Why do we need complex numbers ?
This is the first question that needs to be raised. It is a necessity, while dealing with ac circuits. Before we find out why we need complex numbers, let us look at real numbers first. Let us see what a real number can do, and how it can be represented on a planar surface.
A real number or a scalar variable can handle uni-dimensional quantity such as temperature and distance. It is difficult to use a scalar to represent a multidimensional quantity, such as a point in space. The scope of this text on circuit analysis is restricted to two dimensional quantities. Next we find out how a uni-dimensional quantity can be represented graphically, before the introduction of a complex number.
An equation is presented above. Let OP be a scalar, with a value of 3 assigned to it. We can represent it graphically, as shown in Fig. 1
The scalar number can be represented by a line along the horizontal axis, the distance of the point P, on the horizontal axis from the origin being proportional to the value of O P. Since this line is along the x-axis itself, O P can be assigned a magnitude and an angle. The magnitude is 3, and the angle is zero degrees, since the point P lies on the x-axis.
Let us consider next a negative number, say - 3. It can be called as OR, as shown here. We can represent it graphically, as shown in Fig. 2.
The negative number can be represented by a line along the horizontal axis, the distance of the point R on the horizontal axis from the origin being proportional to the value of OR. Note that this point lies to the left of origin. Points lying to the right of origin along the x-axis are positive real numbers, whereas those to the left of origin on x-axis represent negative real numbers. Distance OR can be assigned a magnitude, and an angle, as shown by the sketch in Fig. 2. The magnitude is 3, and the angle is 180o, since the point R lies on the x-axis, to the left of origin. See the way the angle is measured. It is measured along the counter clockwise direction, with respect to the positive real axis. What we have learned here is that a real number can be represented by an absolute value and an angle. Let us generalize this, as shown next.
Let N be a number with a magnitude and an angle. The magnitude is a positive real value. If the angle is zero degrees, then N is a positive real value. If the angle is 180o, then N is a negative number. If the angle is not equal to either zero or 180o, then N is not a real number. It is called a complex number. It can be seen how a complex number can be represented by a magnitude and an angle. Next we find out some other aspects of a complex number, by studying how the square and the square roots of a real number can be obtained.
Let Y be equal to four. Let the square root of Y be A and B. We know that the square roots of 4 are + 2 and - 2. Let A be equal to + 2 and assign - 2 to B. For each of these numbers, we can assign a magnitude and an angle. We can state that + 2 has a magnitude of 2, and an angle of 0o, and - 2 has a magnitude of 2, and an angle of + 180o or - 180o. It is possible to express the same graphically, as shown by the sketch in Fig. 3.
It is seen both the roots have the same magnitude, but their angles differ from each other by 180o. This exercise has not shed any new light on complex numbers, but what we have learnt here is useful in enhancing our understanding of complex numbers. Next we find out how the square of a number is obtained.
To obtain the square of a number, we get the square of the magnitude and the angle of the square is the twice the angle of the number. It is better to remember this. The magnitude of square of - 2 is 2 times 2, which is 4. Since -2 has a magnitude of 2 and angle of 180o, we obtain the product of -2 by -2 as 4 and the angle associated with 4 is the sum of (180o + 180o), which is 360o. We know that going round a full cycle corresponds to 360o, and we get back to the same point. If we start at 0o and go round for 360o, we get back to the point we started with and the angle at that point is 0o. This exercise has also not shed any light on complex numbers. But we will see that these exercises are not in vain.
Let us generalize what we have learned so far. We know that a number can be represented by a magnitude and an angle. The square of this number is obtained as shown, by getting the square of its magnitude and doubling its angle. The square roots are obtained as shown above. We can obtain the square root of magnitude of number N, and the angle of one of the square roots of N is 0.5q, and the other square root has the same magnitude, but is displaced by 180o. The sketch in Fig. 4 illustrates how the square roots of a number can be obtained.
Let us apply what we have learned. Let us ask the question what the square root of - 1 is. Before, that question is answered, let us find out what the square root of plus one is.
Let A be equal to one. We can say that number A has a magnitude of one and an angle, equal to either 0o or 360o. The magnitude of square root is the square root of the magnitude, and the angle of the square root is half the angle of number one. Since number A can be said to have an angle of either 0o or 360o, the first square root has an angle of 0o, and the second square root has an angle of 180o. The two angle associated with number one are 0o and 360o. We get the angles of two square roots by dividing the angles associated with the number A by 2. Now we are ready to answer the question what the square root of - 1 is.
