An orthogonal system is one in which the coordinates arc mutually perpendicular Examples of orthogonal coordinate systems include the Cartesian (or rectangular), the circular cylindrical, the spherical, the elliptic cylindrical, the parabolic cylindrical, the conical, the prolate spheroidal, the oblate spheroidal, and the ellipsoidal.1 A considerable amount of work and time may be saved by choosing a coordinate system that best fits a given problem A hard problem in one coordi nate system may turn out to be easy in another system. In this text, we shall restrict ourselves to the three best-known coordinate systems: the Cartesian, the circular cylindrical, and the spherical.

1. Unit-1
Coordinate System
1
Mohammad Akram,Assistant Professor,ECE
Department,JIT

2. Introduction
• An orthogonal system is one in which the coordinates arc
mutually perpendicular
• Examples of orthogonal coordinate systems include the
Cartesian (or rectangular), the circular cylindrical, the
spherical, the elliptic cylindrical, the parabolic cylindrical, the
conical, the prolate spheroidal, the oblate spheroidal, and the
ellipsoidal.1
• A considerable amount of work and time may be saved by
choosing a coordinate system that best fits a given problem
• A hard problem in one coordi nate system may turn out to be
easy in another system.
• In this text, we shall restrict ourselves to the three best-known
coordinate systems: the Cartesian, the circular cylindrical, and
the spherical. 2
Mohammad Akram,Assistant Professor,ECE
Department,JIT

3. CARTESIAN COORDINATES (X, Y, Z):
•A point P can be represented as (x, y, z) as illustrated
in Figure 1. The ranges of the coordinate variables x, y,
and z are
•A vector A in Cartesian (or rectangular) coordinates
may be represented as
(Ax, Ay, Az) or Ax ax + Ay ay + Az az
Fig.1 (a) Unit vectors ax, ay, and az (b) components of A along ax, ay and az.



z
y
x
3
Mohammad Akram,Assistant Professor,ECE
Department,JIT

4. CIRCULAR CYLINDRICAL COORDINATES
(ρ, ɸ, z)
•The circular cylindrical coordinate system is very
convenient whenever we are dealing with problems
having cylindrical symmetry.
•A point P in cylindrical coordinates is represented as
(ρ, ɸ, z ) and is as shown in Figure 2.
Fig.2. Point P and unit vectors
in the cylindrical coordinate
system.
4
Mohammad Akram,Assistant Professor,ECE
Department,JIT

5. •Observe Figure 2 closely and note how we define each
space variable:
•ρ is the radius of the cylinder passing through P or the
radial distance from the z-axis.
•ɸ, called the azimuthal angle, is measured from the x-
axis in the xy-plane
•z is the same as in the Cartesian system. The ranges of
the variables are



z
20
0
5
Mohammad Akram,Assistant Professor,ECE
Department,JIT

6. •A vector A in cylindrical coordinates can be
written as
(Aρ, Aɸ, Az) or Aρaρ + Aɸaɸ + Azaz
where aρ , aɸ and az are unit vectors in the ρ,ɸ and z
direction respectively as shown in fig.2
•The magnitude of A is
•Notice that the unit vectors aρ , aɸ and az are
mutually perpendicular because our coordinate
system is orthogonal
•aρ points in the direction of increasing ρ
•aɸ in the direction of increasing ɸ
• azin the positive z-direction.
 AAA
2
z
22
A  
6
Mohammad Akram,Assistant Professor,ECE
Department,JIT

7. Thus,
aρ . aρ = aɸ . aɸ = az . az = 1
aρ . aɸ = aɸ . az = az . aρ = 0
aρ × aɸ = az
aɸ × az = aρ
az × aρ = aɸ
•The relationships
between the variables
(x, y, z) of the
Cartesian coordinate
system and those of the
cylindrical system
(ρ, ɸ, z) are easily
obtained from Figure 3 as
Fig.3. Relationship between (x, y, z) and (ρ,ɸ,z). 7
Mohammad Akram,Assistant Professor,ECE
Department,JIT

8. Transforming a point from Cylindrical(ρ,ɸ,z) to Cartesian (x,y,z)
Transforming a point from Cartesian (x,y,z) to
Cylindrical(ρ,ɸ,z)
8
Mohammad Akram,Assistant Professor,ECE
Department,JIT

