BJT CE Amplifier Frequency Response Analysis

A common-emitter (CE) BJT amplifier circuit is presented using a 2N3904 transistor and associated resistors, capacitors, and biasing values.

The analysis focuses on both low-frequency and high-frequency behaviors.

Low-Frequency Analysis

Key steps:

1. Draw the π-model (ignoring r0r_0r0​) for low-frequency.

2. Considering coupling capacitor C1C_1C1​:

   • Derive voltage gain and corner frequency:

     Av=(RB∣∣rπRB∣∣rπ+Rsig)(−gmRL′)⋅ss+ωp1A_v = \left( \frac{R_B || r_\pi}{R_B || r_\pi + R_{sig}} \right) (-g_m R_L') \cdot \frac{s}{s + \omega_{p1}}Av​=(RB​∣∣rπ​+Rsig​RB​∣∣rπ​​)(−gm​RL′​)⋅s+ωp1​s​

     where RL′=RC∣∣RLR_L' = R_C || R_LRL′​=RC​∣∣RL​

     ωp1=1C1(RB∣∣rπ+Rsig)\omega_{p1} = \frac{1}{C_1 (R_B || r_\pi + R_{sig})}ωp1​=C1​(RB​∣∣rπ​+Rsig​)1​

3. Considering coupling capacitor C2C_2C2​:

   • Corner frequency:

     ωp3=1C2(RC+RL)\omega_{p3} = \frac{1}{C_2 (R_C + R_L)}ωp3​=C2​(RC​+RL​)1​

4. Considering bypass capacitor CEC_ECE​:

   • Using T-model:

     ωp2=1CE(RE∣∣(re+RB∣∣Rsig1+β))\omega_{p2} = \frac{1}{C_E \left( R_E || \left(r_e + \frac{R_B || R_{sig}}{1+\beta} \right) \right)}ωp2​=CE​(RE​∣∣(re​+1+βRB​∣∣Rsig​​))1​

5. Overall low-frequency gain:

   Av=(RBRB+Rsig)(−gmRL′)⋅ss+ωp1ss+ωp2ss+ωp3A_v = \left( \frac{R_B}{R_B + R_{sig}} \right) (-g_m R_L') \cdot \frac{s}{s+\omega_{p1}} \frac{s}{s+\omega_{p2}} \frac{s}{s+\omega_{p3}}Av​=(RB​+Rsig​RB​​)(−gm​RL′​)⋅s+ωp1​s​s+ωp2​s​s+ωp3​s​

   Approximate lower cutoff frequency:

   ωL≈ωp12+ωp22+ωp32\omega_L \approx \sqrt{\omega_{p1}^2 + \omega_{p2}^2 + \omega_{p3}^2}

   High-Frequency Analysis

   1. Draw the π-model including r0r_0r0​ and internal capacitors CπC_\piCπ​, CμC_\muCμ​ and series base resistance rxr_xrx​.

   2. Apply Miller’s theorem:

      • Voltage gain:

        Av=(RBRB+Rsig)(rπrπ+rx+(RB∣∣Rsig))(−gmRL′)⋅11+s/ωHA_v = \left( \frac{R_B}{R_B + R_{sig}} \right) \left( \frac{r_\pi}{r_\pi + r_x + (R_B || R_{sig})} \right) (-g_m R_L') \cdot \frac{1}{1 + s/\omega_H}

        Where:

        ωH=1CinRsig′\omega_H = \frac{1}{C_{in} R_{sig}'}

        with:

        Rsig′=rπ∣∣(rx+RB∣∣Rsig)R_{sig}' = r_\pi || (r_x + R_B || R_{sig})

        and:

        Cin=Cπ+Cμ(1+gmRL′)C_{in} = C_\pi + C_\mu (1 + g_m R_L')
   ​