Answer:
see bold numbers below
Step-by-step explanation:
Given the attached series-parallel circuit, you want to know ...
- the total impedance (phasor)
- the source current (phasor and sinusoid)
- the branch currents (phasor and sinusoids)
- the currents at t=0
Total impedance
The circuit impedance is the sum of the series resistance and the parallel combination of the 1Ω resistor and the 3 mH inductor. For ω = 333, the inductor's impedance is Xl = jωL = j(333)(.003)Ω = j0.999Ω.
The total impedance is the sum ...
Ztot = 10 + 1/(1/1 +1/j0.999) = 10.51∠2.73°
Circuit currents
The total current in the circuit is ...
Is = Vs/Ztot = 10∠45°/10.51∠2.73° = 0.9513∠42.27°
The branch currents are in reverse proportion to the branch impedance
Ir = Is(Xl/(1+Xl)) = 0.6724∠87.30°
Il = Is(1/(1+Xl)) = 0.6730∠-2.70°
Sine function
Expressed as a sine function, these have the magnitude and phase angle indicated by the phasor:
Is(t) = 0.9513·sin(333t +42.27°)
Ir(t) = 0.6724·sin(333t +87.30°)
Il(t) = 0.6730·sin(333t -2.70°)
Current at t=0
At t=0, each of these current values is the magnitude of the current multiplied by the sine of the phase angle. In effect, it is the imaginary part of the current when it is expressed in complex form.
Is(0) = 0.9513·sin(42.27°) = 0.6399 A
Ir(0) = 0.6724·sin(87.30°) = 0.6716 A
Il(0) = 0.6730·sin(-2.70°) = -.0317 A
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Additional comment
Solving these problems is immensely aided by a calculator that easily handles complex numbers. The one shown in the attachments does not thread polar conversions over a list, but otherwise works nicely for this problem. Angle mode is set to degrees. The value of x is 0.999i.