There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Tuesday, July 9, 2013

3.18 - Separating out the radiated and simply reflected components - mathematical

All postings by author previous: 3.17 Resonant scattering with two output channels up: Contents next: 3.19 Applying methods of 3.18 to the circuit of 3.17

3.18 - Separating out the radiated and simply reflected components - mathematical

Summary: Previous postings have discussed and animated the concept of separating out the simply reflected and radiated components of waves powering a resonator. These components help in the understanding of coupling waves to a resonator. This posting explores the math behind the concept of separating out these components.

Keywords: LRC resonator, radiated waves, transmission line, Norton's theorem, canceling waves, system of differential equations, numerical solutions

Topics covered in this posting

  • A review of the defining system of differential equations for a resonant LRC circuit as excited with waves on a transmission line.
  • A look at the numerical solutions for the differential equations.
  • Separation of the circuit into two parts, one appropriate for the radiated waves and one appropriate for the simply reflected waves.
  • A look at the differential equations for both components and numerical solutions for each.
  • A showing that the sum of the radiated and the simply reflected waves equal the total voltage for the initial circuit. This is done algebraically and graphically.
  • A discussion of the concept of canceling waves as a method of absorbing power.

1. The general setup

Fig. 1. LRC resonator excited by waves on a transmission line. The physical size of the components ( i.e. V, R0 , LC , L, C and R ) is assumed to be much smaller than the length of the transmission line and also much smaller than a wavelength of the waves traveling on the transmission line.

This posting has to do with a resonator excited by waves on a transmission line as shown in Fig. 1. In a number of postings 3.12, 3.13, 3.14, 3.15, 3.16, 3.17 we have had animations and discussions of this system. In all of them we have used the concept that we can treat the negative going waves as having two components:

  1. The simply reflected component, and
  2. The radiated component.

In this posting we shall discuss the math behind this decomposition.

2. Equivalent circuit, differential equations, and numerical solution
Modeling circuit for a resonator powered by waves on a transmission line
Fig. 2. Circuit for modeling the excitation of a resonator by waves on a transmission line. The transmission line of characteristic impedance R0 and wave source of amplitude Vinc are replaced with an AC voltage source VS = 2Vinc  and resistor R0  in series. We are interested in calculating the incident, simply reflected and radiated waves at point A.

In the coming sections we will calculate and graph the radiated and simply reflected components. In this section, we establish a baseline by reviewing the previous analysis of the circuit.

For the sake of analysis, the transmission line and driving source can be replaced by an ideal AC voltage source VS with series resistor R0 as shown in Fig. 2 (this assumes the transmission line is lossless). While use of this circuit simplifies the analysis, we need to remember that the actual circuit still involves the transmission line connecting R0 with LC at point A on Fig. 1. The voltage calculated at point A will equal the sum of the incident and negative going waves on the transmission line at the right end (the end connected to LC ). We will first focus on solving for the voltage at point A and then after that, separating out the incident and negative going components.

We copy equations from 3.16f part 1. These are the governing equations for the circuit in Fig. 2. These are:

         (1a)     details of the resonator

         (1b)    details of the drive circuit

     .    (1c)     changes 2nd order into 1st order system of differential equations

Differentiating and rearranging (1a) gives:

     .     (2)

We substitute (1b) and (1c) into (2) to get:

       .     (3)

Equations (1b), (1c) and (3) make up a system of differential equations defining the response of this circuit to the AC voltage source VS . The equations are written in terms of the variables: the source current IS , the resonator voltage Vres  and the derivative of Vres . They can be solved algebraically or numerically.

Below we show numerical solutions to Eqns. (1b), (1c) and (3) as done by Euler Math Toolbox (a free scientific software package which is based on Maxima, an older similar package). For these calculations the component values were the same as in an earlier posting with the addition of LC = 8H and VS = 2V. With these values there is unity coupling, Q0 = 20, QL = 10 and the angular resonant frequency ω = 2.78rad/sec. The frequency of the driving source VS is set to this resonant frequency.

