## Magnetic resonance electrical impedance tomography ppt,what is the national geographic channel on cablevision 2014,emergency management degree programs in texas - Good Point

This page shows how Sage manages complex numbers in the analysis of electronic concepts like impedance.

For a Cartesian vector, we might name the components x and y and think of them as the base and height of a right triangle (Figure 1).

For the resistor, voltage and current are synchronized in time, which means the resistor dissipates power (voltage times current) as heat.

For the inductor, measured current represents the time integral of voltage (current lags voltage by 90°). Because an inductor's voltage and current are out of phase by 90°, an ideal inductor doesn't dissipate any power but stores energy as a magnetic field.

For the capacitor, measured voltage represents the time integral of current (voltage lags current by 90°). Because a capacitor's voltage and current are out of phase by 90°, an ideal capacitor doesn't dissipate any power but stores energy as an electrical field. For an RLC circuit and depending on the connection details, the circuit is appropriately described using equations involving complex numbers.

The Sage function "abs(x)" produces the "absolute value" of a number, which means different things depending on context.

In this case, "abs(x)" produces the magnitude of the provided complex argument (or the hypotenuse of a right triangle with x and y as base and height, see Figure 1). In electronics the term "reactance" refers to the behavior listed above for inductors and capacitors in which they store energy instead of dissipating it.

Remember that reactance is a dynamic property that depends primarily on the rate of change in current in the circuit containing the reactors. Note the presence of "i" (i2 = -1) in both the inductive and capacitive reactance subexpressions. For such a circuit there is a resonant frequency where (in a series circuit as shown in Figure 2) the inductive and capacitive reactances become equal (and cancel out), as a result of which the current peaks and the phase angle between resistance and reactance becomes zero.

The green trace represents complex phase angle in radians, and according to the series RLC equation, at resonance it should become zero. For this example, the numerical root finder "find_root()" was able to find a zero using arg(x), which, as shown in the above graph, crosses zero at resonance.

Another point about the above graph is that the red peak rises to about 10 mhos (units of admittance), which is consistent with our circuit's resistance value of .1 ohm and the fact that our "signal generator" is producing 1 volt. We see a dramatic change in the steepness of the slopes, the height of the resonance peak and the abrupt phase change at resonance (green trace). We see the opposite effect — a broadening of the resonance curve, a decrease in the amplitude of the resonance peak and a more gradual phase change near resonance. About these graphs, it would be nice to have two independent graph scales, one for the amplitude trace and another for the phase angle, so the green trace doesn't disappear when high admittance values are being plotted, but it seem Sage hasn't acquired that feature yet. One more observation about circuits possessing impedance — it is almost never the case than one can determine the power being dissipated in the circuit by measuring voltage and current. First, notice about this plot that the vertical axis is impedance, not admittance as with the series examples above. Again, the reason for this is that an ideal parallel RLC circuit with no resistances should dissipate no power and should therefore have infinite impedance at resonance (and zero impedance everywhere else). Well, that's clearly not right, and I suspect it's because the plot routine isn't providing enough resolution near resonance.

The reason for this result is that, at frequencies much less than f0, the capacitive reactance (Xc) becomes very high and the inductive reactance (Xl) becomes very low, which causes rl to predominate, but at frequencies much greater than f0, it's the opposite. Think about that result — the impedance at resonance is less than the resistance value in either leg. This kind of virtual circuit modeling is a very fruitful way to learn about electronic circuits in a short time, especially when compared to building and testing actual circuits.

I've been using computer algebra systems for years, and what impresses me most about Sage is the speed with which one acquires useful results, as well as the fact that one can easily export a decent graphic of an equation or a plot (for me personally, that's a revelation).

Those familiar with the conventions of electronic calculation may object to my having used i as the complex operator (i2 = -1) — in electrical work, i normally stands for current and this conflicts with its other meaning.

I want to close with a discussion of the symbolic representation for admittance, the reciprocal of impedance, expressed in the unit "mho" (that's ohm spelled backwards, if you missed it) or "Siemens" in SI units. Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. Setting the imaginary part to zero I get \$\omega = 0\$, but I think that in this case, it should corresponds to \$\omega = \frac{1}{\sqrt{LC}}\$ which makes the imaginary part infinite and the transfer function 1.