Let us now find out what the square root of - 1 is. We can express - 1 to have an amplitude of one, and an angle of either + 180o or - 180o . Hence the magnitude of square root of - 1 is one, and the angles of the square root are half of the angles, that can be associated with - 1. The values of angles of square roots are then + 90o and - 90o. You can see that the difference between angles of the two square roots is always 180o. A graphical representation for the square roots of minus one is shown by Fig. 5.
The square roots of - 1have an angle of + 90o and - 90o. Hence they can be located on an axis perpendicular to the horizontal or the real axis. The vertical axis can be given a name and it is called the imaginary axis. We find that the square roots lie not on the real axis, but on the imaginary axis. The square roots of - 1 have imaginary values. Please note that the positive direction for angle is assigned to be anti-clockwise, and hence number one with an angle of + 90o is on the part of vertical axis, above horizontal axis. The unit for imaginary number is indicated by either of the two letters, j or i. Hence, number one with an angle of + 90o is said to be equal + j J or j.
The number one, with an angle of - 90o is - j and it is on part of the vertical axis or the imaginary axis, below the real axis. We can call j as the imaginary operator. Using the imaginary operator, it is possible to express complex numbers in a different way.
The unit for imaginary number is j. The square of j is equal to - 1, as shown above.
The symbol for imaginary number is j, as stated earlier. It is marked on the imaginary axis, as shown in Fig. 6. If the imaginary value is positive, it is marked on the part of the imaginary axis, that lies above the real or the horizontal axis. The imaginary numbers with negative values lie on the part of imaginary axis, below the real or the horizontal axis. The letter j indicates that the number is imaginary, and we use the minus sign to indicate negative values. We can call j as the imaginary operator also. The reason for calling it as an operator is also shown. We can obtain the square of j as illustrated, and it equals - 1. If we multiply 1 by j, we get j which has a magnitude 1 and an angle of + 90o. It is seen that multiplying a number by j changes its angle by 90oand we call j as the imaginary operator. Now we have progressed thus far, and we are now ready to be introduced to complex numbers.
We know what a real number is and what its limitation is. A real number or a scalar can represent only a uni-dimensional quantity such as temperature. In the previous section, we have seen what an imaginary number is and how it can be represented. The combination of a real part and an imaginary part gives rise to a complex number. We have seen earlier that we can represent a complex number can be represented by a magnitude and an angle. Here we find an alternative way of representing a complex number.
Let us take a point on a two dimensional surface or a planar surface as shown in Fig. 7 and let the point be called by letter P. Let the projection of point P on the real axis be A, and let the projection of point P on the imaginary axis or the vertical axis, be represented by letter B. Let the distance of point P from the origin by letter Z.
Then we can express Z, by equation (8). Equation (8) represents the co-ordinates of P as a complex number, with a real part and an imaginary part.You can see that the imaginary part of the complex number is expressed as 3j , where j is the symbol for imaginary number. It is possible to represent this complex number by a magnitude, and an angle, as shown by equation (9) . The magnitude of complex number is obtained using the relationship between the hypotenuse and the sides of a right angled triangle , and the angle is marked as q is obtained as shown in Fig. 7. Note that the two axes are perpendicular to each other. The angle can be obtained from the x and y co-ordinates of point P. Equation (10) shows the expression for magnitude and the expression for angle. We can now see how these two ways of representing a complex number are equivalent to each other.
A complex number can represent a vector with two dimensions. In other words, a complex number can represent a two dimensional quantity, whereas the concept of a vector can be extended to a three dimensional object, like a point in space. Our interest here is restricted to two-dimensional objects.
For example, let us consider a planar surface, like a graph sheet. A planar surface can be said to have two independent dimensions. Any point on the planar surface can be described accurately by two co-ordinates, one co-ordinate along the horizontal axis, or the x-axis, and the other co-ordinate along the vertical axis, or the y-axis and these co-ordinates are expressed with respect to origin, marked as O, in Fig. 7. In Fig. 7, the x-co-ordinate of point P is 4 with respect to origin, and the y-co-ordinate of point P is 3. We can call the planar surface, as the Complex plane, where the x-axis becomes the real, axis, and the y-axis becomes the imaginary axis. Then point P can represent a complex number, marked as Z, in Fig. 7. Equation (11) states the value of complex number Z by an equation. The magnitude of the complex number can be expressed, as shown by equation (12). The value of x-co-ordinate of line OP is the projection of line OP on the horizontal axis, and its value can be expressed by the product of magnitude and cosine of the angle, as shown by equation (13). As you may recall, the line representing the projection on x-axis is perpendicular to x-axis. The value of y-co-ordinate of line OP is the projection of line OP on the vertical axis and its value is obtained as the product of the magnitude and sine of the angle, as expressed by equation (14). Then equation (11) can be re-presented as shown by equation (15). This equation is the third way of representing a complex number, and it is known as the trigonometric form of complex number. The representation of the complex number using the exponential form is based on the Euler's identity, expressed by equation (16).
