Is the Superposition Principle invalid?
 

On Mar 30, 3:44 pm, Cecil Moore wrote:
So does the Superposition Principle give us permission
to analyze the forward wave and the reflected wave
separately, or not?

It would appear from the many posts that the consensus is
that superposition is alive and well. It works for voltages
and currents. It does not work for power.

But then I have two questions.

Firstly, in another thread, the solution for the problem
presented required knowing the impedance that the generator
presented to the reflected wave. This is exactly the sort
of question that superposition handles easily: The impedance
encountered by the reflected wave at the generator is the
same as the generator's source impedance. I am curious as to
why you don't want to use superposition to facilitate solving
this problem?

Secondly, the "directional wattmeter" uses superposition
to compute Vf and Vr from which it computes Pf and Pr. You,
like many others seem willing to subtract Pr from Pf to
obtain Pnet. But this would only seem to be valid if
superposition works for power. So why are people who accept
that superposition does not work for power, prepared to
accept that Pnet = Pf - Pr?

....Keith

Is the Superposition Principle invalid?
 

Keith Dysart wrote:

The impedance
encountered by the reflected wave at the generator is the
same as the generator's source impedance.

No, the generator's source impedance is *NOT* the
impedance encountered by the reflected wave. Please
reference w2du's article again.

http://www.w2du.com/r3ch19a.pdf

Forget about the conjugate match and concentrate on the
non-dissipative source resistance being different from
what you are calling the generator's source impedance.
An *active* source creates a source impedance looking back
into the source that is *different* from what you are
calling the generator impedance.

Secondly, the "directional wattmeter" uses superposition
to compute Vf and Vr from which it computes Pf and Pr. You,
like many others seem willing to subtract Pr from Pf to
obtain Pnet.

One can directly add and subtract powers under certain
conditions. One condition is if two waves are not coherent.
Another condition is if two coherent waves have no effect
on each other. Since the forward wave and the reflected
wave have no effect on each other (except in the human
mind) reflected power can simply be subtracted from from
forward power to obtain power delivered to the load but
that is NOT superposition of powers. It is a simple
addition/subtraction of scalars based on the conservation
of energy principle.

But this would only seem to be valid if
superposition works for power. So why are people who accept
that superposition does not work for power, prepared to
accept that Pnet = Pf - Pr?

You seem to have forgotten the definition and rules of
superposition. Superposition applies to fields and waves.
Superposition doesn't apply to scalars. Power is a scalar.
Or another way to express it is:

V1 + V2 = V3 (vectors or phasors)

(V1 + V2)^2 = V3^2 (scalars)

V1^2 + V2^2 V3^2 (scalars)

It's a pretty simple principle of mathematics. The square
of the sum is NOT equal to the sum of the squares.
--
73, Cecil http://www.w5dxp.com

Is the Superposition Principle invalid?
 

Cecil Moore wrote:
V1 + V2 = V3 (vectors or phasors)

(V1 + V2)^2 = V3^2 (scalars)

In fact, the irradiance (power) equation falls out
directly from the above valid equation. Continuing
the process:

V1^2 + 2(V1)(V2) + V2^2 = V3^2

V1^2 + 2*SQRT(V1^2)*SQRT(V2^2) + V2^2 = V3^2

V1^2 + V2^2 + 2*SQRT(V1^2*V2^2) = V3^2

Dividing both sides of the equation by Z0 yields:

P1 + P2 + 2*SQRT(P1*P2) = P3

There you have it. The mathematical derivation of
the irradiance (power) equation.
--
73, Cecil http://www.w5dxp.com

Is the Superposition Principle invalid?
 

Cecil Moore wrote:
P1 + P2 + 2*SQRT(P1*P2) = P3

There you have it. The mathematical derivation of
the irradiance (power) equation.

This is, of course, for the condition where V1 and
V2 are in phase.
--
73, Cecil http://www.w5dxp.com

Is the Superposition Principle invalid?
 

On Apr 1, 9:34 am, Cecil Moore wrote:
Keith Dysart wrote:
The impedance
encountered by the reflected wave at the generator is the
same as the generator's source impedance.

No, the generator's source impedance is *NOT* the
impedance encountered by the reflected wave. Please
reference w2du's article again.

http://www.w2du.com/r3ch19a.pdf

Forget about the conjugate match and concentrate on the
non-dissipative source resistance being different from
what you are calling the generator's source impedance.
An *active* source creates a source impedance looking back
into the source that is *different* from what you are
calling the generator impedance.

It would appear that you are confusing the possible complexities
of a class C power amplifier with the simplicity of the generator
in the experiment I proposed. A 2 Amp current source
in parallel with a 450 Ohm resistor does not, as far as I can
tell, have a 'non-dissipative source resistance'. It has a
dissipating source resistor. This is not a particularly efficient
implementation, but is certainly a possible one. Now that I
have clarified that there is a dissipating source resistor will
this allow you to use superposition to solve the problem?