Since the above definition for resonance does NOT work in the circuit above, what is the correct one?

In this context (RLC), the resonance frequency is the frequency where the impedance of the inductor and capacitor are equal in magnitude and opposite in sign.

In summary, the frequencies at which the impedance is real, at which the impedance is stationary (max or min), and at which the reactances of the L & C are equal can be the same or different and each is some type of resonance frequency.

What you say about the poles and zeros is interesting but I am not sure if it helps because \$\omega_0\$ is not always a pole or a zero. So if resonance happends - in hypotetical ideal LC circuit there are no power losses on reactance.

For the 2nd part of the question (non-ideal inductor) you have a formula for omega when the impedance of the RLC circuit is purely real i.e. You need to make the denominator real by multiplying top and bottom with the denominator's complex conjugate. If you were to calculate where the pole is (irrespective of the complexity of the impedance it's simpler - you need to equate the denominator to zero and use the solution to a quadratic equation to find the complex s value. This second part of the equation is on the jw axis and represents where the pole's co-ordinate would be along that axis. Conclusion - there are two important frequencies in the case of the lossy inductor parallel resonated with a capacitor - how do you learn to get from A to B. The reason you are having trouble is because setting the imaginary part of the impedance to zero to find the resonant frequency only works for series rlc circuits. When you have an ideal inductor and an ideal capacitor in parallel, the resonant angular frequency is simply \$\frac{1}{\sqrt{LC}}\$. This is a circuit in which the LC "antiresonates" -- at some frequency the combined impedance is infinite (or in a practical circuit, at least maximum).

If we want to use the transferr function, we need 4 nodes (2 for the input and 2 for the output). Again, it intuitively makes sense that the imaginary part of the transfer function should be set to zero because the transfer function compares the input against the output of a circuit.

Effectively we just converted the original problem (without a transfer function) into a problem with a transfer function. Where Z is the transfer function (regardless of whether it's equal to the impedance or not).

I haven't tested it on any other circuits yet so I can't guarantee that it always works, but for now I am happy with this method and I don't see any situation where it doesn't work. Not the answer you're looking for?Browse other questions tagged resonance or ask your own question. Tomography analyzes the inner structure and composition of objects by examining them with excitations such as electricity and radiation in a cross-sectional manner.

In this research, we describe our efforts to create a small, low-powered and low-cost EIT sensor, one that could be integrated into consumer worn devices, such as smartwatches. 13 cm (5") low midrange system mounted in a Euro standard basket and with coated paper cone. Since we're surrounded by electronics in all directions, I thought this would be a useful and educational example of what Sage can do.

Just think of a two-component Cartesian vector — its elements are at right angles to each other and its properties are easy to understand. It's important to emphasize that the three notation systems in Figure 1 are equivalent and interchangeable — it's a matter of deciding which notation is appropriate to a particular problem. Such a circuit is normally analyzed with respect to time-varying applied voltages — a typical example would be an RLC circuit driven by a signal generator. The most important thing to understand about reactance is that voltage and current are orthogonal (at right angles), which explains why ideal inductors and capacitors don't dissipate any power (real-world reactors have some internal resistance, so they dissipate some power). The current flowing in the circuit is proportional to the hypotenuse of a right triangle with the resistance representing the base, and both reactances representing the height. I want to emphasize to my readers that these results are perfectly consistent with physical measurement of actual circuits. The reason is that the voltage and current are nearly always out of phase with each other, and the actual power is roughly proportional to the voltage times the current, times the cosine of the phase angle between voltage and current.

