# Must LC resonant frequency be identical to oscillating frequency in RF circuit?

I am developing an RF circuit for 503.3 MHz. It will be an AM RF transmitter. The LC circuit I built into this circuit consists of L = 0.1 μH and C = 1 pF.

The carrier frequency which carries the message is lower than the resonant frequency. I tested the circuit in LTspice and everything is as it should be, but FFT says the carrier frequency is lower.

Does it matter that the frequencies are not identical when transmitting a message using AM modulation? If the answer is positive, how can I avoid it?

Below is a picture of the circuit without carrying a message. The antenna is between C3 and the L1-C1 tank circuit.

LTspice .asc file

Version 4
SHEET 1 880 680
WIRE 528 -96 224 -96
WIRE 224 -64 224 -96
WIRE 224 -64 112 -64
WIRE 320 -64 224 -64
WIRE 224 0 224 -64
WIRE 320 0 320 -64
WIRE 112 96 112 -64
WIRE 224 96 224 80
WIRE 304 96 224 96
WIRE 320 96 320 64
WIRE 320 96 304 96
WIRE 528 96 528 -96
WIRE 304 112 304 96
WIRE 384 112 304 112
WIRE 304 128 304 112
WIRE 384 144 384 112
WIRE 240 176 112 176
WIRE 304 256 304 224
WIRE 384 256 384 208
WIRE 384 256 304 256
WIRE 112 288 112 176
WIRE 304 288 304 256
WIRE 112 400 112 352
WIRE 224 400 112 400
WIRE 304 400 304 368
WIRE 304 400 224 400
WIRE 224 464 224 400
WIRE 528 464 528 176
WIRE 528 464 224 464
WIRE 224 496 224 464
FLAG 224 496 0
SYMBOL cap 304 0 R0
SYMATTR InstName C1
SYMATTR Value 1p
SYMBOL ind 208 -16 R0
SYMATTR InstName L1
SYMATTR Value 0.1µ
SYMBOL npn 240 128 R0
SYMATTR InstName Q1
SYMATTR Value BC547B
SYMBOL res 288 272 R0
SYMATTR InstName R1
SYMATTR Value 33
SYMBOL res 96 80 R0
SYMATTR InstName R2
SYMATTR Value 100k
SYMBOL cap 96 288 R0
SYMATTR InstName C2
SYMATTR Value 1n
SYMBOL cap 368 144 R0
SYMATTR InstName C3
SYMATTR Value 10p
SYMBOL voltage 528 80 R0
WINDOW 123 0 0 Left 0
WINDOW 39 0 0 Left 0
SYMATTR InstName V1
SYMATTR Value 12
TEXT 94 520 Left 2 !.tran 10u

• Please clarify what your definition of "resonant frequency" is. How familiar are you with oscillator behavior and function? Is your understanding of oscillators compatible with that definition? || How would you impose a message upon the oscillator? Are you familiar with AM modulation concepts? It's not clear to me what you're asking here. (Which may suggest you should ask questions clarifying those subjects, first.) Commented Mar 14 at 20:41
• @TimWilliams I mean LC resonant frequency. Just edited the title for this post. AM modulation is simple voltage dependent modulation. The frequency is high, but it works. I designed AM RF circuit for several MHz frequencies at home. Not only the range of 500kHz-1600kHz can be AM modulated. There are also other frequencies which are enabled.
– user321220
Commented Mar 14 at 20:43
• You should clarify where your inputs and outputs are. If it's a filter, it's OK if the carrier (and sidebands) are within the bandwidth of the filter. If it is (as it looks like) an oscillator, it will oscillate at or near the resonant frequency of the tank. The electrons aren't aware of your intended carrier frequency. Commented Mar 14 at 20:48
• @CristobolPolychronopolis Can you tell me why the LC circuit frequency is other one than the carrier frequency? I will get your answer and go home simply. TY
– user321220
Commented Mar 14 at 20:50
• What about that 10pF capacitor across the transie? That's effectively in parallel with your LC tank circuit. Also, BC547 doesn't look like the right transistor for 500 MHz. I'd go for something more radical: BFR380, BFG425W, BFP650 or some such. Yes it requires a proper PCB design. And it will work at 500 MHz. I don't think this would work on a breadboard.
– frr
Commented Mar 14 at 21:19

Must LC resonant frequency be identical to oscillating frequency in RF circuit?

I'm going to make some assumptions about what you mean, but the answer is no, the oscillating frequency of a sine-wave oscillator is the frequency at which the phase delay of the loop is $$\0^\circ\$$ (or the phase delay of the negative feedback is $$\180^\circ\$$). This is often very close to the natural resonant frequency of an LC tank network, but virtually never exactly equal to it. That is one part of the Barkhausen criteria for oscillation.

Does it matter that the frequencies are not identical when transmitting a message using AM modulation?

Generally, the frequency that is important is the oscillating frequency, and not the LC tank's natural resonant frequency (which is typically slightly different). The LC natural resonant frequency is only important insofar as it may approximately determine the oscillation frequency.

If the answer is positive, how can I avoid it?

Often times, practical oscillators are designed with trimmer capacitors, which allows a technician to make minor adjustments to the oscillator frequency (and consequently also to the tank's natural resonant frequency). It is also often the case that the actual values of the capacitor(s) and inductor(s) only approximate their nominal values anyways, and so trimming is necessary to get the oscillation frequency right, even if the natural resonant frequency did not slightly differ from the oscillation frequency.

