EEE 443/591 Blog Kick-Off
Welcome to my new blog. Hopefully, this will serve as a good forum to discuss all things about electromagnetics, waveguides, and antennas.
EEE 443/591 - Welcome! Please feel free to come to this site and to post whenever you want.
I was thinking this weekend about complex power in circuit analysis. I went and reviewed from my undegrad Circuits I course: real average power, reactive power, apparent power, power factor, leading, lagging, capacitive loads, inductive loads, etc.
When we are dealing with sinusoidal steady state stuff (SSSSS), the effective voltage and current will be approximately 0.707 of the maximum voltage and current. Recall that the impedance of an inductor is j*omega*L and the impedance of a capacitor is 1/(j*omega*C). Thus, when analyzing what is going on at the load of a sinusoidal steady state circuit, the current leads the voltage if the load is capacitive (can't instantly change the voltage across a capacitor), and the current lags the voltage if the load is inductive (can't instantly change the current through an inductor).
The apparent power is effective or RMS voltage times the effective or RMS current. The units are VA. The name "apparent" comes from the fact that we are seeing the familiar p=v*i formula here - which is apparently the power. The apparent power is the magnitude of the complex power - which has a real part (real average power) and an imaginary part (reactive power). Thus, the real average power will always be less than or equal to the apparent power. The reactive power is due to inductors and/or capacitors in the circuit which causes the current and voltage to be out of phase with each other. Inductors and capacitors are both passive devices with the ability to store electrical energy. Having stored energy, the energy can be released (discharging a capacitor - I forget what you would call this for an inductor; we don't "discharge" inductors). The transient response can be rather easily analyzed. The point is though that in a cycle, while some of the power is being absorbed by the load, other power (reactive power) is being stored in inductors and capacitors. This unit for reactive power is VAR (volt-amps-reactive).
The power factor is the real average power divided by the apparent power; so, in SSSSS, the pf is simply the cos(theta-phi). Theta would be the phase angle for the voltage and phi would be the phase angle of the current.
If the current and voltage are in phase, then the power factor angle (theta - phi) is 0, and the pf is 1. The load is purely resistive (at least the thevenin equivalent of the load is purely resistive). If the current leads the voltage by 90 degrees, then the load is purely reactive - and capacitive. If the current lags the voltage by 90 degrees, then the load is purely reactive - and inductive.
I assume that for people in EEE 443/591, all of this is review. My question is this: What is happening on the field level? An electric field exists between the parallel plates in a charged up capacitor. This can be seen fairly easily by Gauss's law. The presence of an electrical charge causes an electric field. The electric field is electric potential energy per unit charge (J/C = V). A moving charge (a current) produces a magnetic field, and the line integral of the magnetic field intensity about any closed path is equal to the direct current enclosed by that path. In time varying fields, a changing magnetic field produces a back emf (Faraday's Law). This is somehow linked to a displacement current that must be accounted for in Amepere's circuital law. That is, the line integral of the magnetic field intensity about any closed path is equal to the conduction current enclosed by that path plus the displacement current enclosed by that path. This is stuff I need to review. Things are starting to get fuzzy for me.
What I am trying to get at though is this: In a transmission line, a waveguide, or an antenna, if the load is capacitive, then do we say that the stored energy is mostly in the electric field? And if the load is inductive, then do we say that the stored energy is in the magnetic field? And how does this effect the radiation pattern transmitted by an antenna? I have a feeling that polarization is linked to all of this - another field concept that I need to review.
EEE 443/591 - Welcome! Please feel free to come to this site and to post whenever you want.
I was thinking this weekend about complex power in circuit analysis. I went and reviewed from my undegrad Circuits I course: real average power, reactive power, apparent power, power factor, leading, lagging, capacitive loads, inductive loads, etc.
When we are dealing with sinusoidal steady state stuff (SSSSS), the effective voltage and current will be approximately 0.707 of the maximum voltage and current. Recall that the impedance of an inductor is j*omega*L and the impedance of a capacitor is 1/(j*omega*C). Thus, when analyzing what is going on at the load of a sinusoidal steady state circuit, the current leads the voltage if the load is capacitive (can't instantly change the voltage across a capacitor), and the current lags the voltage if the load is inductive (can't instantly change the current through an inductor).
The apparent power is effective or RMS voltage times the effective or RMS current. The units are VA. The name "apparent" comes from the fact that we are seeing the familiar p=v*i formula here - which is apparently the power. The apparent power is the magnitude of the complex power - which has a real part (real average power) and an imaginary part (reactive power). Thus, the real average power will always be less than or equal to the apparent power. The reactive power is due to inductors and/or capacitors in the circuit which causes the current and voltage to be out of phase with each other. Inductors and capacitors are both passive devices with the ability to store electrical energy. Having stored energy, the energy can be released (discharging a capacitor - I forget what you would call this for an inductor; we don't "discharge" inductors). The transient response can be rather easily analyzed. The point is though that in a cycle, while some of the power is being absorbed by the load, other power (reactive power) is being stored in inductors and capacitors. This unit for reactive power is VAR (volt-amps-reactive).