This circuit is a bit more complex, having two resistances (rl and rc) to compute, and consistent with real-world parallel resonant circuits where both resistances must be taken into account. Notice that the phase versus frequency relationship is reversed compared to the series case, and that at resonance the impedance is 5Ω. Think about this — at the circuit's resonant frequency we have increased the circuit's overall impedance by decreasing one of the resistances. Remember this about parallel-resonant circuits — the impedance at resonance is determined by the distributed resistances in the circuit, and if they could be eliminated, the impedance at resonance would be infinite, but for an infinitely narrow bandwidth. Let's find out — let's give rl and rc distinctive values and produce numerical results for f much less than, and much greater than, f0. I found somewhere(I think on this site but I cannot find it anymore) a definition that said that the resonance frequency is when the impedance is purely real.

And I am not sure what you mean by your last point because that again is not true for any RLC circuit.

We can do that because we considering this as ideal coil and capacitor (vector angles are -90 and +90).

Impedance vectors for ideal coil and capacitor are opposite so they substract and that makes impedance vector equal zero.

Energy flows from coil (magnetic field) to capacitor (electric field) and it flows back and forth with resonance frequency. I never worked with reactances before, but it seems that what you are doing is the same as setting the impedance of the capacitor equal to that of the inductor.

For parallel circuits, if there is resistance in the circuit resonance occurs where the impedance is maximum, and resonance occurs when the admittance has zero imaginary part. This configuration is used for tuning in AM radio and elsewhere-- as you noted the transfer function becomes 1 at the resonant frequency. When I asked someone for a general method of finding the resonance frequency of a cirquit, I was told to use the transfer function instead of the impedance accross the circuit.

This makes sense, because the phase shift of a signal is equal to zero at resonance and a signal's phase shift can't be evaluated unless the original signal and the shifted (output) signal are known. After setting the imaginary part of the transfer function to zero, I just solve for omega, and it turns out that this method works for the bandpass filter that was posted in the original question.

Electrical Impedance Tomography (EIT), proposed in 1978, uses pair-wise impendence measurements from surface electrodes surrounding an object to recover the impedance distribution of the inner structure. Achieving these design properties comes at the cost of reduced precision and resolution compared to medical EIT systems.

Due to the enormously strong and ultra-light titanium dome and in conjunction with a highly damped sypronyle surround, linear frequency response and no partial oscillations. With strong magnet system and large cone displacement due to elastic rubber surround.No longer available. That in turn means we can treat x and y as a Cartesian vector (Figure 1) and derive some useful values. Because reactances and resistances are orthogonal, we need to express this idea using complex numbers.

I am calling a special function "plt()" that appears above in the worksheet initialization block. The reason for the relatively low impedance is that currents within a parallel resonant circuit flow back and forth between the inductor and capacitor (the inductor stores energy as a magnetic field and the capacitor stores energy as an electrical field), but because of the circuit layout (Figure 3), this flow is resisted by both rl and rc. Remember that, in this idealized circuit, the resistors are the only devices able to dissipate power. Real-world circuits tend to have a finite bandwidth and a finite impedance, as shown by the first few graphs in this section. I won't reveal the result — it defies expectation, but a clue can be seen in the above two plots.

I must say, even though I've designed circuits for years including some aboard the NASA Space Shuttle, I didn't know about the outcome for two resistances of 1Ω in a parallel-resonant circuit. These losses are not affecting resonance frequency but there are some other parasitic losses (inductance in capacitor, capacity in coil etc.), capacity and inductance changes due changes to environment (temperature, magnetic permeability of coil neigbourhood and they may change resonance frequency a bit.

I say there's two frequencies but in fact there is another frequency that is important - the 3dB roll-off point but I'm not going there today. Like other tomographic methods – such as CT scans (x-rays), PET scans (gamma rays) and magnetic resonance imaging – medical EIT devices tend to be large and expensive, precluding integration into consumer electronics.

Due to the enormously high magnetic flux in the ferrofluid-filled air-gap high efficiency and high power handling are possible.

Even though I never realized it before, I can picture the reason (it has to do with internal versus external currents). This ability to non-invasively look inside a user’s body opens many new and interesting application possibilities. For example, muscles change their cross-sectional shape and impedance distribution when flexed.

In Proceedings of the 28th Annual ACM Symposium on User interface Software and Technology (Charlotte, North Carolina, November 8 - 11, 2015). Therefore, as a proof-of-concept application domain, we use our EIT sensor for hand gesture recognition.

