AC Resistance and Impedance

Impedance, measured in Ohms, is the effective resistance to current flow around an AC circuit containing resistances and reactances.

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 Resistancewhen 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

 

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:

 

and the instantaneous value of the current, i will be:

 

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

 

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

 

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

Phase Relationship

 

This shows that a pure resistance within an AC circuit produces a relationship between its voltage and current phasors in exactly the same way as it would relate the same resistors voltage and current relationship within a DC circuit. However, in a DC circuit this relationship is commonly called Resistance, as defined by Ohm’s Law but in a sinusoidal AC circuit this voltage-current relationship is now called Impedance. In other words, in an AC circuit electrical resistance is called “Impedance”.

In both cases this voltage-current ( V-I ) relationship is always linear in a pure resistance. So when using resistors in AC circuits the term Impedance, symbol Z is the generally used to mean its resistance. Therefore, we can correctly say that 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:

AC Impedance

Impedance can also be represented by a complex number as it depends upon the frequency of the circuit, ω when reactive components are present. But in the case of a purely resistive circuit this reactive component will always be zero and the general expression for impedance in a purely resistive circuit given as a complex number will be:

Z = R + j0 = R Ω’s

Since the phase angle between the voltage and current in a purely resistive AC circuit is zero, the power factor must also be zero and is given as: cos 0o = 1.0 , Then the instantaneous power consumed in the resistor is given by:

 

However, as the average power in a resistive or reactive circuit depends upon the phase angle and in a purely resistive circuit this is equal to θ = 0, the power factor is equal to one so the average power consumed by an AC resistance can be defined simply by using Ohm’s Law as:

Which are the same Ohm’s Law equations as for DC circuits. Then the effective power consumed by an AC resistance is equal to the power consumed by the same resistor in a DC circuit.

Many AC circuits such as heating elements and lamps consist of a pure ohmic resistance only and have negligible values of inductance or capacitance containing on impedance.

In such circuits we can use both Ohm’s Law ,Kirchoff’s Law  as well as simple circuit rules for calculating and finding the voltage, current, impedance and power as in DC circuit analysis. When working with such rules it is usual to use RMS values only.

AC Resistance Example No1

An electrical heating element which has an AC resistance of 60 Ohms is connected across a 240V AC single phase supply. Calculate the current drawn from the supply and the power consumed by the heating element. Also draw the corresponding phasor diagram showing the phase relationship between the current and voltage.

1. The supply current:

2. The Active power consumed by the AC resistance is calculated as:

3. As there is no phase difference in a resistive component, ( θ = 0 ), the corresponding phasor diagram is given as:

 

AC Resistance Example No2

A sinusoidal voltage supply defined as: V(t) = 100 x cos(ωt + 30o) is connected to a pure resistance of 50 Ohms. Determine its impedance and the peak value of the current flowing through the circuit. Draw the corresponding phasor diagram.

The sinusoidal voltage across the resistance will be the same as for the supply in a purely resistive circuit. Converting this voltage from the time-domain expression into the phasor-domain expression gives us:

Applying Ohms Law gives us:

The corresponding phasor diagram will therefore be:

Impedance Summary

In a pure ohmic AC Resistance, the current and voltage are both “in-phase” as there is no phase difference between them. The current flowing through the resistance is directly proportional to the voltage across it with this linear relationship in an AC circuit being called Impedance.

Impedance, which is given the letter Z, in a pure ohmic resistance is a complex number consisting only of a real part being the actual AC resistance value, ( R ) and a zero imaginary part, ( j0 ). Because of this Ohm’s Law can be used in circuits containing an AC resistance to calculate these voltages and currents.

In the next tutorial about AC Inductance we will look at the voltage-current relationship of an inductor when a steady state sinusoidal AC waveform is applied to it along with its phasor diagram representation for both pure and non-pure inductance’s.

Be Continued,,,,

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Complex Numbers and Phasors

The mathematics used in Electrical Engineering to add together resistances, currents or DC voltages use what are called “real numbers” used as either integers or as fractions.

But real numbers are not the only kind of numbers we need to use especially when dealing with frequency dependent sinusoidal sources and vectors. As well as using normal or real numbers, Complex Numbers were introduced to allow complex equations to be solved with numbers that are the square roots of negative numbers, √-1.

