Outline: |
Nodal Analysis |
Mesh Analysis |
When Superposition is necessary |
This circuit has two sources with the same ω: Since both sources have the same ω then superpostion is optional. In other words any other analysis technique could be used to find i, such as mesh or nodal analysis. Step 1: To use superpostion, we first 'kill' the current source. Killing a source means making the source zero. A current source with 0A is an OPEN CIRCUIT. To find i' we will simply use Ohm's Law: Zeq. = (4 + 6j -3j) = (4 + 3j) i' = (10:0)/(4 + 3j) (4 + 3j) = (5:37o) i' = (10:0)/(5:37o) = (2:-37o) Step 2: We will now kill the independent voltage source. OV means a SHORT CIRCUIT. To find i'' we will use Current Division: i'' = (3:0)[(4 + 6j)/(4 + 3j)] i'' = (3:0)[( 7.7 :56o)/( 5 :37o)] i'' = (3:0)[( 1.5 :19o)] i'' = ( 4.5 : 19o) Step 3: Now to find the original i: i = i' + i'' = ( 2 :-37o) + ( 4.5 : 19o) Since i' and i'' have the same ω we can add these two numbers: Converting i' to rectangular form: i'= 1.6 - 1.2j Converting i'' to rectangular form: i'' = 4.3 + 1.5j i = i' + i'' = (1.6 + 4.3) + (-1.2 + 1.5)j = 5.9 + 0.3j Now putting i back into the time domain: i = (6:2.9o) i = 6 cos (2t + 2.9o) |
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This circuit has two sources with different ω's: Here it is mandatory to use superpostion because the ω's are different. If we didn't use superposition then we would not know which ω to use. Step 1: ω=2 '/s First kill the current source: i' = 10/(4 + 6j - 3j) = 10/(4 + 3j) = 10/(5:37o) i' = (2:-37o) Step 2: ω=1 '/s Now kill the voltage source: i'' = (3:0)[(4 + 3j)/(4 + 3j - 6j)] = (3:0)[(4 + 3j)/(4 - 3j)] i'' = (3:0)[(5:37o)/(5:-37o)] i'' = (3:0)(1:74o) i'' = (3:74o) Step 3: Now to find the original i: i = i' + i'' i' = 2cos(2t - 37o) i'' = 3cos(t + 74o) Since i' and i'' have different ω's, we can not add up the complex numbers to get one complex number for i. i = 2cos(2t - 37o) + 3cos(t + 74o) |