Calculating current and voltage using superposition

Question

I'm trying to calculate Vo and Io from the following circuit:

I was using node analysis but I don't really know how that worked. A friend of me told me that I could better us superposition. I understand that I have to calculate the current for each generator with the others excluded (open for current, short-circuit for voltage) so i would have 3 different problems to solve in this case. Now I looked up some examples online but they don't include capacitors or imaginary numbers (the j in the circuit is considered the imaginary number i, to avoid conflicts with the current I). Can anyone explain me how I would solve this?

Answer 1

1. You could certainly use superposition to solve this problem, but I think that node voltage would probably be much easier to use in this case due to the fact that you have only two unknown voltages in the circuit, but even then you have a relationship between the unknown voltages (due to the voltage source connected to either unknown node) which makes your life much easier. The voltage at the other node (call it v₁), can be found as a function of v_o due to the fact that, since a voltage source is connected to both of these nodes (call it v_s),we can subtract the voltage at v₁ from the voltage at v_o to determine the v_s. Also this can be written as:$$ v_s = v_o-v_1 $$ Therefore: $$ v_1=v_o-v_s $$ $$ $$

2. Also you said that you used node voltage already, but you aren't sure how that worked. Does this mean that you solved it correctly and are unsure of the philosophy behind the node voltage method or you are unsure of your answer?$$ $$

3. And to answer your question about the imaginary numbers: you calculate the node voltage using complex numbers (which is a combination of both real and imaginary parts). For example, to find the current in the branch of the circuit with the inductor and resistor (call it i₁), you would simply write: $$ i_1 = \frac{v_o}{Z_1}=\frac{v_o}{1+j} $$ Where Z₁ is called the complex impedance of the branch, which is essentially the same as resistance, but also phase shifts the current (in case you haven't already covered complex impedance).

Let me know if you have any questions!