For a Cartesian vector, we might name the components x and y and think of them as the base and height of a right triangle (Figure 1).

For the resistor, voltage and current are synchronized in time, which means the resistor dissipates power (voltage times current) as heat.

For the inductor, measured current represents the time integral of voltage (current lags voltage by 90°). Because an inductor's voltage and current are out of phase by 90°, an ideal inductor doesn't dissipate any power but stores energy as a magnetic field.

For the capacitor, measured voltage represents the time integral of current (voltage lags current by 90°). Because a capacitor's voltage and current are out of phase by 90°, an ideal capacitor doesn't dissipate any power but stores energy as an electrical field. For an RLC circuit and depending on the connection details, the circuit is appropriately described using equations involving complex numbers.

The Sage function "abs(x)" produces the "absolute value" of a number, which means different things depending on context.

In this case, "abs(x)" produces the magnitude of the provided complex argument (or the hypotenuse of a right triangle with x and y as base and height, see Figure 1). In electronics the term "reactance" refers to the behavior listed above for inductors and capacitors in which they store energy instead of dissipating it.

Remember that reactance is a dynamic property that depends primarily on the rate of change in current in the circuit containing the reactors. Note the presence of "i" (i2 = -1) in both the inductive and capacitive reactance subexpressions. For such a circuit there is a resonant frequency where (in a series circuit as shown in Figure 2) the inductive and capacitive reactances become equal (and cancel out), as a result of which the current peaks and the phase angle between resistance and reactance becomes zero.

The green trace represents complex phase angle in radians, and according to the series RLC equation, at resonance it should become zero. For this example, the numerical root finder "find_root()" was able to find a zero using arg(x), which, as shown in the above graph, crosses zero at resonance.

Another point about the above graph is that the red peak rises to about 10 mhos (units of admittance), which is consistent with our circuit's resistance value of .1 ohm and the fact that our "signal generator" is producing 1 volt. We see a dramatic change in the steepness of the slopes, the height of the resonance peak and the abrupt phase change at resonance (green trace). We see the opposite effect — a broadening of the resonance curve, a decrease in the amplitude of the resonance peak and a more gradual phase change near resonance. About these graphs, it would be nice to have two independent graph scales, one for the amplitude trace and another for the phase angle, so the green trace doesn't disappear when high admittance values are being plotted, but it seem Sage hasn't acquired that feature yet. One more observation about circuits possessing impedance — it is almost never the case than one can determine the power being dissipated in the circuit by measuring voltage and current. First, notice about this plot that the vertical axis is impedance, not admittance as with the series examples above. Again, the reason for this is that an ideal parallel RLC circuit with no resistances should dissipate no power and should therefore have infinite impedance at resonance (and zero impedance everywhere else). Well, that's clearly not right, and I suspect it's because the plot routine isn't providing enough resolution near resonance.

The reason for this result is that, at frequencies much less than f0, the capacitive reactance (Xc) becomes very high and the inductive reactance (Xl) becomes very low, which causes rl to predominate, but at frequencies much greater than f0, it's the opposite. Think about that result — the impedance at resonance is less than the resistance value in either leg. This kind of virtual circuit modeling is a very fruitful way to learn about electronic circuits in a short time, especially when compared to building and testing actual circuits.

I've been using computer algebra systems for years, and what impresses me most about Sage is the speed with which one acquires useful results, as well as the fact that one can easily export a decent graphic of an equation or a plot (for me personally, that's a revelation).

Those familiar with the conventions of electronic calculation may object to my having used i as the complex operator (i2 = -1) — in electrical work, i normally stands for current and this conflicts with its other meaning.

I want to close with a discussion of the symbolic representation for admittance, the reciprocal of impedance, expressed in the unit "mho" (that's ohm spelled backwards, if you missed it) or "Siemens" in SI units. Electrical Engineering Stack Exchange is a question and answer site for electronics and electrical engineering professionals, students, and enthusiasts. Setting the imaginary part to zero I get \$\omega = 0\$, but I think that in this case, it should corresponds to \$\omega = \frac{1}{\sqrt{LC}}\$ which makes the imaginary part infinite and the transfer function 1.

