In this second part full input, small signal, impedance characteristics Z_{11} (magnitude and phase) with frequency of a single flat coil are measured using a Vector Nework Analyser (VNA). The SDR-Kits Vector Network Analyser 3E (VNA-SDR) is predominantly used as it provides data directly connected to a computer. Some measurements have also been cross-measured and checked using a Hewlett Packard 4195A Network Analyser (VNA-HP), and particularly when an equivalent circuit function is required to model actual device circuit equivalent values.

The measurements reported in this second part are for Z_{11}, the effective input impedance that the generator will see when connected to the input of the flat coil, and subsequently in part 3 connected with a range of loads and other flat coils. Impedance measurements for Z_{21} the transmission impedance between the input of the primary and the output of the secondary will be reported in future parts.

For network analyser impedance-frequency measurements an adjustable capacitance box was connected across the primary coil at the correct termination point to match the equal weights of copper for the secondary and primary. The unconnected load capacitance of the box when set to 0pF is 30.5pF. The VNA being used was calibrated to the end of the coaxial cable to be connected to the capacitance box and then tested with a 50Ω termination for accuracy over the frequency range. This calibration was then re-checked at the end of the measurement cycle to confirm stable calibration throughout the measurement period.

A wider band frequency scan 0.1MHz – 20MHz was used initially in order to identify the fundamental resonance frequency, any low-order harmonics, and any other impedance features of interest. Subsequently the frequency scan band was reduced to (0.1MHz – 5MHz) to allow for greater detail in the results.

Figures 1. show the measurement arrangement.

**VNA-SDR Measurements for 1P Primary**

Figures 2. show the wide frequency scan VNA impedance results for Z_{11} from calibration and through changing load capacitance on the primary. It is recommended to view the full-size scan images where the detail can be seen much clearer, (click on the image to see the full-size image and navigation icons). Below the figures 2. each individual result is considered and explained. In the explanations standard abbreviations are used as follows:

L_{P} = Inductance of the primary coil.

C_{P} = Capacitance box value connected in parallel with the primary coil.

L_{P}C_{P} = Parallel resonant circuit formed by the primary.

C_{PP} = Self-capacitance of the primary including the parasitic capacitance of the capacitance box when set at 0pF, which in total has been measured to be 30.5pF.

F_{P} = Fundamental resonant frequency of the primary.

L_{S} = Inductance of the secondary coil.

C_{S} = Self-capacitance of the secondary coil.

L_{S}C_{S} = Resonant circuit formed by the inductance of the secondary combined with self-capacitance of the secondary.

F_{S} = Fundamental resonant frequency of the secondary (F_{S1}).

F_{S2} = Second harmonic of F_{S} up to F_{S}_{N }the nth-harmonic.

F_{Ø} = Frequency at which a phase change takes place.

F_{Ø180} = Frequency at which a 180° phase change takes place.

F_{U} = Upper resonant frequency of the flat coil.

F_{L} = Lower resonant frequency of the flat coil.

M_{1} – M_{N} = Frequency markers on the results can be identified with a down pointing arrow on the result curve with a number above it.

Q – The quality factor of an impedance feature. For example, as the Q increases a resonance peak will become sharper and narrower, and as the Q decreases a resonance peak will become more rounded and wider.

|**Z**| – Magnitude of the impedance, (|**Z _{S}**| for secondary, |

**Z**| for primary, |

_{P}**Z**| for the upper frequency of the flat coil, and |

_{U}**Z**| for the lower frequency of the flat coil).

_{L}Ø – Phase of the impedance.

To view the large images in a new window whilst reading the explanations click on the figure numbers below:

Fig 2.1. Shows the calibration to the end of the bnc connected to the primary capacitance box (C_{P}). For this calibration the bnc was terminated with a standard 50Ω load and can be seen to be constant over the calibrated range of 0.1Mc/s and 20Mc/s. The phase in the calibration will swing repeatedly between ±180° indicating the near perfect match between the output impedance of the VNA (50Ω) and the standard 50Ω termination, as expected for a calibration of this type of instrument. M_{1} confirms the impedance magnitude 50.00Ω and phase -1.19° at 2863kc/s.

