# How to Solve in Rectangular and Polar Form

#### mbrooke

##### Batteries Not Included
As a disclaimer this is not home work- or class work- rather an exam sheet I came across posted publicly online. I'm genuinely wondering what type of math is involved and how to go about solving its questions step by step.

#### mbrooke

##### Batteries Not Included
And ultimately for load where Z delta is not equal...

#### __dan

##### Senior Member
The magnitude, angle, representation is the Steinmetz method.

#### GoldDigger

##### Moderator
Staff member
The magnitude, angle, representation is the Steinmetz method.

Which is also referred to on the math side as polar coordinate representation.
The Z=R + iJ complex impedance, on the other hand, uses rectangular coordinates.

#### __dan

##### Senior Member
I'm not sure I would call it polar coordinates, seems like an obfuscation, almost wrong (to my eye).

It is a representation occuring in the complex plane, so each changing of the angle theta is a different instance of time. One full rotation of the angle is a representation of one period of the sine wave.

In polar or rectangular coordinates, I would assume you could plot any point on the plane and they all represent the same instance of time (as long as time t is not one of the axis). In the complex plane there is an underlying fundamental sine wave so you are also carrying a component of time, just not on the axis. It is inherent in the underlying sine wave.

It is a complex plane notation. I would not say it is polar and the textbook should not say that.

#### __dan

##### Senior Member
In the complex plane, any point along the indicated radius, on the same angle theta, happens at the same instance of time.

Any point not on the indicated radius, meaning having a different angle theta, is happening at a different instance of time, or a different time on the underlying sine wave, even as time is not one of the plane axis. The book could teach this.

#### mbrooke

##### Batteries Not Included
The magnitude, angle, representation is the Steinmetz method.

Thanks #### synchro

##### Senior Member
Unless we're talking about modulated waves, a phasor is a complex constant that can be in the polar form A e ʲᵠ , where A is the amplitude and φ is the phase.

A sinusoidal time waveform on the complex plane would be A e ʲ ⁽ ᵚ ⁺ ᵠ = A e ʲᵠ ⋅ e ʲ = A e ʲᵠ ⋅ [cos(2πft) + j sin (2πft)], where ω = 2πft with frequency f, time t, and phase shift φ.
I view the phasor A e ʲᵠ as the value of the rotating complex vector that _dan mentioned if it was captured or "strobed" synchronously with the period of the waveform. The value of φ in an absolute sense is somewhat arbitrary, because usually it's only the phase difference between circuit conductors or "nodes" that is important.

The physical time waveform that can be measured, such as a voltage or current, is the component of the complex waveform along the real (horizontal) axis which is A cos( 2πft + φ).

• mbrooke

#### mbrooke

##### Batteries Not Included
φ = phi?

cosine phi= power factor?

#### synchro

##### Senior Member
φ = phi?

cosine phi= power factor?
That is correct when we're interested in the phase difference between the current drawn by a load and the voltage applied to this load. So as you know with a purely resistive load this phase difference is zero and PF = cosine(0) = 1.
In my description of phasors above, φ was the phase of one particular sinusoidal voltage (or current) waveform. It was not a relative phase between current and voltage as when you are evaluating a power factor. But good question!

#### paulengr

##### Senior Member
If you need to know phase angle or differences or to compare magnitudes for instance polar form is the way to go. I’d say most of the time this tends to be the “final answer” form.

But if you just need to do the math convert everything to rectangular form then treat it as standard Ohms Law or Kirkhoffs voltage and current laws except it’s all in complex numbers but very easy to do the math. It is much easier are far less error prone than doing the math directly in polar form. Then just do one simple trig calculation at the end to get back to polar form.

Smith charts are from back in the slide rule days. Mostly displaced by either complex math on calculators or Bode plots,

#### mbrooke

##### Batteries Not Included
That is correct when we're interested in the phase difference between the current drawn by a load and the voltage applied to this load. So as you know with a purely resistive load this phase difference is zero and PF = cosine(0) = 1.
In my description of phasors above, φ was the phase of one particular sinusoidal voltage (or current) waveform. It was not a relative phase between current and voltage as when you are evaluating a power factor. But good question!
Arccos x PF= cosine? I want to see what the "reciprocal" is.

