How Frequency Modulation is achieved with BJT Transistor

 Shown below is circuit diagram of a simple one BJT transistor based FM transmitter circuit. Frequency modulation (FM) in the transistor FM modulator circuit occurs due to the variation in the capacitance of the transistor (specifically, the base-collector capacitance, $C_{bc}$) caused by the audio signal applied to the base. This variation in capacitance changes the resonant frequency of the LC tank circuit, resulting in frequency modulation.

For design of the air core inductor use the online air core inductor calculator. Here the inductor is 5 turn, 5mm in length and 4mm diameter air core inductor having inductance of 80nH.

Let’s break this down step by step with equations and an example calculation.

1. How Frequency Modulation Happens

Key Components:

• LC Tank Circuit: The inductor ($L$) and capacitor ($C$) form a resonant circuit that determines the FM oscillator frequency.

• Transistor Capacitance: The base-collector capacitance ($C_{bc}$) of the transistor acts as a variable capacitor, influenced by the audio signal. Amplitude variation causes capacitance variation and thereby the oscillation frequency.
  

• Audio Signal: The audio signal from the microphone modulates the base voltage, causing $C_{bc}$ to vary.

Mechanism:

1. The LC tank circuit(use online LC calculator) generates a carrier wave at its resonant frequency:

   $$f_c=\frac{1}{2\pi \sqrt{LC}}$$

2. The audio signal applied to the base of the transistor causes the base-collector capacitance ($C_{bc}$) to vary.

3. This variation in $C_{bc}$ changes the total capacitance ($C_{\text{total}}$) of the LC tank circuit:

   $$C_{\text{total}}=C+C_{bc}$$

4. As $C_{\text{total}}$ changes, the resonant frequency of the LC tank circuit also changes:

   $$f(t)=\frac{1}{2\pi \sqrt{L{C}_{\text{total}}(t)}}$$

5. The result is a frequency-modulated signal, where the carrier frequency ($f_c$) varies in proportion to the audio signal.

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2. Mathematical Representation of FM

The frequency-modulated signal can be expressed as:

$$f(t)=f_c+\mathrm{\Delta }f\cdot \cos (2\pi f_{m}t)$$

Where:

• $f(t)$ = instantaneous frequency of the FM signal.

• $f_c$ = carrier frequency (center frequency of the LC tank circuit).

• $\mathrm{\Delta }f$ = frequency deviation (maximum change in frequency caused by the audio signal).

• $f_m$ = frequency of the modulating audio signal.

The frequency deviation ($\mathrm{\Delta }f$) depends on the amplitude of the audio signal and the sensitivity of the transistor's capacitance to voltage changes.

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3. Example Calculation

Let’s assume the following values for your circuit:

• Inductance, $L=80\text{nH}=80\times 10^{-9}\text{H}$

• Capacitance, $C=30\text{pF}=30\times 10^{-12}\text{F}$

• Base-collector capacitance, $C_{bc}=5\text{pF}=5\times 10^{-12}\text{F}$(typical for a 2N3904 transistor)

• Audio signal frequency, $f_m=1\text{kHz}$

• Audio signal amplitude causes $C_{bc}$ to vary by $\pm 2\text{pF}$.

Step 1: Calculate the Carrier Frequency ($f_c$)

The carrier frequency is determined by the LC tank circuit:

$$f_c=\frac{1}{2\pi \sqrt{LC}}$$$$f_c=\frac{1}{2\pi \sqrt{80\times 10^{-9}\times 30\times 10^{-12}}}$$$$f_c=\frac{1}{2\pi \sqrt{2.4\times 10^{-18}}}$$$$f_c=\frac{1}{2\pi \times 1.55\times 10^{-9}}$$$$f_c\approx 102.7\text{MHz}$$

Step 2: Calculate the Total Capacitance ($C_{\text{total}}$)

The total capacitance of the LC tank circuit includes the trimmer capacitor ($C$) and the base-collector capacitance ($C_{bc}$):

$$C_{\text{total}}=C+C_{bc}$$

At rest (no audio signal):

$$C_{\text{total}}=30\text{pF}+5\text{pF}=35\text{pF}$$

When the audio signal modulates $C_{bc}$ by $\pm 2\text{pF}$:

• Maximum capacitance:

  $$C_{\text{total, max}}=30\text{pF}+7\text{pF}=37\text{pF}$$

• Minimum capacitance:

  $$C_{\text{total, min}}=30\text{pF}+3\text{pF}=33\text{pF}$$

Step 3: Calculate the Frequency Deviation ($\mathrm{\Delta }f$)

The frequency deviation is the difference between the maximum and minimum frequencies caused by the variation in $C_{\text{total}}$:

$$f_{\text{max}}=\frac{1}{2\pi \sqrt{L{C}_{\text{total, min}}}}$$​$$f_{\text{max}}=\frac{1}{2\pi \sqrt{80\times 10^{-9}\times 33\times 10^{-12}}}$$$$f_{\text{max}}\approx 107.5\text{MHz}$$$$f_{\text{min}}=\frac{1}{2\pi \sqrt{L{C}_{\text{total, max}}}}$$​$$f_{\text{min}}=\frac{1}{2\pi \sqrt{80\times 10^{-9}\times 37\times 10^{-12}}}$$$$f_{\text{min}}\approx 98.2\text{MHz}$$

The frequency deviation is:

$$\mathrm{\Delta }f=\frac{f_{\text{max}}-f_{\text{min}}}{2}$$$$\mathrm{\Delta }f=\frac{107.5-98.2}{2}\approx 4.65\text{MHz}$$

Step 4: FM Signal Representation

The instantaneous frequency of the FM signal is:

$$f(t)=f_c+\mathrm{\Delta }f\cdot \cos (2\pi f_{m}t)$$

Substituting the values:

$$f(t)=102.7\text{MHz}+4.65\text{MHz}\cdot \cos (2\pi \times 1\text{kHz}\times t)$$

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FM transmitter circuit:

• The LC tank circuit generates the carrier wave at $102.7\text{MHz in the <a href="https://www.ee-diary.com/2023/06/guide-to-building-fm-transmitter.html">FM transmitter</a>}$.

• The audio signal modulates the base-collector capacitance ($C_{bc}$) of the transistor, causing the total capacitance ($C_{\text{total}}$) to vary.

• This variation in capacitance changes the resonant frequency of the LC tank circuit, resulting in frequency modulation.

• The frequency deviation ($\mathrm{\Delta }$) in this example is approximately $4.65\text{MHz}$, which is typical for FM broadcasting.