Complex number is defined as a number which including two parts such as real part and imaginary part. These numbers are consisted the real numbers as ordinary. It can be extended by adding the extra numbers.
The general form of complex number is (a + bi). Here a represents real part of complex number, b represents imaginary part of complex number with the standard imaginary unit of i.
Graphing of complex number:
The complex numbers are graphing with complex plane. On that plane, the horizontal axis denotes the real part of complex number and vertical axis denotes the imaginary part of the complex number.
Basic properties of complex numbers:
Usually complex numbers have some basic properties as follows,
The group of every complex numbers is generally indicated by C.
If z = x + yi, the real part x is indicated Re (z), and the imaginary part y is indicated Im (z). Here the imaginary unit i gives the property as i^2 = -1.
The real number x is combined with the complex number as x + 0i.
The imaginary number y is combined with the complex number as 0 + yi.
The equality property of complex numbers states that, let two complex numbers are a + bi and c + di. From the equality property it can be written as a = c and b = d.
Application:
Generally, the complex numbers are used in following fields,
Engineering
Electromagnetism
Quantum physics
Applied mathematics
Chaos theory
Examples:
1) Graphing the following complex number: 2 +3i
Solution:
Plot the real part in x axis and plot the imaginary part is y axis, so
2) Solve (-2 + 4i) + (4 + i) and graphing the result:
Solution:
(-2 + 4i) + (4 + i) = (-2 + 4) + (4 + 1)i
= (2 + 5i)
Therefore the graphing of the result is,
3) Solve (2 + i) + (4 + 2i) and graphing the result:
Solution:
(2 + i) + (4 + 2i) = (2 + 4) + (1 + 2)i
= (6 + 3i)
Therefore the graphing of the result is,
The electrical circuit is the path on which the electric current flows through the circuit. There are some simple circuits and some complex, which contains some of the electrical components. Let us discuss that how to solve the complex circuits.
Solving complex circuit problems
Let us take some complex circuits and the try to solve them.
Example 1:
Six equal resistances each of 4 W are connected to form a network as shown in diagram. What are the equivalent resistance between A and B?
Image of the equivalent circuit
Solution:
The upper five resistances between A and B will form a balanced wheat stone bridge. Hence, there will be no current in arm CO. The equivalent circuit is as:
Here resistances of the arm ACB, AOB and AB are in parallel. The effective resistance R is
1 / R = 1/(4 + 4) + 1/(4 + 4)
1 / R = (1 + 1 + 2) / 8 = 4 / 8 = 1 / 2
R = 2 W.
Example 2:
In the circuit shown below, find the potential difference across the capacitor.
Solution:
When the capacitor is fully charged, it draws zero current, then no current flows in arm EF. The potential difference across the capacitor is equal to the potential difference across C and F. the effective resistance between A and F of the circuit is
R = (3 'xx' 6) / (3 + 6) + 3 = 2 + 3 = 5 W
Main current in the circuit, I = 15 / 5 = 3 A
Current through arm BCD = I 'xx' 3 / (6 + 3) = (3 'xx' 3) / 9 = 1 A
Potential difference across C and D = 1 'xx' 3 = 3 V
Current through the arm DF = 3 A
Potential difference across D and F = 3 'xx' 3 = 9 V
Potential difference across C and F = 3 + 9 = 12 V
Thus, the potential difference across the capacitor is 12 V.
conclusion on complex circuits
Solving complex circuits involves higher accuracy in relating with laws of combination of resistors, which we have seen above, either connected parallel, or in series, we have used different measures for them. Also while calculating the voltage; we follow certain set of rules.
The general form of complex number is (a + bi). Here a represents real part of complex number, b represents imaginary part of complex number with the standard imaginary unit of i.
Graphing of complex number:
The complex numbers are graphing with complex plane. On that plane, the horizontal axis denotes the real part of complex number and vertical axis denotes the imaginary part of the complex number.
Basic properties of complex numbers:
Usually complex numbers have some basic properties as follows,
The group of every complex numbers is generally indicated by C.
If z = x + yi, the real part x is indicated Re (z), and the imaginary part y is indicated Im (z). Here the imaginary unit i gives the property as i^2 = -1.
The real number x is combined with the complex number as x + 0i.
The imaginary number y is combined with the complex number as 0 + yi.
The equality property of complex numbers states that, let two complex numbers are a + bi and c + di. From the equality property it can be written as a = c and b = d.
Application:
Generally, the complex numbers are used in following fields,
Engineering
Electromagnetism
Quantum physics
Applied mathematics
Chaos theory
Examples:
1) Graphing the following complex number: 2 +3i
Solution:
Plot the real part in x axis and plot the imaginary part is y axis, so
2) Solve (-2 + 4i) + (4 + i) and graphing the result:
Solution:
(-2 + 4i) + (4 + i) = (-2 + 4) + (4 + 1)i
= (2 + 5i)
Therefore the graphing of the result is,
3) Solve (2 + i) + (4 + 2i) and graphing the result:
Solution:
(2 + i) + (4 + 2i) = (2 + 4) + (1 + 2)i
= (6 + 3i)
Therefore the graphing of the result is,
The electrical circuit is the path on which the electric current flows through the circuit. There are some simple circuits and some complex, which contains some of the electrical components. Let us discuss that how to solve the complex circuits.
Solving complex circuit problems
Let us take some complex circuits and the try to solve them.
Example 1:
Six equal resistances each of 4 W are connected to form a network as shown in diagram. What are the equivalent resistance between A and B?
Image of the equivalent circuit
Solution:
The upper five resistances between A and B will form a balanced wheat stone bridge. Hence, there will be no current in arm CO. The equivalent circuit is as:
Here resistances of the arm ACB, AOB and AB are in parallel. The effective resistance R is
1 / R = 1/(4 + 4) + 1/(4 + 4)
1 / R = (1 + 1 + 2) / 8 = 4 / 8 = 1 / 2
R = 2 W.
Example 2:
In the circuit shown below, find the potential difference across the capacitor.
Solution:
When the capacitor is fully charged, it draws zero current, then no current flows in arm EF. The potential difference across the capacitor is equal to the potential difference across C and F. the effective resistance between A and F of the circuit is
R = (3 'xx' 6) / (3 + 6) + 3 = 2 + 3 = 5 W
Main current in the circuit, I = 15 / 5 = 3 A
Current through arm BCD = I 'xx' 3 / (6 + 3) = (3 'xx' 3) / 9 = 1 A
Potential difference across C and D = 1 'xx' 3 = 3 V
Current through the arm DF = 3 A
Potential difference across D and F = 3 'xx' 3 = 9 V
Potential difference across C and F = 3 + 9 = 12 V
Thus, the potential difference across the capacitor is 12 V.
conclusion on complex circuits
Solving complex circuits involves higher accuracy in relating with laws of combination of resistors, which we have seen above, either connected parallel, or in series, we have used different measures for them. Also while calculating the voltage; we follow certain set of rules.