to the figure:
Example: multiply (5 ∠ 30°) · (3 ∠ 25°)
(5 ∠ 30°) · (3 ∠ 25°) = (5·3) ∠ (30+25)° = 15 ∠ 55°
Division of phasors in Polar Form
To divide together two vectors in polar form, we must divide the two modulus, and then subtract their angles as shown:
Example: divide 15 ∠ 32° by 3 ∠ 25°
NOTE: As mentioned, division and multiplication will be carried out in polar form. So, if you have been provided with rectangular form, you need to change it into polar form and proceed. The answer then may be changed to rectangular form in the end.
Example: divide 5 + 3i by 2 − 4i
Reciprocal property
1/C=1/M< -Ø
Example:
Complex Conjugation
Complex conjugation is denoted with an asterisk above the phasor to be conjugated. In general, for polar notation, we have:
In rectangular form, we can express complex conjugation as:
Notice the only difference in the complex conjugate of C is the sign change of the imaginary part.
Some basic terms:
Impedance: (Symbol Z)
A measure of the total opposition to current flow in an alternating current circuit, made up of two components, ohmic resistance and reactance, and usually represented in complex notation as Z = R + iX, where R is the ohmic resistance and X is the reactance.
Reactance:
Opposition to the flow of alternating current caused by the inductance and capacitance in a circuit rather than by resistance.
With AC, the linear circuit elements behave like phasors.
Phasor relationship to basic passive circuit elements:
Ø Phasor Relationship of AC Resistance
There is no phase difference between the current and the voltage when using an AC resistance. Thus, the phase difference is none. Both, voltage and current are “in-phase”.
Sinusoidal Waveforms for AC Resistance
For a purely resistive circuit the AC current flowing through the resistor varies in proportion to the applied voltage across it following the same sinusoidal pattern. 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.
Phasor Diagram for AC Resistance
The voltage and current are both in-phase with each other, as there is no phase difference ( θ = 0 ).
For a resistor, DC resistance = AC impedance , or R = Z.
· The impedance vector is represented by the letter, (Z) for an AC resistance value with the units of Ohm's (Ω) the same as for DC. Then Impedance ( or AC resistance ) can be defined as:
Ø Phasor Relationship of AC Inductance
Voltage applied to a purely inductive circuit "LEADS" the current by a quarter of a cycle or 90o.
Sinusoidal Waveforms for AC Inductance
A purely inductive circuit the voltage "LEADS" the current by 90o. But by using the voltage as our reference, we can also say that the current "LAGS" the voltage by one quarter of a cycle or 90o
Phasor Diagram for AC Inductance
For a pure loss less inductor, VL "leads" IL by 90o, or we can say that IL "lags" VL by 90o.
Ø Phase Relationship for an AC Capacitance
A voltage applied to a purely capacitive circuit "LAGS" the current by a quarter of a cycle or 90o.
In a pure AC Capacitance circuit, the voltage and current are both "out-of-phase" with the current leading the voltage by 90o.
Sinusoidal Waveforms for AC Capacitance
In a purely capacitive circuit the voltage "LAGS" the current by 90o. But, by using the voltage as our reference, we can also say that the current "LEADS" the voltage by one quarter of a cycle or 90o
Phasor Diagram for AC Capacitance
For a pure capacitor, VC "lags" IC by 90o, or we can say that IC "leads" VC by 90o. Using, phasor diagram it could be represented as:
Conclusion for three basic Passive elements:
In the Resistance the phase angle is 0o, in the Inductance it is +90o while in the Capacitance it is -90o. While, the other analysis of these three passive elements can be summarized as:
Circuit Element
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Resistance, (R)
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Reactance, (X)
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Impedance, (Z)
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Resistor
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R
|
0
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Inductor
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0
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ωL
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Capacitor
|
0
|  | |
Note that inductance and capacitance are frequency dependant
We can use this to determine the voltage across an linear circuit with AC: V = IZ
EXAMPLE 1:
CONDITION: Consider three circuit elements in series:
With
R = 2 W
L = 50 mH
C = 750 mF
The total impedance is:
Now, if the current across this circuit is I = 10 cos(t) A. we may compute the voltage:
OrV = 23.76 cos(t – 32.69°) V
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EXAMPLE 2: If the current across this circuit is I = 10 cos(377t) A. (Assume same values of R, C AND L). Find Voltage.
Ans: we need to recalculate the impedance. Since,
Now,
Or V = 189.5 cos(377t + 83.94°) V
IMPORTANCE OF PHASORS:
Why Phasors are Important?
1. Linear, time-invariant (LTI) systems can be said to perform only four operations on a signal: copying, scaling, delaying, and adding. The operation of the LTI system on a complex sinusoid is thus reduced to a calculation involving only phasors, which are simply complex numbers.
2. Sinusoids can be expressed as phasors and calculations can be performed easily. Thus, reducing the work.
3. Phasors allow a simple and easy method to represent and perform calculations on a waveform.
4. Phasors give information of the wave at that instant i.e. amplitude (rms value) and direction (phase angle).
5. By treating polyphase AC circuit quantities as phasors, balanced circuits can be simplified and unbalanced circuits can be treated as an algebraic combination of symmetrical circuits. This approach greatly simplifies the work required in electrical calculations of voltage drop, power flow, and short-circuit currents.
6. The technique of synchrophasors (A phasor measurement unit (PMU) or synchrophasor is a device which measures the electrical waves on an electricity grid, using a common time source for synchronization) uses digital instruments to measure the phasors representing transmission system voltages at widespread points in a transmission network. Small changes in the phasors are sensitive indicators of power flow and system stability.