Fig 2.2. Shows the 1P primary with C_{P}=0pF (30.5pF from the self-capacitance of the primary turns and parasitic capacitance combined, C_{PP}). There is a strong parallel self-resonance of the primary coil which results from combination L_{P}C_{P}. From marker M_{1} the fundamental resonant frequency F_{P}=9718kc/s with the 180° phase change, characteristic of a resonant circuit, shifted to a much higher frequency above the upper limit of the scan (20Mc/s). The large shift between F_{Ø180} and F_{P} results from the large imbalance between the inductance of the coil L_{P} and the very small self-capacitance C_{P} . As C_{P }starts to rise in the following figures F_{Ø180} will start to fall in frequency.

Fig 2.3. Shows the effect of increasing C_{P}=250pF. F_{P} has now dropped considerably to 3654kc/s and F_{Ø180} has just entered at the far end of the scan at 19767kc/s. The magnitude of the impedance has increased as the resonance has strengthened, and the Q of the coil has also increased as lumped element capacitance stabilises the electrical properties of the circuit and dominates over the self-capacitance of the coil.

Fig 2.4. Increasing C_{P}=500pF continues to reduce F_{P} and F_{Ø180}. The magnitude of the impedance |**Z**| continues to fall and the Q reduces slightly.

Fig 2.5. Increasing C_{P}=750pF continues to reduce F_{P} and F_{Ø180}. |**Z**| continues to fall as the parallel resonance weakens, and F_{P} passes through what will be F_{Ø180} of the secondary coil when added to the primary.

Fig 2.6. Increasing C_{P}=1000pF continues to reduce F_{P} and F_{Ø180}. |**Z**| continues to fall, and F_{P} comes into the fundamental band of operation (1810-2000kc/s).

Fig 2.7. Increasing C_{P}=1500pF continues to reduce F_{P} and F_{Ø180}. |**Z**| continues to fall, and F_{P} goes below the fundamental band of operation and into the medium wave (MW) band.

Overall the effect of increasing the primary capacitance C_{P} is to progressively reduce the primary’s fundamental resonant frequency F_{P}. As a better balance between L_{P}C_{P }is established the wide gap between F_{P} and F_{Ø180} reduces. F_{P} appears to go through an optimal point of resonance where the impedance |**Z**| is maximum and the Q is maximum at a resonant frequency F_{P} ~ 4500kc/s and C_{P} ~ 195pF. The shifting of F_{P} with C_{P} will allow the complete flat coil 3S-1P to be tuned, as the resonant circuit in the primary interacts with the resonant circuit in the secondary. These two coupled resonant circuits form the overall impedance characteristics of the flat coil as investigated below. No harmonics of the fundamental where observed in the impedance scans of the primary.

**VNA-SDR Measurements for 3S-1P**

Figures 3. show the wide frequency scan VNA impedance results for Z_{11} with changing load capacitance on the primary.

To view the large images in a new window whilst reading the explanations click on the figure numbers below:

Fig 3.1. Shows the secondary and the primary combined together in frequency and with C_{P} = 0pF. Here the two resonant characteristics appear superimposed on one another. The self-resonance of the primary is very clearly defined at M_{4}, and is very similar to that measured in the primary only results of Fig 2.2. The self-resonance of the secondary has generated the fundamental F_{S} = 2254kc/s at M_{1 }and F_{Ø180} = 2437kc/s at M_{2}. For the secondary F_{Ø180} is defined by the effective wire length used in λ/4 mode with the addition of an impedance lowering extension at the bottom-end of the coil. In Part 1 of the design the wire length was to be arranged to give F_{Ø180} = 2400kc/s which is very close to the result measured. The fundamental operating frequency was designed to fall into the 160m amateur band (1810-2000kc/s), and currently F_{S} = 2254kc/s slightly above this band. In operation C_{P} in the primary will be adjusted in order to tune the resonant operating frequency into the required band. The balance between L_{S}C_{S} is much better in the secondary as the self-capacitance of the many turns is much bigger and more stable than that for the primary, and so the gap between F_{P} and F_{Ø180} is smaller, and in this case also contributed to by the series resistance of the secondary coil. Q appears reasonable at this stage and the impedance |**Z**| is much lower than that for the primary which most likely results from the inter-winding capacitive network of the secondary. There are 2 odd harmonics above the fundamental F_{S2 } at M_{3} and F_{S3} at M_{5} which occur at 3λ/4 and 5λ/4 respectively. It maybe by chance but it is interesting to note that F_{P }is not that far away from 4λ/4 for the secondary. Whether this will have any impact on the performance of the flat coil remains to be established.

