Alternating Current

The term

%5Cfrac%7Bj%7D%7B%5Comega%20C%7D

represents the impedance of the capacitor in the circuit. In AC circuits, capacitive impedance is given by

Z_C%3D%5Cfrac%7Bj%7D%7B%5Comega%20C%7D

where

j%3D%5Csqrt%7B-1%7D

and

%5Comega

is the angular frequency of the source.Similarly,$

Z_L%3Dj%20%5Comega%20L

We'll%20apply%20Kirchhoff's%20Current%20Law%20(KCL)%20to%20both%20loops%20of%20the%20circuit.%20By%20analyzing%20the%20phasor%20(complex)%20impedances%20and%20equating%20real%20and%20imaginary%20parts%2C%20we%20determine%20the%20condition%20for%20in-phase%20current.KCL%20in%20loop%201%3A

%5Cbegin%7Bgathered%7D%20%5Cleft.i_2%20R-%5Cfrac%7Bi_2%20j%7D%7B%5Comega%20C%7D-i_1%20R%3D0%20%5Cquad%20%5Ctext%20%7B%20(equation%20%7D%201%5Cright)%20%5C%5C%20i_1%20R%3Di_2%5Cleft(R-%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%5Cright)%20%5C%5C%20i_1%3Di_2%5Cleft(1-%5Cfrac%7Bj%7D%7B%5Comega%20R%20C%7D%5Cright)%20%5Cend%7Bgathered%7D

KCL%20in%20loop%202%3A

%5Cbegin%7Bgathered%7D%20i_2%20R%2Bi_2%5Cleft(%5Cfrac%7B-j%7D%7B%5Comega%20C%7D%5Cright)%2B%5Cleft(i_1%2Bi_2%5Cright)%20j%20%5Comega%20L%3DV_%7B%5Ctext%20%7Bavg%20%7D%7D%3D0%20%5C%5C%20i_2%5Cleft(R-%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%5Cright)%3D%5Cleft(i_1%2Bi_2%5Cright)%20j%20%5Comega%20L%20%5Cend%7Bgathered%7D

On%20substituting%2C

i_2%5Cleft(R-%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%5Cright)%3D%5Cleft(i_2%5Cleft(1-%5Cfrac%7Bj%7D%7B%5Comega%20R%20C%7D%5Cright)%2Bi_2%5Cright)%20j%20%5Comega%20L

Cancelling%20out

i_2

%2C

R-%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%3D%5Cleft(2-%5Cfrac%7Bj%7D%7B%5Comega%20R%20C%7D%5Cright)%20j%20%5Comega%20L

Since

j=\sqrt{-1}, j^2=-1$$

R-%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%3D2%20j%20%5Comega%20L%2B%5Cfrac%7BL%7D%7BR%20C%7D

In%20this%20context%2C%20'in%20phase'%20means%20that%20the%20voltage%20and%20current%20waveforms%20align%20perfectly%20over%20time%2C%20without%20any%20phase%20shift%20between%20them.%20For%20this%20to%20happen%20in%20an%20AC%20circuit%2C%20the%20imaginary%20(reactive)%20components%20of%20the%20circuit's%20impedance%20must%20cancel%20each%20other%20out.%20When%20this%20cancellation%20occurs%2C%20the%20circuit%20behaves%20as%20if%20it%20only%20has%20resistive%20components%2C%20leading%20to%20the%20current%20being%20in%20sync%20with%20the%20voltage

%5Cbegin%7Bgathered%7D%20%5CRightarrow%20%5Cfrac%7Bj%7D%7B%5Comega%20C%7D%3D2%20j%20%5Comega%20L%20%5C%5C%20%5Comega%5E2%3D%5Cfrac%7B1%7D%7B2%20L%20C%7D%20%5C%5C%20%5Comega%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%20L%20C%7D%7D%20%5Cend%7Bgathered%7D

Chapter Questions
1 MCQs