Exponential forcing function to an RL circuit -- poles & zeros

[Joseph M. Zias]

For reference: Engineering Circuit Analysis, Hayt & Kemmerly, 4th ed, 1986, page 345.

Given a series RL circuit, the phasor current is I(s) = Vm/(R+sigma L), where S = sigma, w (omega) = 0. Thus we are dealing with only a exponential forcing function. Obviously I(s) goes to zero as sigma approaches infinity.

However, apply an exponential forcing function of Vm e^sigma t. t = time. Assuming we allow the transient to decay and then just look at the forced function we have: i(t) = (Vm/(R + sigma L) ) x e^sigma t.

Now, my question:
given some time longer than transient does the current i(t) approach infinity as sigma approaches infinity since e^sigma outranks sigma (basic limits). This essentially is questioning looking at only the phasor current.

 

[berkeman]

For reference: Engineering Circuit Analysis, Hayt & Kemmerly, 4th ed, 1986, page 345.

Given a series RL circuit, the phasor current is I(s) = Vm/(R+sigma L), where S = sigma, w (omega) = 0. Thus we are dealing with only a exponential forcing function. Obviously I(s) goes to zero as sigma approaches infinity.

However, apply an exponential forcing function of Vm e^sigma t. t = time. Assuming we allow the transient to decay and then just look at the forced function we have: i(t) = (Vm/(R + sigma L) ) x e^sigma t.

Now, my question:
given some time longer than transient does the current i(t) approach infinity as sigma approaches infinity since e^sigma outranks sigma (basic limits). This essentially is questioning looking at only the phasor current.

Could you post a picture of that page in the textbook to help clarify the question? I'm not familiar with exponential forcing functions with positive exponents...

 

[donpacino]

For reference: Engineering Circuit Analysis, Hayt & Kemmerly, 4th ed, 1986, page 345.

Given a series RL circuit, the phasor current is I(s) = Vm/(R+sigma L), where S = sigma, w (omega) = 0. Thus we are dealing with only a exponential forcing function. Obviously I(s) goes to zero as sigma approaches infinity.

However, apply an exponential forcing function of Vm e^sigma t. t = time. Assuming we allow the transient to decay and then just look at the forced function we have: i(t) = (Vm/(R + sigma L) ) x e^sigma t.

Now, my question:
given some time longer than transient does the current i(t) approach infinity as sigma approaches infinity since e^sigma outranks sigma (basic limits). This essentially is questioning looking at only the phasor current.

Generally its not called sigma its called s.

s=jw
so our new input to the RL filter is Vm * e^(st)

well this is equal to VM*e^(jwt)

we can use eulers identity to convert this to ...

Vm * ( cos(wt) + isin(wt) )What happens as t -> inf?

 

[donpacino]

I would also like to point out that this is a classic case of something we see a lot of on these forums. By trying to improve upon the data you are giving us a writing out sigma, you likely made it harder for people to help you. Often including EXACTLY what is mentioned in the text, along with your interpretation will give the best results.Note: This is not meant as dig at you, just a suggestion for the future! :)

 

[berkeman]

By trying to improve upon the data you are giving us a writing out sigma, you likely made it harder for people to help you.

Thanks Don!

 

[Joseph M. Zias]

Trying again.

 

[donpacino]

ahhhh. so sigma = real(s)

that changes things. Also you left a crucial piece of the puzzle.
When the forcing function is defined as Vm*e^(sigma * t), sigma is defined as a negative number.

With a negative exponent, what happens as t goes to inf?

 

[Joseph M. Zias]

I understand using a negative exponent but I specifically wanted a positive exponent. The pole/zero diagrams include positive values. A positive exponential voltage is possible to implement. My basic question is to resolve the difference between the phasor current and the real time current, at least as a theoretical concept. In the
example given the phasor current goes to zero as the neper frequency increases toward infinity, however the real time current would seem to increase. Thus looking only at a phasor current in studying a circuit may be misleading. I no longer have a circuit simulator to play games with this. So I ask others for their opinion on this.
mz

 

[donpacino]

If you have poles in the right half plane, you're going to have an unstable system, which is what happens when you have a positive exponent. I want to reiterate that the text specifies a negative sigma for good reason, that's the only time the system is stable with the included forcing function

 

[Joseph M. Zias]

I believe the poles in the right half plane generally refers to a feedback system, one classic example is the switching power supply. This is just an a series RL circuit with no feedback. For small values of neper frequency and time a very real current is calculated. Hayt and Kemmerly also show a positive neper frequency in their depiction of the amplitude of the phasor current with no mention of positive values not being permitted when calculating real current.

 

[donpacino]

I believe the poles in the right half plane generally refers to a feedback system, one classic example is the switching power supply. This is just an a series RL circuit with no feedback. For small values of neper frequency and time a very real current is calculated. Hayt and Kemmerly also show a positive neper frequency in their depiction of the amplitude of the phasor current with no mention of positive values not being permitted when calculating real current.

Poles in the right have plane refer to any system, its a mathmatical analysis tool, feedback systems is one application. Its not that positive neper frequencies are not "permited", its that with that forcing function you specified a positive neper frequency will cause the system to be unstable...

 

[Joseph M. Zias]

Well in this case you could consider the circuit unstable in the sense that if you let time continue to run the circuit would eventually have an infinite current. But we could always terminate the application of the forcing function before that. In the classic analog circuits that are unstable you either oscillate or lock up with just
a brief application of an input, perhaps noise. This investigation was to show the difference in looking at only a phasor current vs the real time current.

 

[donpacino]

if you want to know the difference between instantaneous current and phasor current, this may be the wrong way to go about it.
It should be clear that if you make neper frequency positive in your example the amplitude will go to inf as time increases. I'm fairly certain you realize that. I'm not sure what you want us to help you with. Below is a link to a definition of phasors and the link between instantaneous current and phasor current.

http://www.electronics-tutorials.ws/accircuits/phasors.html