In electrical engineering this type of number is called an “imaginary number” and to distinguish an imaginary number from a real number the letter “ j ” known commonly in electrical engineering as the j-operator, is used. Thus the letter “j” is placed in front of a real number to signify its imaginary number operation.

Examples of imaginary numbers are: j3, j12, j100 etc. Then a complex number consists of two distinct but very much related parts, a “ Real Number ” plus an “ Imaginary Number ”.

Complex Numbers represent points in a two dimensional complex or s-plane that are referenced to two distinct axes. The horizontal axis is called the “real axis” while the vertical axis is called the “imaginary axis”. The real and imaginary parts of a complex number are abbreviated as Re(z) and Im(z), respectively.

Complex numbers that are made up of real (the active component) and imaginary (the reactive component) numbers can be added, subtracted and used in exactly the same way as elementary algebra is used to analyse DC Circuits.

The rules and laws used in mathematics for the addition or subtraction of imaginary numbers are the same as for real numbers, j2 + j4 = j6 etc. The only difference is in multiplication because two imaginary numbers multiplied together becomes a negative real number. Real numbers can also be thought of as a complex number but with a zero imaginary part labelled j0.

The j-operator has a value exactly equal to √-1, so successive multiplication of “ j “, ( j x j ) will result in j having the following values of, -1, -j and +1. As the j-operator is commonly used to indicate the anticlockwise rotation of a vector, each successive multiplication or power of “ j “, j2, j3 etc, will force the vector to rotate through a fixed angle of 90o in an anticlockwise direction as shown below. Likewise, if the multiplication of the vector results in a  -j  operator then the phase shift will be -90o, i.e. a clockwise rotation.

Vector Rotation of the j-operator

So by multiplying an imaginary number by j2 will rotate the vector by  180o anticlockwise, multiplying by j3 rotates it  270o and by j4 rotates it  360o or back to its original position. Multiplication by j10 or by j30 will cause the vector to rotate anticlockwise by the appropriate amount. In each successive rotation, the magnitude of the vector always remains the same.

In Electrical Engineering there are different ways to represent a complex number either graphically or mathematically. One such way that uses the cosine and sine rule is called the Cartesian or Rectangular Form.

Complex Numbers using the Rectangular Form

In the last tutorial about Phasors, we saw that a complex number is represented by a real part and an imaginary part that takes the generalised form of:

• 
  
• Where:
•   Z  -  is the Complex Number representing the Vector
•   x  -  is the Real part or the Active component
•   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. So for example, Z = 6 + j4 represents a single point whose coordinates represent 6 on the horizontal real axis and 4 on the vertical imaginary axis as shown.

Complex Numbers using the Complex or s-plane

But as both the real and imaginary parts of a complex number in the rectangular form can be either a positive number or a negative number, then both the real and imaginary axis must also extend in both the positive and negative directions. This then produces a complex plane with four quadrants called an Argand Diagram as shown below.

Four Quadrant Argand Diagram

On the Argand diagram, the horizontal axis represents all positive real numbers to the right of the vertical imaginary axis and all negative real numbers to the left of the vertical imaginary axis. All positive imaginary numbers are represented above the horizontal axis while all the negative imaginary numbers are below the horizontal real axis. This then produces a two dimensional complex plane with four distinct quadrants labelled, QI, QII, QIII, and QIV.

The Argand diagram above can also be used to represent a rotating phasor as a point in the complex plane whose radius is given by the magnitude of the phasor will draw a full circle around it for every 2π/ω seconds.

Then we can extend this idea further to show the definition of a complex number in both the polar and rectangular form for rotations of 90o.

Complex Numbers can also have “zero” real or imaginary parts such as: Z = 6 + j0 or Z = 0 + j4. In this case the points are plotted directly onto the real or imaginary axis. Also, the angle of a complex number can be calculated using simple trigonometry to calculate the angles of right-angled triangles, or measured anti-clockwise around the Argand diagram starting from the positive real axis.