It is possible to express a complex number in four forms. The cartesian form of expressing a complex number is displayed by equation (17). In this form, the real part and the imaginary part of the complex number are specified explicitly, and the magnitude and the angle of the complex form have to be deduced. The polar form of representing a complex number is displayed by equation (18). This form states explicitly the magnitude and the angle, from which the real part and the imaginary part can be obtained. The exponential form of representing a complex number is displayed by equation (19), and this form is similar to the polar form, in the sense that only the magnitude and the angle of a complex number are explicitly stated, but the exponential function contains more information. It shows that we can get the real part and the imaginary part of the complex number, using Euler's identity, The polar form does contain this information in an implicit way. The polar form implies that the angle specified is the angle expressed in equation (19). It can be stated that the polar form is an abridged version of exponential form. The exponential function in equation (19) can be expanded using the Euler’s identity, and then we get the trigonometric form, as displayed by equation (20). This form shows the real part and the imaginary part, but these values have to be computed from the magnitude and cosine and sine of the angle.
The pictorial representation of a complex number is shown in Fig. 8.
Equations (21) and (22) define two complex numbers. Two complex numbers are equal, if both have equal real parts and equal imaginary parts. The magnitude and the polarity of real parts should be the same, and similarly the magnitude and the polarity of imaginary parts should be equal. Equations (23) and (24) express these statements. Alternatively, if both complex numbers have the same magnitude, and if their angles are equal, then the two complex numbers are equal to each other, as expressed by equation (25).
Let us see how we can perform basic arithmetic operations on complex numbers. Let us try adding two complex numbers.
Equations (26) and (27) define two complex numbers. Their sum is represented by equation (28). The real part of the sum is the algebraic sum of the real parts of the two complex numbers defined in equations (26) and (27). The imaginary part is the algebraic sum of the imaginary parts of the two complex numbers defined in equations (26) and (27). The keyword here is algebraic. It means that the polarity of the parts of complex number should be taken into account. For example, if x1 is 3, and x2 is 4, then the sum is 7. On the other hand, if x1 is - 3, and x2 is 4, then the sum is 1. The sketch in Fig. 9 shows the pictorial representation of addition.
For addition, you should use the cartesian form of complex numbers. It is the simplest way to carry out addition. If the complex numbers are in some other form, convert them to cartesian form and then carry out addition.
Now let us see how we can subtract one complex number from another complex number. Equations (30) and (31) define two complex numbers. The result of subtracting Z2 from Z1 is represented by equation (32). The real part of the result is the algebraic difference of the real parts of the two complex numbers defined in equations (30) and (31). The imaginary part is the algebraic difference of the imaginary parts of the two complex numbers defined in equations (30) and (31). The keyword here is algebraic. It means that the sign of the parts of complex number should be taken into account. For example, if x1 is 3 and x2 is 4, the result of subtracting x2 from x1 is - 1. On the other hand, if x1 is - 3, and x2 is 4, the result is - 7.
The sketch in Fig. 10 on the right hand side shows the pictorial representation of subtraction. For subtraction, you should use the cartesian form of complex numbers. It is the simplest way to carry out subtraction.
Multiplication of two complex numbers is presented now. The value of Z1 and Z2 are presented by equation (34). Z3 equals the product of these complex numbers and the value of Z3 is obtained as shown by equation (35). The product is obtained using the distributive law. The process can be extended to show how any two complex numbers can be multiplied. This process is illustrated by equations (36) and (37). In this example, the complex numbers are expressed in the cartesian form. The cartesian form is not very convenient for multiplication and division of complex numbers. It is better to use polar form, as illustrated next. Before multiplication using polar form is introduced, we need a better understanding of Euler's identity. You can recall how the square of a number has been obtained in the previous page, by getting the square of the magnitude and doubling the angle. Multiplication is carried out in a similar manner.
The pictorial representation, of how the result of multiplication of two complex numbers is obtained, is shown in Fig. 11. This representation reflects the use of polar form to for multiplying two complex numbers.
It is necessary to understand Euler's identity well. Equation (38) states Euler's identity, and the sketch in Fig. 12 is a pictorial representation.