9. •The relationships between (ax , ay , az ) and
(aρ , aɸ , az ) are obtained geometrically from Figure 3:
ax =cosɸ aρ − sinɸ aɸ
ay =sinɸ aρ + cosɸ aɸ
az = az
Fig.3 Unit vector transformation: (a) cylindrical components of ax,
(b) cylindrical components of ay
9
Mohammad Akram,Assistant Professor,ECE
Department,JIT

10. Or
aρ =cosɸ ax + sinɸ ay
aɸ = −sinɸ ax + cosɸ ay
az = az
•Finally, the relationship between (Ax, Ay, Az) and
(Aρ, Aɸ, Az) can be obtained as
A = (Ax cosɸ + Aysinɸ) aρ + (− Ax sinɸ + Ay cosɸ) aɸ +Azaz
Aρ = Ax cosɸ + Aysinɸ
Aɸ = − Ax sinɸ + Ay cosɸ
Az = Az
In matrix form, we have the transformation of vector A from
(Ax, Ay, Az) to (Aρ, Aɸ, Az) as
10
Mohammad Akram,Assistant Professor,ECE
Department,JIT

11. •The inverse of the transformation (Aρ, Aɸ, Az) to (Ax, Ay,
Az) is obtained as
An alternative way is using the dot product. For example:
11
Mohammad Akram,Assistant Professor,ECE
Department,JIT

12. SPHERICAL COORDINATES (r, θ, φ)
Fig.4 Point P and unit vectors in spherical coordinates.
•The spherical
coordinate system is
most appropriate
when dealing with
problems having a
degree of spherical
symmetry.
•A point P can be
represented as
(r, θ, φ) and is
illustrated in Figure 4.
12
Mohammad Akram,Assistant Professor,ECE
Department,JIT

13. •From Figure 4, we notice that “r” is defined as the distance
from the origin to point P or the radius of a sphere centered
at the origin and passing through P.
•“θ” (called the colatitude) is the angle between the z-axis
and the position vector of P.
•“φ” is measured from the x-axis (the same azimuthal angle
in cylindrical coordinates)
•According to these definitions, the ranges of the variables
are


20
0
0


 r
13
Mohammad Akram,Assistant Professor,ECE
Department,JIT

14. •A vector A in spherical coordinates may be written as
(Ar ,Aθ,Aφ) or Ar ar + Aθ aθ + Aφ aφ
where ar , aθ and aφ are unit vectors along the r, θ, and φ
directions.
•The magnitude of A is
•The unit vectors ar , aθ and aφ are mutually orthogonal.
•ar being directed along the radius or in the direction of
increasing r.
•aθ in the direction of increasing θ.
•aφ in the direction of increasing φ.
2
1222
)( AAAr
A 

14
Mohammad Akram,Assistant Professor,ECE
Department,JIT

15. Thus,
15
Mohammad Akram,Assistant Professor,ECE
Department,JIT

16. •The space variables (x, y, z) in Cartesian coordinates can
be related to variables (r, θ, φ) of a spherical coordinate
system
•From Figure 5 it is easy to notice that
Fig.5 Relationships between space variables
(x, y, z), (r, θ,φ) and (ρ,φ,z).
)(
222
zyxr 
z
yx )(
22
1
tan




x
y
tan
1

16
Mohammad Akram,Assistant Professor,ECE
Department,JIT

17. or x= r sinθ cosφ, y= r sinθ sinφ, z=r cos θ
In matrix form, the (Ax ,Ay,Az) to (Ar ,Aθ,Aφ) vector
transformation is performed according to
•The inverse transformation (Ar ,Aθ,Aφ) to (Ax ,Ay,Az) is
obtained as
17
Mohammad Akram,Assistant Professor,ECE
Department,JIT

18. Mohammad Akram,Assistant Professor,ECE
Department,JIT
18
Reference
1. “Principles of Electromagnetics” Book by “Matthew
N.O. Sadiku”, “S.V Kulkarni”

19. 19
Mohammad Akram,Assistant Professor,ECE
Department,JIT