The power factor is the real average power divided by the apparent power; so, in SSSSS, the pf is simply the cos(theta-phi). Theta would be the phase angle for the voltage and phi would be the phase angle of the current.
If the current and voltage are in phase, then the power factor angle (theta - phi) is 0, and the pf is 1. The load is purely resistive (at least the thevenin equivalent of the load is purely resistive). If the current leads the voltage by 90 degrees, then the load is purely reactive - and capacitive. If the current lags the voltage by 90 degrees, then the load is purely reactive - and inductive.
I assume that for people in EEE 443/591, all of this is review. My question is this: What is happening on the field level? An electric field exists between the parallel plates in a charged up capacitor. This can be seen fairly easily by Gauss's law. The presence of an electrical charge causes an electric field. The electric field is electric potential energy per unit charge (J/C = V). A moving charge (a current) produces a magnetic field, and the line integral of the magnetic field intensity about any closed path is equal to the direct current enclosed by that path. In time varying fields, a changing magnetic field produces a back emf (Faraday's Law). This is somehow linked to a displacement current that must be accounted for in Amepere's circuital law. That is, the line integral of the magnetic field intensity about any closed path is equal to the conduction current enclosed by that path plus the displacement current enclosed by that path. This is stuff I need to review. Things are starting to get fuzzy for me.
What I am trying to get at though is this: In a transmission line, a waveguide, or an antenna, if the load is capacitive, then do we say that the stored energy is mostly in the electric field? And if the load is inductive, then do we say that the stored energy is in the magnetic field? And how does this effect the radiation pattern transmitted by an antenna? I have a feeling that polarization is linked to all of this - another field concept that I need to review.
posted by Dan at 2:38 PM
1 Comments:
I'm thinking more about this:
Firstly, we know that the poynting vector, the radiated power density, is E x H. The units are W/m^2.
Digression:
On Capacitance:
The units for capacitance are coulombs/volt, but this is somewhat deceptive, because the capacitance of a capacitor has nothing to do with the charge on the plates or the voltage across the plates. It is rather a function of the dielectric between the plates and the dimensions. C=(epsilon)*A/d. The capacitance is directly proportional to the permittivity of the dielectric material. Both the capacitance and the permittivity are inversely proportional to the voltage across the plates and directly proportional to the charge on the plates.
The energy stored in a capacitor is 1/2 of the capacitance times the voltage squared, which is 1/2 of the charge times the voltage, which is 1/2 times the charge squared divided by the capacitance. If we look at charge as the most fundamental thing in our study of EM (and I think we should), then we would say that the enery stored in the electric fields of the capacitor is directly proportional to charge on the plates squared and inversely proportional to the capacitance (and thus inversely proportional to the permitivvity). I think this is the best way to look at it, for charge seems more fundamental than electric fields. Electric fields are somehow caused by the presence of charges (just as magnetic fields are caused by the presence of moving charges).
Duality:
On Inductance:
The energy stored in an inductor is stored in magnetic fields. The energy stored in magnetic fields is equal to 1/2 times the volume integral of the permeability times the square of the magnetic field intensity. Inductance is directly proportional to the permeability - just as capacitance is directly proportional to permitivvity. The energy stored in an inductor is 1/2 times the square of the current flowing in the closed path. The units of permeability are Wb/(A*m) or H/m. 1 H = 1 Wb/A. Dimensionally, a Wb is a V*s or a J/A.
At this point, I'm getting a little lost in the units and the math. I need to go back and reconsider what is physically going on here.
But in any case, back to the poynting vector. The power density is E x H. The total power into a volume (that is, the negative of the surface integral of the poynting vector) is equal to the ohmic power dissipated in that volume (the summation of the current density times the electric field intensity throughout the volume) plus the summation of the time derivative of the energy stored in the electric and magnetic fields within the volume. The energy in the electric fields is the 1/2 the permittivity times the electric field intesnity sqaured, and the energy in the magnetic fields is 1/2 times the permeability times the magnetic field intensity squared.
I take it that when we say ohmic power is "dissipated" that 100% of the ohmic power dissipated is transferred into mechanical heat energy. If harnessed, a portion of this mechanical heat energy can be converted back into electrical energy. However, in EM problems this is referred to as a "loss." Meanwhile, the energy stored in the electric and magnetic fields is essentially the energy stored in the capacitors and the inductors. But more fundamentally, the energy is stored in the fields themselves. For EM radiation does not need any medium through which to travel. The presence of a charge induces an electric field. A moving charge induces a magnetic field. But once the wave starts, a changing electric field induces a magnetic field and a changing magnetic field induces an electric field. The presence of a charge is not necessary any longer.
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