As q changes from zero radian to 2p radians, the locus of Euler's identity moves along the periphery of a circle, with a radius equal to unity. It can be seen that Euler's identity represents a complex number of magnitude one, and angle q. For any angle q, cos(q) is the real part, and j times sin(q) is the imaginary part. We know that the radius equals one from equation (39). Equation (39) represents the square of the magnitude, but since the magnitude is 1, the square root also has a value of 1. Equation (38) is explained next. It can be seen that when q equals zero, the value of Euler's identity is 1, as expressed by equation (40). . If q equals p/2 radians, the value of Euler's identity is + j, as expressed by equation (41). When q equals p radians, the value of Euler's identity is - 1, as shown by equation (42). From equation (43), it can be seen that the value of Euler's identity is - j, when q equals - (p/2) radians.
Given a complex number, its conjugate is also a complex number. Let the magnitude of complex number be some value, say M, and let the angle be q. The conjugate has the same magnitude, which is M, and its angle is - q, since it is the negative of the angle the given complex number. The conjugate of Euler's identity is expressed by equations (44) and (45). The product of Euler's identity and its conjugate is obtained as shown by equation (46). The exponents can be added, and the result is zero. Hence the product of Euler's identity and its conjugate is equal to one. We get the same value from the product of trigonometric representations of Euler's identity and its conjugate as displayed by equation (47).
The sketch in Fig. 13 presents a pictorial representation of Euler's identity, and its conjugate. The product of a complex number and its conjugate is always a positive real number, equal to the square of the magnitude of the complex number.
Given a complex number in cartesian form, its conjugate is denoted by a bar over the letter, as shown by equation (48). The conjugate has the same real part as Z, but the imaginary part of conjugate of Z has the same magnitude, but opposite sign, compared with the imaginary part of Z. If the imaginary part of Z is jy, then the imaginary part of the conjugate is - jy, as expressed by equation (49). The product of Z and its conjugate is obtained as shown by equation (50). It can be seen that the product of a complex number and its conjugate is always a real number. A pictorial representation is provided in Fig. 14. Next we have an example for multiplication using polar form.
Given complex numbers in cartesian form, they can be represented in polar form as shown by equations (51) and (52). Polar form is derived from exponential form. In exponential form, multiplication means that we multiply the magnitudes of complex numbers with each other, and that we add the two angles of complex numbers. In polar form, we do the same, as shown by equation (53). To generalize, we can express Z1 and Z2, as shown by equation (54), where A and B are the magnitudes. Both are A and B are positive real values. Angle of complex number Z1 is q, and the angle of Z2 is j, and each lies within the range from - p to p radians. The angle of Z3 is the algebraic sum of angles, q and j. It means that we have to take into account their sign, while performing addition.
The pictorial representation of multiplication is shown in Fig. 15. The sketch is self-explanatory. The two numbers are (1 + j2) and (2 + j), and their product equals Z, which is equal to j5.
Given a complex number, its reciprocal is also a complex number. If the complex number is in polar form, its reciprocal has a magnitude, which is the reciprocal of the magnitude of the complex number, as shown by equations (56) and (57). For example, if the complex number has a magnitude of 2, the reciprocal of complex number has a magnitude of 0.5, as shown by the sketch in Fig. 16.
The angle of the reciprocal of complex number is the negative of the angle of the complex number. If the complex number is in the cartesian form as shown by equation (58), its reciprocal is obtained by multiplying both the numerator and the denominator by the conjugate of the complex number, and then simplifying the result. Remember that the product of a complex number and its conjugate, yields a real number equaling the square of the magnitude of the complex number.
Given two complex numbers Z1 and Z2, as shown by equation (59), division of Z1by Z2 yields a complex number, let us say, Z3. Equation (60) illustrates how division is performed. If both Z1 and Z2 are in polar form, the magnitude of Z3 is obtained by dividing A by B, where A is the magnitude of Z1, and B is the magnitude of Z2. The angle of Z3 is obtained by subtracting j from q, where q is the angle of Z1, and j is the angle of Z2.
The sketch in Fig. 17 shows a pictorial representation of the process. If both Z1 and Z2 are in cartesian form, multiply both numerator and denominator by the conjugate of Z2. Then the denominator contains the square of magnitude of Z2, and the numerator can be expanded, and the terms can be re-grouped to yield a real part and an imaginary part. For division, it is better to use the polar form. Remember that polar form is derived from the exponential form. When we multiply numbers in exponential form, the exponents get added. If we divide, the exponent of divisor or the denominator, gets subtracted from the exponent of the dividend or the numerator.
This page has described what a complex number is and how arithmetic operations can be performed when using complex numbers. In addition, we have learnt what a sinusoidal signal is in the previous page. Now we are ready to take on phasors. However, before the topic of presented, two pages containing interactive applets are presented.