Impedance Question Homework: Solving for Z with Complex Numbers

[Enochfoul]

Homework Statement

Hi would somebody please be able to check to see if my workings look good?

Homework Equations

1/Z=1/Z₁+1/Z₂

IF Z₁ = 3 + J2 and Z₂ = 1 - j3, calculate Z giving your answer in the form r(cosθ + jin son θ) where θ is in radians

The Attempt at a Solution

Attached as a scanned image

 

[Joppy]

Are you sure its Z₁ = 3 + j² and not Z₁ = 3 + 2j ?

Either way, your working looks good.

 

[Enochfoul]

Are you sure its Z₁ = 3 + j² and not Z₁ = 3 + 2j ?

Either way, your working looks good.

Thanks for the reply here is a screen shot of the question

 

[Joppy]

Thanks. While your method looks correct, you've made a simple error in translation.

That is, in your working you've taken Z₁ as 3 + j² and not 3 + 2j. Similarly with Z₂, you've written it down as 1 - j³ instead of 1 - 3j.

Try again but this time with Z_eq = ( (3 + 2j) (1 - 3j) / (1 - 3j) + (3 + 2j) )

Excuse the brackets, having some trouble with Latex.

 

[Enochfoul]

Thanks. While your method looks correct, you've made a simple error in translation.

That is, in your working you've taken Z₁ as 3 + j² and not 3 + 2j. Similarly with Z₂, you've written it down as 1 - j³ instead of 1 - 3j.

Try again but this time with Z_eq = ( (3 + 2j) (1 - 3j) / (1 - 3j) + (3 + 2j) )

Excuse the brackets, having some trouble with Latex.

Hi Joppy on careful inspection of my work I have written J^2 and J^3, however I have calculated the solution based on J2 and J3. I have altered the values and have come up with the same answer as in my original scanned image. Do you deem the answers to be correct?

Ive just realized my answer in radians should have been negative. I have changed the solution to
Z= 2.765(cos(-0.416) - Jsin(-0.416)) Does that seem correct?

 

[Joppy]

So you have! I was wondering why things didn't reduce sooner in your working : ).

Yes that's correct. Well done!

 

[Enochfoul]

For my final answer I have a negative tangent, hence the solution will be in the 2nd and 4th quadrant of the circle where the tangent is negative.
My solution of -23.83 deg is correct but in positive format the answer would be 156.17 deg or 336.17 deg.

In the context of the question I suspect the correct solution would be 156.17° or its radian equivalent.

i.e

156.17xpi/180 = 2.735 rads.

Should my answer not be:
Z= 2.765(cos(2.735) - Jsin(2.735))

or do you believe I got it right as Z= 2.765(cos(-0.416) - Jsin(-0.416))

Thanks

 

[gneill]

For my final answer I have a negative tangent, hence the solution will be in the 2nd and 4th quadrant of the circle where the tangent is negative.
My solution of -23.83 deg is correct but in positive format the answer would be 156.17 deg or 336.17 deg.

In the context of the question I suspect the correct solution would be 156.17° or its radian equivalent.

The atan() function takes a single number as an argument, so when using it to find the angle associated with a Cartesian vector (or the angle of a complex number) it cannot distinguish whether a negative argument should place the angle in quadrants 2 or 4, or 1 or 3. In other words an argument comprised of (-y)/(+x) is indistinguishable from (+y)/(-x), and (+y)/(+x) is indistinguishable from (-y)/(-x). So the signs of the sine and cosine of an angle can be confused when using atan. (There is another function called atan2(y,x) which takes two arguments and always returns a result in the correct quadrant, but you can research that separately).

The best approach when looking for the angle associated with a vector or complex number is to make a quick sketch of the vector or point to locate the correct quadrant. Then adjust the angle returns by atan() if required.

As you can see in the figure your result of -23.8° is good. 336° would also work, but by convention these angles are always normalized to lie within the range -180° ≤ θ ≤ +180° . 156° would be incorrect because it would place the point or vector in the second quadrant rather than the third.

 

[Enochfoul]

The atan() function takes a single number as an argument, so when using it to find the angle associated with a Cartesian vector (or the angle of a complex number) it cannot distinguish whether a negative argument should place the angle in quadrants 2 or 4, or 1 or 3. In other words an argument comprised of (-y)/(+x) is indistinguishable from (+y)/(-x), and (+y)/(+x) is indistinguishable from (-y)/(-x). So the signs of the sine and cosine of an angle can be confused when using atan. (There is another function called atan2(y,x) which takes two arguments and always returns a result in the correct quadrant, but you can research that separately).

The best approach when looking for the angle associated with a vector or complex number is to make a quick sketch of the vector or point to locate the correct quadrant. Then adjust the angle returns by atan() if required.

View attachment 98115

As you can see in the figure your result of -23.8° is good. 336° would also work, but by convention these angles are always normalized to lie within the range -180° ≤ θ ≤ +180° . 156° would be incorrect because it would place the point or vector in the second quadrant rather than the third.

Thanks again an excellent answer I will stick with my solution surrounding -23.8