Since the above definition for resonance does NOT work in the circuit above, what is the correct one?

In this context (RLC), the resonance frequency is the frequency where the impedance of the inductor and capacitor are equal in magnitude and opposite in sign.

In summary, the frequencies at which the impedance is real, at which the impedance is stationary (max or min), and at which the reactances of the L & C are equal can be the same or different and each is some type of resonance frequency.

What you say about the poles and zeros is interesting but I am not sure if it helps because \$\omega_0\$ is not always a pole or a zero. So if resonance happends - in hypotetical ideal LC circuit there are no power losses on reactance.

For the 2nd part of the question (non-ideal inductor) you have a formula for omega when the impedance of the RLC circuit is purely real i.e. You need to make the denominator real by multiplying top and bottom with the denominator's complex conjugate. If you were to calculate where the pole is (irrespective of the complexity of the impedance it's simpler - you need to equate the denominator to zero and use the solution to a quadratic equation to find the complex s value. This second part of the equation is on the jw axis and represents where the pole's co-ordinate would be along that axis. Conclusion - there are two important frequencies in the case of the lossy inductor parallel resonated with a capacitor - how do you learn to get from A to B. The reason you are having trouble is because setting the imaginary part of the impedance to zero to find the resonant frequency only works for series rlc circuits. When you have an ideal inductor and an ideal capacitor in parallel, the resonant angular frequency is simply \$\frac{1}{\sqrt{LC}}\$. This is a circuit in which the LC "antiresonates" -- at some frequency the combined impedance is infinite (or in a practical circuit, at least maximum).

If we want to use the transferr function, we need 4 nodes (2 for the input and 2 for the output). Again, it intuitively makes sense that the imaginary part of the transfer function should be set to zero because the transfer function compares the input against the output of a circuit.

Effectively we just converted the original problem (without a transfer function) into a problem with a transfer function. Where Z is the transfer function (regardless of whether it's equal to the impedance or not).

I haven't tested it on any other circuits yet so I can't guarantee that it always works, but for now I am happy with this method and I don't see any situation where it doesn't work. Not the answer you're looking for?Browse other questions tagged resonance or ask your own question. Tomography analyzes the inner structure and composition of objects by examining them with excitations such as electricity and radiation in a cross-sectional manner.

In this research, we describe our efforts to create a small, low-powered and low-cost EIT sensor, one that could be integrated into consumer worn devices, such as smartwatches. 13 cm (5") low midrange system mounted in a Euro standard basket and with coated paper cone. Since we're surrounded by electronics in all directions, I thought this would be a useful and educational example of what Sage can do.

Just think of a two-component Cartesian vector — its elements are at right angles to each other and its properties are easy to understand. It's important to emphasize that the three notation systems in Figure 1 are equivalent and interchangeable — it's a matter of deciding which notation is appropriate to a particular problem. Such a circuit is normally analyzed with respect to time-varying applied voltages — a typical example would be an RLC circuit driven by a signal generator. The most important thing to understand about reactance is that voltage and current are orthogonal (at right angles), which explains why ideal inductors and capacitors don't dissipate any power (real-world reactors have some internal resistance, so they dissipate some power). The current flowing in the circuit is proportional to the hypotenuse of a right triangle with the resistance representing the base, and both reactances representing the height. I want to emphasize to my readers that these results are perfectly consistent with physical measurement of actual circuits. The reason is that the voltage and current are nearly always out of phase with each other, and the actual power is roughly proportional to the voltage times the current, times the cosine of the phase angle between voltage and current.

This circuit is a bit more complex, having two resistances (rl and rc) to compute, and consistent with real-world parallel resonant circuits where both resistances must be taken into account. Notice that the phase versus frequency relationship is reversed compared to the series case, and that at resonance the impedance is 5Ω. Think about this — at the circuit's resonant frequency we have increased the circuit's overall impedance by decreasing one of the resistances. Remember this about parallel-resonant circuits — the impedance at resonance is determined by the distributed resistances in the circuit, and if they could be eliminated, the impedance at resonance would be infinite, but for an infinitely narrow bandwidth. Let's find out — let's give rl and rc distinctive values and produce numerical results for f much less than, and much greater than, f0. I found somewhere(I think on this site but I cannot find it anymore) a definition that said that the resonance frequency is when the impedance is purely real.