Then angles between 0 and 90o will be in the first quadrant ( I ), angles ( θ ) between 90 and 180o in the second quadrant ( II ). The third quadrant ( III ) includes angles between 180 and 270o while the fourth and final quadrant ( IV ) which completes the full circle, includes the angles between 270 and 360o and so on. In all the four quadrants the relevant angles can be found from:

tan-1(imaginary component ÷ real component)

Addition and Subtraction of Complex Numbers

The addition or subtraction of complex numbers can be done either mathematically or graphically in rectangular form. For addition, the real parts are firstly added together to form the real part of the sum, and then the imaginary parts to form the imaginary part of the sum and this process is as follows using two complex numbers A and B as examples.

Complex Addition and Subtraction

Complex Numbers Example No1

Two vectors are defined as, A = 4 + j1 and B = 2 + j3 respectively. Determine the sum and difference of the two vectors in both rectangular ( a + jb ) form and graphically as an Argand Diagram.

Mathematical Addition and Subtraction

Addition

Subtraction

Graphical Addition and Subtraction

Multiplication and Division of Complex Numbers

The multiplication of complex numbers in the rectangular form follows more or less the same rules as for normal algebra along with some additional rules for the successive multiplication of the j-operator where: j2 = -1. So for example, multiplying together our two vectors from above of A = 4 + j1 and B = 2 + j3 will give us the following result.

Mathematically, the division of complex numbers in rectangular form is a little more difficult to perform as it requires the use of the denominators conjugate function to convert the denominator of the equation into a real number. This is called “rationalising”. Then the division of complex numbers is best carried out using “Polar Form”, which we will look at later. However, as an example in rectangular form lets find the value of vector A divided by vector B.

The Complex Conjugate

The Complex Conjugate, or simply Conjugate of a complex number is found by reversing the algebraic sign of the complex numbers imaginary number only while keeping the algebraic sign of the real number the same and to identify the complex conjugate of z the symbol z is used. For example, the conjugate of z = 6 + j4 is z = 6 – j4, likewise the conjugate of z = 6 – j4 is z = 6 + j4.

The points on the Argand diagram for a complex conjugate have the same horizontal position on the real axis as the original complex number, but opposite vertical positions. Thus, complex conjugates can be thought of as a reflection of a complex number. The following example shows a complex number, 6 + j4 and its conjugate in the complex plane.

Conjugate Complex Numbers

The sum of a complex number and its complex conjugate will always be a real number as we have seen above. Then the addition of a complex number and its conjugate gives the result as a real number or active component only, while their subtraction gives an imaginary number or reactive component only. The conjugate of a complex number is an important element used in Electrical Engineering to determine the apparent power of an AC circuit using rectangular form.

Complex Numbers using Polar Form

Unlike rectangular form which plots points in the complex plane, the Polar Form of a complex number is written in terms of its magnitude and angle. Thus, a polar form vector is presented as:  Z = A ∠±θ, where: Z is the complex number in polar form, A is the magnitude or modulo of the vector and θ is its angle or argument of A which can be either positive or negative. The magnitude and angle of the point still remains the same as for the rectangular form above, this time in polar form the location of the point is represented in a “triangular form” as shown below.

Polar Form Representation of a Complex Number

As the polar representation of a point is based around the triangular form, we can use simple geometry of the triangle and especially trigonometry and Pythagoras’s Theorem on triangles to find both the magnitude and the angle of the complex number. As we remember from school, trigonometry deals with the relationship between the sides and the angles of triangles so we can describe the relationships between the sides as:

Using trigonometry again, the angle θ of A is given as follows.

Then in Polar form the length of A and its angle represents the complex number instead of a point. Also in polar form, the conjugate of the complex number has the same magnitude or modulus it is the sign of the angle that changes, so for example the conjugate of 6 ∠30o would be 6 ∠– 30o.

Converting between Rectangular Form and Polar Form

In the rectangular form we can express a vector in terms of its rectangular coordinates, with the horizontal axis being its real axis and the vertical axis being its imaginary axis or j-component. In polar form these real and imaginary axes are simply represented by “A ∠θ“. Then using our example above, the relationship between rectangular form and polar form can be defined as.

Converting Polar Form into Rectangular Form, ( P→R )

We can also convert back from rectangular form to polar form as follows.