Graphs based on numerical solutions to Eqns. (1b), (1c) and (3)
Graphs of Vinc and Vres versus time
Graph of VA versus time

Fig. 3a. Voltage of incident sinusoidal burst (in blue) and voltage of the resonator Vres  (in red) as functions of time. Note that the vertical scale of this graph has a wider range than does that of Figs. 3b and 3c, meaning that the resonator voltage Vres  has a much larger amplitude than does the incident voltage or the voltage in the other two figures because of resonant build up.

To calculate Vinc we assumed that Vinc = VS/2 , a fact that is algebraically shown to be true lower down in the current posting.

Fig. 3b. Voltage at point A as a function of time as calculated with: VA = VS − ISR0 .
Graph of VnegGoing versus time

The differential equations (1b), (1c) and (3) are sufficient to calculate all voltages and currents in the circuit of Fig. 1 above, including Vres  and VA  as plotted in Figs. 3a and 3b.

To calculate Vinc we assumed that Vinc = VS/2 , as noted in the caption for fig. 3a just above. We then used this to calculate the negative going wave voltage. That is, we used the fact that the voltage at point A is the sum of the incident wave and the negative going wave meaning that VnegGoing = VA − Vinc .

Fig. 3c. Voltage of the negative going wave at point A as a function of time.

3. Separating out the wave components

In this section we separate the total voltage at point A into two components. These are:

  1. The simply reflected component which equals the incident wave with the appropriate phase shift, and
  2. The radiated component that originates from the resonator's oscillations.

These components are probably easiest to understand when the resonator is subjected to a burst of oscillations. In such a case we see its oscillations build up in time, and then decay after the burst is finished. The simply reflected component is proportional to the incident wave while the radiated component is proportional to the resonator's oscillation strength.

We now wish to add some algebra to these concepts. Because we will be dealing with a number of wave components we have a relationship chart (Chart 1) of these components below.

The simply reflected wave Vrefl  is the negatively going wave that exists at point "A" when there is an incident wave at that point and when the resonator is not excited. The resonator not excited means Vres = 0 which is equivalent to shorting out the resonator and then eliminating it as is shown in Fig. 4a. The component V1 is the part of VA due to the source under this condition. V1 is the sum of the incident wave Vinc  and the simply reflected wave Vrefl , as indicated in the Chart 1 below.

The radiated wave Vrad  is the wave at point A when the driving source is turned off or eliminated as shown in Fig. 4b and the circuit is powered only by remnant energy left in the resonator (also shown in Fig. 4b).

This process of splitting up a circuit based on its responses to various sources is the core of Norton's theorem, a widely used principle in electrical engineering.

Circuit for modeling the incident and simply reflected waves Circuit for modeling the radiated waves
↑ Fig. 4a. Circuit for modeling the simply reflected waves, with the resonator shorted out and deleted. Note that we define the current in this circuit as I1 and the voltage at point A as V1 . ↑ Fig. 4b. Circuit for modeling the radiated waves with source voltage set to zero and deleted. The resonator powers the circuit in a decay mode. Note that we define the current in this circuit as I2 and the voltage at point A as V2 .

The defining differential equations for the above circuit fragment (for the incident and simply reflected wave components) are:

   .   (5)

   .   (6)

We might have combined (5) and (6) to eliminate one variable (I1 ) however the resulting derivate of VS causes problems with the differential equation solver. We chose to use (5) to solve for I1  and then use (6) to calculate V1 .

The defining differential equations for the above circuit fragment (for the radiated wave component) are:

   .   (7)

   .   (8)

We can solve (8) for I2 and substitute it into (7) to eliminate I2.

   .   (9)

Graphs of V1 and V2

← Fig. 4c. V1 in blue and V2 in red as a function of time as per the equations just above. For Vres we used the solution obtained in Section 2 above.

V2 equals the radiated signal. We need to do more work to unravel the two components of V1 . These two components are the incident and simply reflected signals.

The reason for the relatively large V1 is that at ω = 2.78rad/sec, the impedance of LC is  j 22.2Ω . This impedance dominates over R0 = 5Ω and sees almost the whole voltage drop due to VS, except for the initial settling out times at the beginning and end of the burst.

