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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.
Contents of this posting
 The general setup
 Equivalent circuit, differential equations and numerical solution
 Separating out the wave components
 Obtaining the incident and the simply reflected components from V_{1}
 Showing that the components V_{1} and V_{2} add to the voltage V_{A} of the complete circuit.
 Power absorption via canceling radiation
1. The general setup 

Fig. 1. LRC resonator excited by waves on a transmission line. The physical size of the components ( i.e. V, R_{0} , L_{C} , 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:
In this posting we shall discuss the math behind this decomposition. 
2. Equivalent circuit, differential equations, and numerical solution  

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 V_{S} with series resistor R_{0} 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 R_{0} with L_{C} 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 L_{C} ). 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: (1b) details of the drive circuit . (1c) changes 2nd order into 1st order system of differential equations Differentiating and rearranging (1a) gives: 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 V_{S} . The equations are written in terms of the variables: the source current I_{S} , the resonator voltage V_{res} and the derivative of V_{res} . 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 L_{C} = 8H and V_{S} = 2V. With these values there is unity coupling, Q_{0} = 20, Q_{L} = 10 and the angular resonant frequency ω = 2.78rad/sec. The frequency of the driving source V_{S} is set to this resonant frequency.

3. Separating out the wave components  

In this section we separate the total voltage at point A into two components. These are:
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 V_{refl} 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 V_{res} = 0 which is equivalent to shorting out the resonator and then eliminating it as is shown in Fig. 4a. The component V_{1} is the part of V_{A} due to the source under this condition. V_{1} is the sum of the incident wave V_{inc} and the simply reflected wave V_{refl} , as indicated in the Chart 1 below. The radiated wave V_{rad} 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.

4. Obtaining the incident and the simply reflected components from V_{1}  

We still need to tease the incident and simply reflected voltages out of V_{1}. The "incident" and "reflected" have no relevance in the circuit (Fig. 2) unless we remember that the wire connecting the source (V_{S} plus R_{0} ) 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:
where V_{refl} is the "simply" reflected voltage, that reflected with the resonator shorted out, and not the V_{negGoing} , the voltage of the total negatively propagating wave. Equation (10) can be solved for the currents: Because I_{1} = I_{inc} + I_{refl} we can write: This can be solved for V_{refl}: Eqn (6) above implies that I_{1}R_{0} = V_{S} − V_{1} which makes (13) become: Eqn. (14) gives us the reflected wave. We now need the incident signal. Because V_{1} = V_{inc} + V_{refl} means that V_{inc} = V_{1} − V_{refl} , we write: As we did for (13), we can use I_{1}R_{0} = V_{S} − V_{1} to simplify (15): This agrees with the notion that V_{S} plus the resistor R_{0} represent a source launching a wave of amplitude V_{S} / 2 down a transmission line of impedance R_{0} . We used this fact in producing the graphs in Fig. 3 above. 
5. Showing that the components V_{1} and V_{2} add to the voltage V_{A} of the complete circuit 

We start with Equations (5) through (8) above. First subtracting (7) from (5) yields: Substituting I_{1} + I_{2} = I_{S} into (17a) yields: 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: which becomes: While not a defining equation, (18b) is correct for the behavior of the whole circuit. The resonator voltage V_{res} 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 V_{res} . In order to calculate V_{1} , V_{2} , V_{inc} , V_{refl} and V_{rad} in Figs. 4 and 5 above, we started with the solution for V_{res} from Section 2 above, then used the equations from Sections 3 and 4 along with this V_{res} to calculate V_{1} , V_{2} , V_{inc} , V_{refl} and V_{rad} . Thus we start with a V_{res} that is a correct solution to the equations in Section 2. The above work in the current section shows that the sum of V_{1} and V_{2} 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 (V_{1} + V_{2} ) and V_{A} . 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  

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: where the power in the output channel is given by: In order for the resonator to absorb power, it must reduce the power in the output channel which means reducing the amplitude of V_{negGoing} . The two components in V_{negGoing} are the simply reflected component V_{refl} and the radiated component V_{rad} . 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.
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. 
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