Why do the imaginary parts of phasors in Kirchoff's Laws add to zero?

In summary, the conversation discusses the application of Kirchhoff's Voltage Law (KVL) in sinusoidal circuits, specifically when using phasors. The group of real terms in the phasor equation is equal to zero, but the question arises whether the group of imaginary terms also adds to zero. A proof is provided in the link provided, where taking the derivative and dividing by -w shows that the imaginary parts do indeed add to zero. The issue of removing the "R" in the KVL equation for phasors is also addressed.
  • #1
learningphysics
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Something's bugging me. Suppose we take kvl around a loop in a circuit, we have:

v1(t)+v2(t)+...=0

Suppose v1, v2, v3(t) are all sinusoidal (they can be written as Acos(wt+s)).

So we have
A1cost(wt+s1)+A2cost(wt+s2)+...=0

Suppose we replace all of them by their phasors, this should also equal zero but why? I'll write it out here (without suppressing e^jwt, but just adding the imaginary parts)

(A1cos(wt+s1)+A2cos(wt+s2)+...) + j(A1sin(wt+s1) + A2sin(wt+s2)+...)

If I know the group of real terms add to zero, does that necessary imply that the group of imaginary terms add to zero? Is there a proof of this?
 
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  • #2
http://www.csupomona.edu/~zaliyazici/f2001/ece209/ece209-02.pdf [Broken]

I just saw a very unsatisfying derivation of kvl for phasors here. How do you just remove the "R", and assume kvl holds for the entire phasor (not just the real part). I saw a similar technique elsewhere. They write kvl for the reals, and then R {V1+V2+...}=0 (where V1, V2 are phasors), then just remove the R. This removing of the R is bothering me. I don't see how it is justified.
 
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  • #3
Figured out the answer. It's simple. I wish they just put it in the proof in my book:

If we know
A1cos(wt+s1)+A2cos(wt+s2)+...=0

then take the derivative of both sides

-A1wsin(wt+s1)-A2wsin(wt+s2)+...=0

divide by -w on both sides:

A1sin(wt+s1)+A2sin(wt+s2)+...=0

so the imaginary parts in phasor notation add to zero.
 

1. What are Kirchoff's laws for phasors?

Kirchoff's laws for phasors are principles that describe the behavior of electrical circuits operating in the frequency domain. These laws state that the algebraic sum of currents entering and leaving a node in a circuit must equal zero, and the sum of voltage drops around a closed loop must also equal zero.

2. How do Kirchoff's laws apply to AC circuits?

Kirchoff's laws apply to AC circuits in the same way they apply to DC circuits. However, in AC circuits, the currents and voltages are represented using phasors, which are complex numbers that describe the magnitude and phase of the AC signal.

3. What is the difference between Kirchoff's current law and voltage law?

Kirchoff's current law states that the sum of currents entering and leaving a node must equal zero, while Kirchoff's voltage law states that the sum of voltage drops around a closed loop must equal zero. In other words, Kirchoff's current law deals with currents at a single point in a circuit, while Kirchoff's voltage law deals with voltages around a closed loop in a circuit.

4. How are Kirchoff's laws used in circuit analysis?

Kirchoff's laws are used in circuit analysis to determine the values of currents and voltages in a circuit. By applying these laws to a circuit, we can create a system of equations that can be solved to find the unknown values.

5. Can Kirchoff's laws be applied to non-linear circuits?

Yes, Kirchoff's laws can be applied to non-linear circuits. However, in non-linear circuits, the equations become more complex and may require numerical methods to solve them.

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