4. Obtaining the incident and the simply reflected components from V1

We still need to tease the incident and simply reflected voltages out of V1. The "incident" and "reflected" have no relevance in the circuit (Fig. 2) unless we remember that the wire connecting the source (VS plus R0 ) and the coupling resistor is really a transmission line (shown in Fig. 1) on which waves exist and propagate and that the above simple circuit (Fig. 2) is a simplification that is only appropriate at the right hand end of this transmission line.

In the transmission line (at point A) we have two equations that must hold concerning the waves:

      and           ,    (10)

Graphs of Vinc and Vrefl

Fig. 5. Graph of Vinc  (red) and Vrefl  (blue), the incident wave and the simply reflected wave from Eqns. (14) and (15). These are both components of V1, the circuit with the resonator shorted out or removed as shown in Fig. 4a above. The radiated wave Vrad is not part of these components and equals V2 .   Vrad is graphed above in Fig. 4c.

Note that except for the very beginning, the two sets of oscillations are of equal amplitude, both having an amplitude of 1, i.e. equal to the assumed incident wave amplitude. The equal amplitudes is a result of the simply reflected wave being reflected off a pure inductance (see Fig. 4a) which causes a phase shift in the reflected wave, but no amplitude change (no ability to absorb power). The oddity of Vrefl at the very beginning and end of the sinusoidal burst is a transient effect due to energizing and de-energizing the inductor.

where Vrefl is the "simply" reflected voltage, that reflected with the resonator shorted out, and not the VnegGoing , the voltage of the total negatively propagating wave.

Equation (10) can be solved for the currents:

      and           .    (11)

Because I1 = Iinc + Irefl  we can write:

    .    (12)

This can be solved for Vrefl:

    .    (13)

Eqn (6) above implies that  I1R0 = VS − V1  which makes (13) become:

    .    (14)

Eqn. (14) gives us the reflected wave. We now need the incident signal. Because V1 = Vinc + Vrefl means that Vinc = V1 − Vrefl , we write:

    .    (15)

As we did for (13), we can use  I1R0 = VS − V1  to simplify (15):

    .    (16)

This agrees with the notion that VS  plus the resistor R0  represent a source launching a wave of amplitude VS / 2  down a transmission line of impedance R0 .  We used this fact in producing the graphs in Fig. 3 above.

5. Showing that the components V1  and V2  add to the voltage VA  of the complete circuit
Comparison of (V1 + V2) and VA

Fig. 6. Comparison of the sum of the component voltages (V1 + V2), in blue, and the total circuit voltage VA calculated with Eqns. (1b) through (3), in red. The sum voltage has been vertically offset so it isn't totally covered by the other waveform.

We start with Equations (5) through (8) above. First subtracting (7) from (5) yields:

    .    (17a)

Substituting I1 + I2 = IS  into (17a) yields:

    ,    (17b)

which is a simple rearrangement of Equation (1b), one of the defining differential equations of the original complete circuit.

Similarly, adding equations (6) and (8) yields:

    ,    (18a)

which becomes:

    .    (18b)

While not a defining equation, (18b) is correct for the behavior of the whole circuit.

The resonator voltage Vres is not defined by either equation (17b) or (18b) and must come from the Eqn. (1b) of the initial circuit. Neither of the component circuits Figs. 4a and 4b help in calculating Vres .

In order to calculate V1 , V2 , Vinc , Vrefl  and Vrad  in Figs. 4 and 5 above, we started with the solution for Vres  from Section 2 above, then used the equations from Sections 3 and 4 along with this Vres  to calculate V1 , V2 , Vinc , Vrefl  and Vrad . Thus we start with a Vres  that is a correct solution to the equations in Section 2. The above work in the current section shows that the sum of V1  and V2  is correct also. This shows conclusively that the components are consistent with the defining equations of the original circuit.

As further proof, in Fig. 6 we graphically compare numerical solutions of (V1  + V2 ) and VA . One of these has been slightly shifted so that one curve doesn't totally cover the other one. They are identical.

