How best to match and ground antenna?

Question

An example of RF AM modulation:

https://www.ee-diary.com/2021/12/am-modulator-using-bjt-transistor.html

It is clear that the output is on point C4 and the circuit has a RF function, so I need an antenna. The antenna, say has 10 ohms impedance, but the output impedance of this circuit above, has 10kohms. The impedance matching takes place and I have to match 10kohms to 10ohms. Right? I would prefer a L-Match network here. After matching, the output will be at 10ohms from antenna outside transmitted. My question is about wiring a copper antenna, where the 10kohms resistor is and if it would need grounding like 10kohms resistor labeled as RL. The resonant frequency of the circuit above is about 10kHz and a Q-Factor of 3.185. When I match the impedance from 10kohms to 10 ohms, then it would become as following circuit at that branch:

As in picture above described, is the grounding of RL with the antenna (10 ohms) set on really necessary?

Additional info. Edit:

The picture below is about the resistor coming from RF modulation for Q-Factor experience. The resistor in RLC circuit for Q-Factor must be impedance matched for the power transmission to antenna. Say, the resistor is 500ohms and is grounded at the end of this circuit. This circuit needs an antenna for RF signaling which has 50ohms, so I match them by L-Match networking from 500ohms to antenna's 50ohms. The question is, the antenna which has for example the full-wave length at given frequency, must it be grounded into the circuit's ground like the 500ohms before or does it ground the RF-signals into air with the electromagnetic field that surrounds the antenna or better are both of them needed?

Answer 1

The impedance matching takes place and I have to match 10kohms to 10ohms.

No.
The 10k resistor that you add doesn't define the output impedance of the output node. The transistor collector node has very much lower impedance. The reactance of C4 is significant as well.

It is useful to examine the collector node of BC107 (Q2) near resonance. To do this, transform series RL (0.5 ohm, 25.36uH) to its parallel equivalent:

^{simulate this circuit – Schematic created using CircuitLab}

You might see that the resonant point has shifted a bit from 10kHz. An LTspice frequency response of this network shows resonance at 9.903 kHz, where impedance peaks at about 5.3 ohms.

The 1uf coupling capacitor (C4) has significant reactance at this frequency (-j16.07 ohms), so a 10 ohm antenna will see a higher impedance looking back at the transistor collector. The transmission line connecting antenna to collector will be very short as far as wavelength is concerned.

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At 10 kHz, one might try to raise collector impedance by making the coil multi-turn. As suggested in another answer, this coil might serve as the antenna itself.

Answer 2

As in picture above described, is the grounding of RL with the antenna (10 ohms) set on really necessary?

Yes, the grounding at point RL is important to close the circuit. The length of the antenna is very huge at frequency 10kHz, but I consider the fact that it is possible to get around this problem obtaining that long wiring. The impedance matching makes the copper wiring possible to swing at this frequency (10kHz) with proper power transmission. The fact that the transmission is AC voltage, will make the antenna signaling during being connected to ground.

Answer 3

At 10kHz, a resonant 1/4 wave dipole is 7.5km in length. I suspect you don't have an antenna that long.

For these low frequencies you are better off with a tuned loop antenna, and accepting a largely inductive (so \$1/r^3\$) field.

I've made such a system for a local timing signal repeater with 20 turns of wire on a 15cm loop former, with a cap sized to resonate at 60kHz. It works well for clocks upto 2m away.

In your case you'd tune the loop to 10kHz.