Fig 3.2. Increasing C_{P} = 250pF has mainly, and as expected from figures 1, reduced the primary resonant frequency F_{P} down to a frequency much closer to the secondary. Here we can see the start of the formation of the flat coil upper and lower resonant frequencies. The F_{Ø180} point at M_{2} stays constant as the effective wire length of the secondary is not changing and dominates the fundamental F_{S }of the secondary. The lower frequency F_{S }at M_{1} from the secondary forms the lower frequency of the flat coil, and starts to move away from F_{Ø180}_{ }as C_{P} is increased, which allows tuning of F_{S} to the required frequency using C_{P}. The upper frequency F_{P} at M_{3} from the primary forms the upper frequency of the flat coil, and moves progressively down towards F_{Ø180}_{ }as C_{P} is increased. As two coupled circuits cannot resonate at exactly the same frequency when C_{P} is continued to be increased > 1000pF F_{S} and F_{P }will appear to swap position, with F_{P} emerging below F_{S} and whilst close to F_{S} appearing to push F_{S} slightly above F_{Ø180}. When two or more coupled resonant circuits interact the energy exchange between the modes of vibration creates beating and a particular mode becomes dominant (drives) coupling energy from one resonant circuit to the other. It is in this region that the flat coil is most interesting to investigate, and an important factor in the study of the displacement and transference of electric power. These coupled modes of F_{P} and F_{S} will be investigated in more detail on Figures 4., and practically within the experiments.

Fig 3.3. Increasing C_{P}=500pF starts to bring the upper and lower frequencies of the flat coil into closer balance. |**Z _{S}**| at M

_{1 }is increasing in impedance as the parallel resonance in the secondary is strengthened by coupling from the primary, whilst |

**Z**| at M

_{P}_{3 }is reducing. The overall effect is to bring the electric and magnetic fields of induction, across the primary and the secondary, towards a more balanced point. It is this point of balance (harmony between the two induction fields) that is conjectured to be the optimal point to trigger a non-linear event. It is conjectured that a non-linear event at this point of balance, and dependent on the form of the load connected, will generate a coherent displacement event between the generator (source) and the load(s). This consideration will be developed further during the experimental reporting, and in conjunction with actual results obtained.

Fig 3.4. Increasing C_{P}=750pF has now passed through the balance point between the fields of induction and to where the lower frequency starts to dominate the resonance of the flat coil, and the upper frequency will now continue to diminish. The lower frequency F_{S} at M_{1} is now within the 160m of operation. F_{Ø180} remains unchanged. Harmonics are diminishing as the resonance of the secondary starts to be suppressed by the high capacitive loading of the primary.

Fig 3.5. Increasing C_{P}=1000pF the primary resonance F_{P} is dominating the overall resonance at the lower frequency of the flat coil. Secondary coil harmonics have almost completely been suppressed by the high capacitive loading of C_{P}. The lower resonant frequency at M_{1} has now moved out of the lower end of the 160m amateur band.

Fig 3.6. Increasing C_{P}=1500pF the primary resonance F_{P} is now totally dominating the overall resonance at the lower frequency of the flat coil. The lower resonant frequency at M_{1} has now moved into the medium wave band at 1515kc/s.

When C_{P }is lower, and in the range ~200 – 450pF, the overall resonance of the flat coil is dominated by F_{P} the fundamental resonant frequency of the primary, and what has formed the upper resonant frequency of the flat coil ~ 2700kc/s – 4500kc/s. When C_{P }is larger, and >950pF, the overall resonance of the flat coil is again dominated by F_{P} the fundamental resonant frequency of the primary at the lower resonant frequency of the flat coil < 1750kc/s. In between, in the range 450pF < C_{P} < 950pF, there is a more established balance between F_{S} and F_{P} and the upper and lower resonant frequencies of the flat coil are determined by the interaction and energetic exchange between the secondary and the primary. It is this region that is most interesting to experiments in the displacement and transference of electric power, and whose impedance characteristics are investigated in more detail below.

Figures 4. show the narrow frequency scan VNA impedance results for Z_{11} at a key set of different load capacitance.