6. Power absorption via canceling radiation
Scattering channels

Fig. 7. Abstract view of the process of a resonator being driven by waves, illustrating the concept of scattering channels. The setup in this posting would be classified as having one input channel and one output channel.

In the abstract sense, the setup in this posting (see Figs. 1 and 2) can be drawn as shown in Fig. 7 with waves coming in through one "channel" and leaving through another "channel". The power that is absorbed internally by the resonator is the difference between that in the input channel and the output channels:

    ,    (19)

where the power in the output channel is given by:

    .    (20)

In order for the resonator to absorb power, it must reduce the power in the output channel which means reducing the amplitude of VnegGoing .

The two components in VnegGoing  are the simply reflected component Vrefl  and the radiated component Vrad . While the simply reflected component is a simple reflection of incident wave and therefore has the same amplitude as the incident channel, the radiated component comes from inside the resonator and is under the resonator's control.

The graphs in Fig. 8 show the various wave components. Of particular note is Fig. 8b which shows in an expanded view the phasing between the simply reflected and radiated wave components. We see that these two components are phased exactly 180degrees out of phase, a phasing that allows them to cancel to the maximum extent possible. Study the captions for more details of these graphs.

Vrefl , Vrad and VnegGoing
← Vrefl (blue) and Vrad (red)

Fig. 8a. Shows the two components that make up the negative going wave. The simply reflected wave Vrefl  (blue) is a simple reflection of the incident wave and closely resembles the incident sinusoidal burst. See Fig. 5 for more on this.

The radiated component (red) is proportional to the resonator oscillations (See Fig. 3a) so we see the expected build up of oscillations followed by the decay after the burst is turned off.

← Expanded view: Vrefl (blue) and Vrad (red)

Fig. 8b. An expanded view of Fig. 8a above.

The most striking feature of this graph is that the two sinusoids are perfectly out of phase. This allows the radiated component to cancel the simply reflected component to the maximum extent possible given the amplitudes of the two waves. In a sense this is one of the conditions for having the excitation right on resonance ... to minimize the negative going wave and thus maximize the power transfer to the resonator.

The odd overshoot on the first oscillation (in the negative direction) was verified by a simulation of the circuit in SPICE so appears to be a real transient effect. We also see similar behavior in Fig. 8a in the graph of Vrefl .

← VnegGoing, i.e. sum of Vrefl and Vrad

←Fig. 8c. This is a graph of the negative going wave. This is the sum of the two components shown in Fig. 8a. Because we have picked components so as to have unity coupling, the simply reflected and radiated components cancel each other after the resonator is "filled". Other coupling strengths would result in other patterns as shown in a previous posting.

← Off resonance, expanded view: Vrefl (blue) and Vrad (red)

←Fig. 8d. A similar graph to Fig. 8b except that the drive frequency is not tuned exactly to the resonant frequency (which is 2.78rad/sec). The drive frequency is set at 2.6rad/sec. We see the two sets of oscillations are not perfectly phased opposite each other as they were in Fig. 8b which means the oscillations will not cancel each other as well resulting in a larger negatively going wave and larger outgoing power. The resonator will not be excited to the degree it would have been. For more on driving resonators off resonance see earlier postings [1] [2].

← Over coupled, expanded view: Vrefl (blue) and Vrad (red)

←Fig. 8e. This is similar to Fig. 8b and 8d except that the coupling inductor LC was changed to 4H (from 8H) and the drive frequency was readjusted from 2.78rad/sec to 3.0rad/sec, the new resonant frequency at this different coupling inductance. The lower value of LC means that we now have an overcoupled condition, where the radiated oscillations become too large and actually add to the negatively going wave. The result is that power is sent down the output channel and the resonator sees a smaller level of excitation than it would otherwise. For more on undercoupling and overcoupling see earlier postings [1] [2].

CONCLUSION: We see above that when driven right at resonance, the radiated wave is phased to cancel the simply reflected wave to the maximum extent possible and maximize the power available to the resonator. We call the radiated wave a canceling wave.

All postings by author previous: 3.17 Resonant scattering with two output channels up: Contents next: 3.19 Applying methods of 3.18 to the circuit of 3.17