Hey guys, this may seem a bit strange but I am trying to prove to myself that the impedance of a capacitor to a sinusoidal input signal is 1/ωC using the complex exponential form of cos(θ) and sin(θ) but i cannot solve it. Could someone tell me where i am going wrong or what i must do to complete it.
Let Vin be sin(ωt) = (e^(jωt)-e^(-jωt))/2j
Due to the formula I = Cdv/dt
Iin will be Cωcos(ωt) = Cjω(e^(jωt)+e^(-jωt))/2
Xc is then Vin/Iin
= ((e^(jωt)-e^-(jωt))/2j) / Cjω(e^(jωt)+e^-(jωt))/2)
it is here that i get bogged down and am not sure how to arrive at the expected result of 1/JωC
I have tried looking this up and and i can only find solutions using Cos and sin (which makes perfect sense)
Let Vin be sin(ωt) = (e^(jωt)-e^(-jωt))/2j
Due to the formula I = Cdv/dt
Iin will be Cωcos(ωt) = Cjω(e^(jωt)+e^(-jωt))/2
Xc is then Vin/Iin
= ((e^(jωt)-e^-(jωt))/2j) / Cjω(e^(jωt)+e^-(jωt))/2)
it is here that i get bogged down and am not sure how to arrive at the expected result of 1/JωC
I have tried looking this up and and i can only find solutions using Cos and sin (which makes perfect sense)