Converting Rectangular Form into Polar Form, ( R→P )

Polar Form Multiplication and Division

Rectangular form is best for adding and subtracting complex numbers as we saw above, 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 in Polar Form

Multiplying together 6 ∠30o and 8 ∠– 45o in polar form gives us.

Division in Polar Form

Likewise, to divide together two vectors in polar form, we must divide the two modulus and then subtract their angles as shown.

Fortunately today’s modern scientific calculators have built in mathematical functions (check your book) that allows for the easy conversion of rectangular to polar form, ( R → P ) and back from polar to rectangular form, ( R → P ).

Complex Numbers using Exponential Form

So far we have considered complex numbers in the Rectangular Form, ( a + jb ) and the Polar Form, ( A ∠±θ ). But there is also a third method for representing a complex number which is similar to the polar form that corresponds to the length (magnitude) and phase angle of the sinusoid but uses the base of the natural logarithm, e = 2.718 281.. to find the value of the complex number. This third method is called the Exponential Form.

The Exponential Form uses the trigonometric functions of both the sine ( sin ) and the cosine ( cos ) values of a right angled triangle to define the complex exponential as a rotating point in the complex plane. The exponential form for finding the position of the point is based around Euler’s Identity, named after Swiss mathematician, Leonhard Euler and is given as:

Then Euler’s identity can be represented by the following rotating phasor diagram in the complex plane.

We can see that Euler’s identity is very similar to the polar form above and that it shows us that a number such as Ae jθ which has a magnitude of 1 is also a complex number. Not only can we convert complex numbers that are in exponential form easily into polar form such as: 2e j30 = 2∠30, 10e j120 = 10∠120 or -6e j90 = -6∠90, but Euler’s identity also gives us a way of converting a complex number from its exponential form into its rectangular form. Then the relationship between, Exponential, Polar and Rectangular form in defining a complex number is given as.

Complex Number Forms

Phasor Notation

So far we have look at different ways to represent either a rotating vector or a stationary vector using complex numbers to define a point on the complex plane. Phasor notation is the process of constructing a single complex number that has the amplitude and the phase angle of the given sinusoidal waveform.

Then phasor notation or phasor transform as it is sometimes called, transfers the real part of the sinusoidal function: A(t) = Am cos(ωt ± Φ) from the time domain into the complex number domain which is also called the frequency domain. For example:

Please note that the √2 converts the maximum amplitude into an effective or RMS value with the phase angle given in radians, ( ω ).

Summary of Complex Numbers

Then to summarize this tutorial about Complex Numbers and the use of complex numbers in electrical engineering.

• Complex Numbers consist of two distinct numbers, a real number plus an imaginary number.
• Imaginary numbers are distinguish from a real number by the use of the j-operator.
• A number with the letter “ j ” in front of it identifies it as an imaginary number in the complex plane.
• By definition, the j-operator j ≡ √-1
• Imaginary numbers can be added, subtracted, multiplied and divided the same as real numbers.
• The multiplication of “ j ” by “ j ” gives j2 = -1
• In Rectangular Form a complex number is represented by a point in space on the complex plane.
• In Polar Form a complex number is represented by a line whose length is the amplitude and by the phase angle.
• In Exponential Form a complex number is represented by a line and corresponding angle that uses the base of the natural logarithm.
• A complex number can be represented in one of three ways:
  • Z = x + jy   »  Rectangular Form
  • Z = A ∠Φ   »  Polar Form
  • Z = A e jΦ   »  Exponential Form
• Euler’s identity can be used to convert Complex Numbers from exponential form into rectangular form.

In the previous tutorials including this one we have seen that we can use phasors to represent sinusoidal waveforms and that their amplitude and phase angle can be written in the form of a complex number. We have also seen that Complex Numbers can be presented in rectangular, polar or exponential form with the conversion between each complex number algebra form including addition, subtracting, multiplication and division.

In the next few tutorials relating to the phasor relationship in AC series circuits, we will look at the impedance of some common passive circuit components and draw the phasor diagrams for both the current flowing through the component and the voltage applied across it starting with the AC Resistance.

Be Continued,,,,

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