Calculating current in a parallel AC RL circuit

I have this for Q4. a):

I have been struggling to calculate this as my numbers do not seem to be correct. When do I include the 90° lead that voltage has over current due to the inductor? Here's my attempt:

I understand that part b is pretty much the same but Xc = 1/wC and I want just the branch, not total current.

Any help or pointers with this would be great as I am a bit stuck on this one.

Thanks

• w value used in formula for XL is wrong. w=1000 – Harshad D Aug 25 '16 at 17:39
• Um. Just curious. Isn't your step (1) in 4a wrong? – jonk Aug 25 '16 at 17:40
• Maybe this will help you understand all the places you have gone wrong. electrical4u.com/rl-parallel-circuit – Harshad D Aug 25 '16 at 17:47
• I was under the impression that t = time? If so wouldn't I need 1/t = frequency for 2*pi*frequency? – Christopher Dyer Aug 25 '16 at 18:00

You attempt has many issues. First, let's clear up what $\omega$ is. With an expression in the form of $\sin(at)$, one period or one cycle is when: $$at_{period} = 2\pi$$ $$\Rightarrow a = 2\pi\frac{1}{t_{period}} = 2\pi\times{freq} = \omega$$ Therefore, $\omega$ is simply the coefficient in front of $t$, which is $10^3$ in 4a).
For the circuit 4a): $$I_{total} = \frac{V}{Z_{total}}$$ $$Z_{total} = \frac{1}{\frac{1}{Z_R}+\frac{1}{Z_L}}$$ ($Z_{total}$ is not $Z_R + Z_L$ as in your step 3.)
The equation is equivalent to: $$I_{total} = \frac{V}{Z_R} + \frac{V}{Z_L} = I_R + I_L$$ I will use this alternative representation because the intermediate quantities are slightly more interesting. $$I_R = \frac{100\angle 50^\circ}{5} = 20\angle 50^\circ$$ You are looking for an answer with time dependence, use the impedance of the inductor $Z_L = j\omega L$ which has the time related phase information (don't use reactance). $$I_L = \frac{100\angle 50^\circ}{(j\omega L = \omega L\angle 90^\circ)} = \frac{100}{1000\times0.02} \angle(50-90)^\circ = 5\angle{-40}^\circ$$ Finally, $$I_{total} = 20\angle 50 + 5\angle {-40} = 20.6\angle 35.96$$ (You need to look up how to add two numbers with phase angles. You cannot just add the amplitudes and the phase angles.) $$I_{total} = 20.6\sin(10^3t+35.96^\circ)\space A$$