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Phasor and Element Circuit Relationship for AC Electric Circuits

In the previous post, we have learned how to write a current or voltage in the frequency or phasor domain. After understand how to convert the current or voltage in time domain to frequency domain or vice versa, now we will learn how to implement it to circuits consisting of elements R, L, and C.
phasor diagram
Make sure to read what is ac circuit first.

Phasor Relationship for Circuit Elements

What we need to do first is to convert the voltage-current relationship from the time domain to the frequency domain for every element. Passive sign convention will still be used in this process. For a starter, we will use the resistor. Assume the current flowing through a resistor R is i = Im cos(ωt + ), then the voltage across it with Ohm's law is
phasor diagram
Its phasor form of the voltage is
phasor diagram
But the phasor representation of the current is IIm∠∅. Thus,
phasor diagram
it shows us that the voltage-current relationship for a resistor in the phasor domain continues to be Ohm's law, as in the time domain. Figure.(1) draws the voltage-current relationship of a resistor.
phasor diagram
Figure 1. Relationship of voltage-current for a resistor in the : (a) time domain, (b) frequency domain
We should notice from Equation.(3) that the voltage and current are in phase, as drawn in the phasor diagram of Figure.(2).
phasor diagram
Figure 2. Phasor diagram for the resistor
Now we try with inductor L, assume the current through it is i = Im cos(ωt + ). The voltage across the inductor is
phasor diagram
Remember from the previous post about sinusoidal, that -sin A = cos(A + 90o). We can rewrite the voltage as
phasor diagram
and transforms to the phasor
phasor diagram
But Im∠∅ = I, and ej90o = j. Hence,
phasor diagram
showing that the voltage has a magnitude of ωLIm and a phase of  + 90o. The voltage and current are 90o out of phase. Specifically, the current lags the voltage by 90o. Figure.(3) draws the voltage-current relations for the inductor and the Figure.(4) draws the phasor diagram.
phasor diagram
Figure 3. Relationship of voltage-current for an inductor in the : (a) time domain, (b) frequency domain
phasor diagram
Figure 4. Phasor diagram for the inductor; I lags V 
Move on to capacitor C, assume the voltage across it is v = Vm cos(ωt + ). Its flowing current through the capacitor is
phasor diagram
Using the previous steps, we apply the transformation of derivative v(t) to the phasor domain as jωV as in here (Equation.15) and Equation.(8) above to obtain
phasor diagram
and shows us that the current and voltage are 90o out of phase. To be more precise, the current leads the voltage by 90o.
phasor diagram
Figure 5. Relationship of voltage-current for a capacitor in the: (a) time domain, (b) frequency domain
From Figure.(5) we notice the relationship of voltage-current for a capacitor.
phasor diagram
Figure 6. Phasor diagram for the capacitor ; I leads V
Figure.(6) shows the phasor diagram of the capacitor. Table.(1) gives a brief summary of the time domain and phasor domain representative of the circuit elements.
phasor diagram
Table 1

Phasor Relationship for Circuit Elements Example

For a better understanding let us review the example below :
1. Voltage v = 12 cos(60t + 45o) is applied to an inductor with 0.1 H. Find the steady-state current through the inductor.
Solution :
For the inductor, V = jωLI, where ω = 60 rad/s and V = 1245o V. Thus,
phasor diagram
Let us convert the result above and we get the time domain form,
phasor diagram

Have you understood what is phasor relationship with elements? Don't forget to share and subscribe! Happy learning!
Reference:  Fundamentals of electric circuits by Charles K. Alexander and Matthew N. O. Sadiku

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