And I am not sure what you mean by your last point because that again is not true for any RLC circuit.

We can do that because we considering this as ideal coil and capacitor (vector angles are -90 and +90).

Impedance vectors for ideal coil and capacitor are opposite so they substract and that makes impedance vector equal zero.

Energy flows from coil (magnetic field) to capacitor (electric field) and it flows back and forth with resonance frequency. I never worked with reactances before, but it seems that what you are doing is the same as setting the impedance of the capacitor equal to that of the inductor.

For parallel circuits, if there is resistance in the circuit resonance occurs where the impedance is maximum, and resonance occurs when the admittance has zero imaginary part. This configuration is used for tuning in AM radio and elsewhere-- as you noted the transfer function becomes 1 at the resonant frequency. When I asked someone for a general method of finding the resonance frequency of a cirquit, I was told to use the transfer function instead of the impedance accross the circuit.

This makes sense, because the phase shift of a signal is equal to zero at resonance and a signal's phase shift can't be evaluated unless the original signal and the shifted (output) signal are known. After setting the imaginary part of the transfer function to zero, I just solve for omega, and it turns out that this method works for the bandpass filter that was posted in the original question.

Electrical Impedance Tomography (EIT), proposed in 1978, uses pair-wise impendence measurements from surface electrodes surrounding an object to recover the impedance distribution of the inner structure. Achieving these design properties comes at the cost of reduced precision and resolution compared to medical EIT systems.

Due to the enormously strong and ultra-light titanium dome and in conjunction with a highly damped sypronyle surround, linear frequency response and no partial oscillations. With strong magnet system and large cone displacement due to elastic rubber surround.No longer available. That in turn means we can treat x and y as a Cartesian vector (Figure 1) and derive some useful values. Because reactances and resistances are orthogonal, we need to express this idea using complex numbers.

I am calling a special function "plt()" that appears above in the worksheet initialization block. The reason for the relatively low impedance is that currents within a parallel resonant circuit flow back and forth between the inductor and capacitor (the inductor stores energy as a magnetic field and the capacitor stores energy as an electrical field), but because of the circuit layout (Figure 3), this flow is resisted by both rl and rc. Remember that, in this idealized circuit, the resistors are the only devices able to dissipate power. Real-world circuits tend to have a finite bandwidth and a finite impedance, as shown by the first few graphs in this section. I won't reveal the result — it defies expectation, but a clue can be seen in the above two plots.

I must say, even though I've designed circuits for years including some aboard the NASA Space Shuttle, I didn't know about the outcome for two resistances of 1Ω in a parallel-resonant circuit. These losses are not affecting resonance frequency but there are some other parasitic losses (inductance in capacitor, capacity in coil etc.), capacity and inductance changes due changes to environment (temperature, magnetic permeability of coil neigbourhood and they may change resonance frequency a bit.

I say there's two frequencies but in fact there is another frequency that is important - the 3dB roll-off point but I'm not going there today. Like other tomographic methods – such as CT scans (x-rays), PET scans (gamma rays) and magnetic resonance imaging – medical EIT devices tend to be large and expensive, precluding integration into consumer electronics.

Due to the enormously high magnetic flux in the ferrofluid-filled air-gap high efficiency and high power handling are possible.

Even though I never realized it before, I can picture the reason (it has to do with internal versus external currents). This ability to non-invasively look inside a user’s body opens many new and interesting application possibilities. For example, muscles change their cross-sectional shape and impedance distribution when flexed.

In Proceedings of the 28th Annual ACM Symposium on User interface Software and Technology (Charlotte, North Carolina, November 8 - 11, 2015). Therefore, as a proof-of-concept application domain, we use our EIT sensor for hand gesture recognition.

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