We can understand the circuit as a negative [incremental] resistance, i.e., as voltage changes by some small increment from quiescent, current varies in the opposite direction.

I have labeled the node of interest VC. My model may not be exactly the same as yours (I'm using the one published by onsemi here, "BC547B Lib Model"), but to the extent I can adjust values to observe various behavior, anything will do, more or less.

Notable modifications:

• I have changed the bias network to a resistor divider. This allows more specificity on setting the bias. The values shown right now are equivalent (R3 approx. infinite), but I would recommend e.g. 10k and 4.7k. R4 is also rather small for a transistor of this rating.
• C4 is added. Typically a common-base Colpitts oscillator like this needs a well-defined capacitor divider from collector to emitter to ground.
• The resonant tank is removed, replaced with ideal bias tee L1-C1. This node is being probed with the AC source V2, of 50 ohm source impedance.

Here is the plot, showing the impedance looking into the VC node, from V2. We observe negative resistance from about 20MHz up, indicating instability, and ensuing oscillation, at least as long as the attached LC resonator has low enough losses not to swamp this negative resistance (the parallel combination must still be negative).

However, we also see reactance, which is dominant over much of the range, and negative indicating capacitance. Which we expect, given that C2 is mostly shunting the VC node, and the collector has capacitance (the model specifies about 4pF zero-bias, though it will be more like 1-2pF at this bias voltage).

In particular, at 500MHz, the resistance and reactance are -7.6 and -28.67j ohms, respectively. This is equivalent to a 1.11pF capacitor in series with -7.6Ω.

Thus, connecting an LC resonator in parallel with this node, is connecting an additional 1.11pF with it, shifting its resonant frequency lower than that given by the explicit L and C placed there.

If we change L1 to 91nH and switch to transient analysis, we find it does in fact oscillate, but that the frequency of oscillation is around 200MHz. It seems the Nyquist criteria are satisfied at a lower frequency, and this makes more sense, as C2 + Ccb dominates, and we would expect 11pF and 91nH to resonate at 160MHz; it seems neither simplified calculation is quite an adequate prediction here, but a yet more nuanced calculation would notice that Cbe is in series with C2.

As for better values, if we change R1 = 10k, R3 = 4.7k, R4 = 330Ω and C2 = 2.2pF, we get -17Ω of negative resistance at 500MHz, a clear improvement.

Mind, it's not clear that the real device will even oscillate at this frequency at all: there are real device effects not modeled by SPICE, not to mention package and layout strays, which affect operation at this frequency. A MMBTH10, MMBT5179, BF193, etc. would be vastly preferable to BC547.

Making some more adjustments, these values seem reasonable:

Note the addition of a step pulse to ensure oscillator startup.

Transient response:

Zoom:

You will have a hard time constructing an accurate 23nH inductance; more likely a shorted-stub resonator will be more effective.

Note that supply voltage modulation varies device capacitance and bias current, varying frequency in turn. A stable oscillator cannot be direct modulated in this way. It's not clear what your aims are, but it won't work for precisely defined, narrowband UHF channels, at least.

These results are from a more-or-less normal XSPICE based engine, with settings:

.OPTIONS ABSTOL=1E-9 CHGTOL=1E-9 GMIN=1E-8 ITL4=5000 PIVTOL=1E-9 RELTOL=0.0001
.OPTIONS RSHUNT=1E8 TRTOL=4 METHOD=GEAR MAXORD=2
.TRAN 1E-10 2E-7 0 1E-10


2nd order GEAR, and max timestep of 100ps, are probably the most influential settings here. Oscillation amplitude and stability are strongly affected by numerical resolution, and I wouldn't expect that these values are representative, even if the device model is accurate at this frequency. YMMV with default (or modified) LTSpice settings.

Your circuit is NOT a LC-resonator, I see a transistor oscillator. Every capacitor, capacitive effects inside the transistor, the inductor and the total behaviour of the quite tricky feedback amplifier circuit together determine does it oscillate and what's the oscillating frequency. In a practical (=built) circuit there's in addition the properties of the wiring or circuit board which also have substantial effects.

I'm afraid you should work at much lower frequencies, say at 1MHz to be able to determine the oscillation frequency with the resonant frequency of an LC resonator. High Q crystals may do the job at few tens of megahertzes.

• What would happen when the oscillating frequency is not identical to resonant frequency during transmission? Will the message be transmitted as resonant frequency of LC circuit or as oscillating frequency? TY for reply.
– user321220
Commented Mar 14 at 21:12
• The oscillation frequency IS the resonant frequency of the whole circuit. If you swing the operating voltage of this oscillator up and down around the idle DC value of V1 with audio or data signal you get amplitude modulated signal at the oscillation frequency. Some parts, for ex. L1 and C1 alone without the rest of the drawn circuit may have other resonant frequency, but it's totally meaningless for your output from the drawn circuit. Commented Mar 14 at 21:22
• @lastime (continued) I do not believe you'll succeed to get the oscillation frequency up to 503 MHz without having a well designed miniature size circuit board where the wiring is designed to make the reactive parts. Using ordinary wires you may succeed if you can build it smaller than max. diameter about half an inch or less. The transistor and other parts must be much smaller than for example the standard TO-92 case of the transistor.BC547. That type is designed for much lower frequencies than 503MHz. Commented Mar 14 at 21:41