To view the large images in a new window whilst reading the explanations click on the figure numbers below:

Fig 4.1. Here the primary capacitance C_{P }= 830pF has been adjusted so that the lower resonant frequency of the flat coil F_{L }at M_{1} is at the nominal designed point of 1850kc/s in the 160m amateur band. When setup to self-resonate with feedback from the flat coil to the generator the flat coil will stably oscillate at F_{L}, and will form a base starting point for the experiments in the displacement and transference of electric power.

Fig 4.2. Shows the resonant frequency of the primary F_{P} when the secondary is removed from the flat coil, and all other conditions and setup are kept the same. F_{P} is closer to the lower resonant frequency of the flat coil F_{L }than the upper F_{U}, which corresponds with a stronger resonance at F_{L}, and a weaker one at F_{U}, and hence |**Z _{L}**| > |

**Z**|.

_{U}Fig 4.3. Here the primary capacitance C_{P }= 515pF has been adjusted so that the magnitude of the impedance at the upper resonant frequency is equal to the magnitude of the impedance at the lower resonant frequency |**Z _{U}**| = |

**Z**|. It is conjectured that at this point there is balanced interaction between the secondary and primary resonance points which is optimal for the balanced energetic inter-exchange between the electric and magnetic fields of induction between the two coils. It is at this point where best coherence between the two fields of induction can be established, and hence a significant pre-condition to displacement established. It is also conjectured that the initiation of a displacement event requires a non-linear trigger within the system being tested whether that originates from the generator, the coils, or is stimulated as a response (pulled by) the load. It is the purpose of the experimental measurements to establish if this or another mechanism is the case, and the properties and characteristics under which they occur.

_{L}Fig 4.4. Again shows the resonant frequency of the primary F_{P} when the secondary is removed from the flat coil, and all other conditions and setup are kept the same. Here we can see that F_{P} at M_{1} (2577kc/s) occurs almost exactly equi-distant between F_{S }and F_{P }with the secondary added to the flat coil. From Fig. 4.1. (F_{P} – F_{S}) / 2 + F_{S} = 2548.5kc/s, and from Fig. 4.2. F_{P} = 2577kc/s (<2% difference). Here the resonance of the primary F_{P} has inter-acted with the resonance of the F_{S} so that both contribute equally to the overall flat-coil characteristic and hence establishing the balance between the electric and magnetic fields of induction as discussed in Fig. 4.1.

Fig 4.5. Here the primary capacitance C_{P }= 650pF has been adjusted so that the upper F_{U} and lower F_{L} resonant frequencies of the flat-coil are equi-distant from the 180° phase change frequency of the secondary, F_{Ø180}. The resonant frequency of the secondary F_{S} (F_{L}) has just moved into the 160m amateur band at 1956kc/s, and |**Z _{S}**| wil start to dominate the resonance of the flat coil. When allowed to self-resonate with feedback to the generator the flat coil will stably oscillate at F

_{S}.

Fig 4.6. Again shows the resonant frequency of the primary F_{P} when the secondary is removed from the flat coil, and all other conditions and setup are kept the same. F_{P }has progressed down slightly in frequency with increased C_{P }from Fig. 4.4. as expected.

Fig 4.7. Calibration test at the end of the measurement period using a standard 50Ω load, with M_{1} confirming 50Ω at 1850kc/s.

**VNA-HP Measurements for 3S-1P
**

Figures 5. show a selection of frequency results to confirm and check the accuracy of the results from two different VNAs, and also a basic equivalent circuit analysis for the primary S3 the narrow frequency scan VNA impedance results for Z_{11} at a key set of different load capacitance.

Fig 5.1. Shows the 1P primary with C_{P}=0pF as per the measurement of Fig. 2.2. F_{P} at the MKR frequency is 10647kc/s as measured by the VNA-HP, and was previously measured as 9718kc/s as measured by the VNA-SDR which represents ~ 9% variation in this measurements between the two methods. The phase curve indicated by the VNA-HP also shows a large variation corresponding to F_{P} indicating the fundamental resonant frequency of the primary, and some further small impedance variation and phase change at ~18Mc/s. This highlights the difficulty of making measurements directly on the primary where there is a large imbalance between L_{P} and C_{P} which easily leads to varying measurement conditions easily influenced by the surrounding factors such as earthing structures, conductors, and other electrical loading influences. As L_{P} and C_{P} come into better balance this stabalises and a much greater measurement accuracy is obtained between both measurement machines.