#### synchro

##### Senior Member
Arccos x PF= cosine? I want to see what the "reciprocal" is.

Since PF= cos (φ), then to get the angle φ from the power factor you would apply the inverse function of the cosine: arccos(PF) = φ. But since the output "range" of the arccos function is limited to 0 to +180 degrees, you would also need to specify whether the power factor is leading or lagging to determine the sign of the angle (whether it is positive or negative).
By the way, mathematically a reciprocal of a number x is 1/x.

• mbrooke

#### dkarst

##### Senior Member
I haven't seen anyone take a crack at the request from OP. To answer first part I would note that Vbn + Vna + (I x Z ) = 0. You only have one unknown you need so just just do some arithmetic and solve for I. Pay attention you are given Van and not Vna.

Edit sorry, I was looking at the 1.2 question asking for phase current...

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#### Hv&Lv

##### Senior Member
I haven't seen anyone take a crack at the request from OP. To answer first part I would note that Vbn + Vna + (I x Z ) = 0. You only have one unknown you need so just just do some arithmetic and solve for I. Pay attention you are given Van and not Vna.

Edit sorry, I was looking at the 1.2 question asking for phase current...

#### mbrooke

##### Batteries Not Included

I don't want to just see the answer. I want to know how its done step by step, and why.

#### __dan

##### Senior Member
I don't want to just see the answer. I want to know how its done step by step, and why.
Besoeker (years ago) posted a Steinmetz pdf treatment of pendulum motion that was great. I grabbed a copy when I could, to go through later, but I cannot find it now. I would be grateful if he sees this and can post it again. That is a very hard to find pdf.

You can also work out the falling body problem in the complex plane with the Steinmetx notation. Working out both pendulum motion and the falling body, in the complex plane with the Steinmetz notation, that will make you a magician.

Having been through the university training program for this type of stuff, I can say they absolutely do not teach it. They may test you to see if you have copies of the old test and homework, but they do not teach it.

#### mbrooke

##### Batteries Not Included
Besoeker (years ago) posted a Steinmetz pdf treatment of pendulum motion that was great. I grabbed a copy when I could, to go through later, but I cannot find it now. I would be grateful if he sees this and can post it again. That is a very hard to find pdf.

You can also work out the falling body problem in the complex plane with the Steinmetx notation. Working out both pendulum motion and the falling body, in the complex plane with the Steinmetz notation, that will make you a magician.

Having been through the university training program for this type of stuff, I can say they absolutely do not teach it. They may test you to see if you have copies of the old test and homework, but they do not teach it.

I'm thinking there could be a modern (easy) way to solve this? It doesn't look complex though, and vector math is good enough explanation IMO.

#### __dan

##### Senior Member
I'm thinking there could be a modern (easy) way to solve this? It doesn't look complex though, and vector math is good enough explanation IMO.
As you go from falling body, pendulum motion, spring and shock absorber, to inductance and capacitance, the math is the same.

In the modern analytical methods, differential equations, it is very abstracted away from the underlying physical reality. I had just grabbed one of the textbooks, "Diifferential equations" Blanchard and was looking for a free online pdf.

At the time I took the courses it was all pencil and paper with Laplace transform substitutions, linear algebra was a different course (the teacher casted sleeping spells). Surprisingly the university did not use vector math in the complex plane "at any time". They had a 50% graduation rate when I was there, so it is not easier, and there was no tie ever to the underlying physical reality. I don't know if you could stop in the middle of an analytical method and pick out the value for time t or the fundamental underlying sinewave. You could probably do it backwards working back from the Steinmetz notation, but that's not what they teach.

In the Steinmetz method, vector math in the complex plane, all of the values have a direct analogue in the observable physical reality.

#### mbrooke

##### Batteries Not Included
Thanks you for this, I will start reading more into it.