Just for your convenience, a reminder of the problem:
- generator with 450 Ohm source resistance drives
- a line with 450 Ohm characteristic impedance
- terminated by a 75 Ohm load
- the generator is set such that it would output
450 Volts into a 450 Ohm load

Question:
- Will there be ghosts?
If the answer is yes...
- What is the magnitude of the first re-reflection?
Ancillary question:
- What 'forward power' will a directional wattmeter in the
450 Ohm line indicate?
- What 'reverse power' will a directional wattmeter in the
450 Ohm line indicate?

If necessary for answering the question:
- The line can be assumed to be 31 wavelengths long.
- The generator can be assumed to be a 2 Amp current source
in parallel with a 450 Ohm resistor.

....Keith

Is the Superposition Principle invalid?
 

Keith Dysart wrote:
It would appear that you are confusing the possible complexities
of a class C power amplifier with the simplicity of the generator
in the experiment I proposed.

I am differentiating between what is possible in the
real world and what is possible in your mind. Presumably,
technical miracles are possible in your mind. The miracle
of rendering every circulator in the world obsolete by
adding a ten cent resistor is a miracle that you should
definitely pursue.

- Will there be ghosts?
If the answer is yes...
- What is the magnitude of the first re-reflection?

The answer is yes, but you have not given enough
information to solve the problem. Again, please
furnish a math model of a real world source.
--
73, Cecil http://www.w5dxp.com

Is the Superposition Principle invalid?
 

On Apr 2, 5:14 pm, Cecil Moore wrote:
Keith Dysart wrote:
It would appear that you are confusing the possible complexities
of a class C power amplifier with the simplicity of the generator
in the experiment I proposed.

The miracle
of rendering every circulator in the world obsolete by
adding a ten cent resistor is a miracle that you should
definitely pursue.

You raise an interesting question. What are the design tradeoffs
that drive choosing between a ten cent resistor and a circulator?
I suspect it relates mostly to the powers involved. The ten cent
resistor seems to see application in signal generators where the
extra power it dissipates is not of concern and it really is a
ten cent resistor. In high power applications, it definitely
would not be a ten cent resistor and having an amplifier which
can provide twice the normally required output voltage would be
far from efficient. It is perhaps here that a circulator finds
application driven by simple engineering cost-benefit tradeoffs.

- Will there be ghosts?
If the answer is yes...
- What is the magnitude of the first re-reflection?

The answer is yes, but you have not given enough
information to solve the problem. Again, please
furnish a math model of a real world source.

So why don't you prove my contention that the impedance
encountered by the reflected wave is 450 Ohms incorrect.

Here is how. Using my 450 Ohm value for the impedance
compute the reflection coefficient at the generator
using RC = (Z2-Z1)/(Z2_Z1). Using the various equations
for Vthrough, Vreflected, Ithrough and Ireflected, along
with superposition compute the various currents and
voltages within the system.

Then using Kirchoff's voltage and current laws, validate
all the nodal voltages and loop currents. If 450 Ohms is
not the appropriate impedance, then the validity check
will fail.

The beauty of this approach is that you do not need to
know what the actual value is, you merely need to prove
that it is not 450 Ohms.

I await the analysis.

....Keith

Is the Superposition Principle invalid?
 

Cecil Moore wrote:


You seem to have forgotten the definition and rules of
superposition. Superposition applies to fields and waves.
Superposition doesn't apply to scalars. Power is a scalar.


Cecil,

Superposition has nothing whatsoever to do with scalars vs. vectors.
Superposition has everything to do with linearity.

73,
Gene
W4SZ

Is the Superposition Principle invalid?
 

Keith Dysart wrote:
On Apr 2, 5:14 pm, Cecil Moore wrote:
Keith Dysart wrote:
It would appear that you are confusing the possible complexities
of a class C power amplifier with the simplicity of the generator
in the experiment I proposed.
The miracle
of rendering every circulator in the world obsolete by
adding a ten cent resistor is a miracle that you should
definitely pursue.

You raise an interesting question. What are the design tradeoffs
that drive choosing between a ten cent resistor and a circulator?

The design tradeoff is that the ten cent resistor will
not work.

So why don't you prove my contention that the impedance
encountered by the reflected wave is 450 Ohms incorrect.

Walter Maxwell did it for me.

Here is how. Using my 450 Ohm value for the impedance
compute the reflection coefficient at the generator
using RC = (Z2-Z1)/(Z2_Z1).

Invalid! If it were that easy my ten year old grandson
would already be an RF engineer.

I await the analysis.

http://www.w2du.com/r3ch19a.pdf
--
73, Cecil http://www.w5dxp.com