Fig 5.2. Increasing C_{P}=750pF as per the measurement in Fig., 2.5. the 1P primary resonant frequency has moved to 2239.25kc/s, and was previously measured as 2233kc/s by the VNA-SDR showing < 0.5% variation in the frequency. |**Z**| measures as 8.02kΩ, and was previously 9.072kΩ showing ~12% variation. F_{Ø180} measures as 11000kc/s and was previously 11744kc/s ~7% variation.

Fig 5.3. Shows the computed equivalent circuit of the 1P primary conditions as per Fig. 5.2. The equivalent circuit is calculated by the HP4195A based on a fit model to the measured curve. In part 1 of the impedance measurements the inductance of the 1P primary was measured at 6.453µH, and here is modelled as 6.163µH ~5% variation. The 1P capacitance C_{P} + C_{pp }=750 + 30.5 = 785.5pF, and here is modelled as 815.91pF ~9% variation. The dc resistance from part 1 was 1.31mΩ and here is modelled as 144.47mΩ. The resonant circuit L_{P}C_{P} shows a reasonably good fit between the measured values and the modelled values. The series resistance of the coil modelled does not correspond to that measured, although it is considered that this difference does not significantly effect the quality factor of the flat coil, or assessment of the small signal impedance measurements thus far.

Fig 5.4. Shows the secondary added to form the flat coil 3S-1P where C_{P}=0pF. As per Fig. 3.1 for the secondary and primary impedance-frequency responses become superimposed on one another. The primary resonance F_{P} can be identified clearly at 11000kc/s along with the gradual phase chanhge, along with the secondary resonance F_{S} at 2189.5kc/s with a variation of ~3% from the VNA-SDR measurement of 2254kc/s. Secondary harmonics can be identified in a similar way in a corresponding frequencies as per Fig. 3.1. The greater sensitivity and more finally tuned input circuits of the VNA-HP also show a strong additional resonance at F_{S4} at ~17.5Mc/s which is not identified in Fig. 3.1.

Fig 5.5. Shows primary capacitance C_{P}= 830pF as per Fig. 4.1 where the lower resonant frequency of the flat coil F_{L} has been adjusted to be at the designed frequency of 1850kc/s. The VNA-HP shows a close correlation at 1851.75kc/s ~1% variation, and corresponsing close correlation of F_{Ø180} and F_{U}.

Fig 5.6. Shows primary capacitance C_{P}= 515pF as per Fig. 4.3. Here the F_{Ø180} point is compared at 2390.75kc/s and 2418kc/s a variation of ~3%.

Fig 5.7. Shows primary capacitance C_{P}= 650pF as per Fig. 4.5. Here the F_{U} point is compared at 2819.5kc/s and 2875kc/s a variation of ~2%.

It has been shown that there is good correspondence of the key impedance features of flat coil 3S-1P when measured on both VNA-SDR and VNA-HP. The variation between measured parameters is acceptable for the intended purpose of the flat coil, and all the various measurements correlate well in drawing key conclusions regarding the impedance-frequency properties in part 2.

**Summary of the VNA results and conclusions so far:**

1. The fundamental resonant frequency impedance characteristics of the primary F_{P}, have been shown to interact with that of the secondary F_{S} to produce an upper and lower resonant frequency for the flat coil, F_{U} and F_{L}.

2. F_{U }and F_{L} can be adjusted in frequency by adjusting C_{P}, which also leads to changes in |**Z _{U}**| and |

**Z**|. Adjustment of C

_{L}_{P}allows the frequency band of operation to be selected, and occurs for F

_{L }within the target operation band, the 160m amateur band.

3. The balance of |**Z _{U}**| and |

**Z**| leads to several important operating points for the experiments in displacement and transference of electric power. Most particularly when |

_{L}**Z**| = |

_{U}**Z**| and it is conjectured that the magnetic and electric fields of induction are in balance between the primary and secondary, which will lead to the best operating point for coherence between the induction fields and hence displacement events stimulated by non-linear events in the system.

_{L}4. The correspondence between measurements using different VNAs is good with variations in most key parameters being < 5%.

5. Equivalent circuit elements yield circuit values in reasonable correlation to those expected and those measured in part 1.

6. Small signal input impedance measurements Z_{11} have provided greater understanding and insight into the mechanisms governing the characteristics of the flat coil, how best to experiment using the flat coil, and how best to drive and match the coil to the various generators and loads.

Click here to continue to the flat coil impedance measurements part 3.