BJT Frequency Response

Every BJT amplifier has a limited frequency range where it provides useful gain. BJT frequency response determines the lower and upper -3 dB points of this range, and understanding it is essential for designing audio amplifiers, RF front ends, and wideband video circuits.

Core Concept

At low frequencies, coupling capacitors (CC1, CC2) and bypass capacitors (CE) have high reactance. They block the signal or fail to short the emitter resistor, reducing gain. The lower -3 dB frequency fL is determined by the RC time constants formed by these capacitors with their surrounding resistances.

At high frequencies, internal BJT junction capacitances take over. The base-emitter capacitance Cπ and the base-collector capacitance Cµ (also called CBC) limit bandwidth. The Miller effect multiplies Cµ by (1 + gm*RC), making it appear much larger at the input. For a BC547 with gm=38 mA/V and RC=3.3 kΩ, Miller capacitance = Cµ*(1 + 125).

The transition frequency fT is the frequency at which the common-emitter current gain drops to 1. For a 2N3904, fT = 300 MHz. The beta-cutoff frequency fβ = fT/beta. For beta=150 and fT=300 MHz, fβ = 2 MHz. This means the small signal beta begins falling at 2 MHz, reaching unity at 300 MHz.

Key Equations

Lower -3 dB frequency: fL = 1 / (2*pi*R*C) computed separately for each capacitor, then combined as square root of sum of squares of individual fL values.

Upper -3 dB frequency: fH = 1 / (2*pi*Req*Ceq) where Ceq = Cπ + Cµ*(1 + gm*RC) (Miller approximation) and Req is the Thevenin resistance at the base node.

Transition frequency: fT = gm / (2*pi*(Cπ + Cµ)). Datasheets list fT directly; use it to find Cπ + Cµ = gm/(2*pi*fT).

Miller capacitance: CM = Cµ * (1 + gm*RC). This appears in parallel with Cπ at the input, severely limiting high-frequency gain.

Given:
  2N3904 CE amplifier
  IC = 1 mA, RC = 3.3 kΩ, Rs = 1 kΩ (source resistance)
  beta = 150, fT = 300 MHz, Cµ = 4 pF
  VT = 26 mV

Why this formula:
  Find fH using Miller approximation for high-frequency analysis.

Step 1 - gm:
  gm = IC / VT = 1 mA / 26 mV = 38.46 mA/V

Step 2 - rπ:
  rπ = beta / gm = 150 / 0.03846 = 3900 Ω = 3.9 kΩ

Step 3 - Cπ from fT:
  Cπ + Cµ = gm / (2*pi*fT)
          = 0.03846 / (2 * 3.1416 * 300×10⁶)
          = 0.03846 / (1884.96×10⁶)
          = 20.4 pF
  Cπ = 20.4 - 4 = 16.4 pF

Step 4 - Miller capacitance:
  CM = Cµ * (1 + gm*RC)
     = 4 pF * (1 + 38.46×10⁻³ * 3300)
     = 4 pF * (1 + 126.9)
     = 4 * 127.9 = 511.6 pF

Step 5 - Total input capacitance:
  Cin = Cπ + CM = 16.4 + 511.6 = 528 pF

Step 6 - Req at base node:
  Req = Rs || rπ = 1000 || 3900 = (1000*3900)/(4900) = 795 Ω

Step 7 - fH:
  fH = 1 / (2*pi*Req*Cin)
     = 1 / (2 * 3.1416 * 795 * 528×10⁻¹²)
     = 1 / (2.636×10⁻⁶)

Calculation:
  fH = 379 kHz

Final Answer:
  fH ≈ 379 kHz
  Bandwidth is limited primarily by the Miller capacitance (511 pF >> 16 pF).

Exam Tip: The Miller effect is the most tested concept in BJT frequency response. Remember: CM = Cµ*(1 + gm*RC), and this adds to Cπ at the input. A cascode amplifier eliminates the Miller effect because the CB stage above the CE stage presents a low-impedance load (≈ 1/gm) to the first transistor's collector, making the Miller multiplication factor nearly 1 instead of 1 + gm*RC.

Key Properties

• fL is set by coupling and bypass capacitors: larger capacitors give lower fL, extending the low-frequency response.
• fH is set by Cπ and Cµ. Miller effect makes Cµ the dominant limit in high-gain CE amplifiers.
• fT (transition frequency) for 2N3904 is 300 MHz; for BC547 it is 300 MHz; for RF transistors like BFR90 it exceeds 5 GHz.
• Gain-bandwidth product: in the Miller approximation, AV * BW ≈ fT/2π*(Cπ+Cµ)*Req is approximately constant.
• The emitter bypass capacitor CE must satisfy 1/(2π*fL*CE) << RE. For fL = 20 Hz and RE = 1 kΩ, CE > 8 µF, so a 10 µF electrolytic is used.
• Cascode configuration (CE + CB) reduces Miller effect and extends fH to near fT of the device.

Quick Revision

• fL set by coupling/bypass capacitors: fL = 1/(2π*RC).
• fH set by Cπ and Miller-multiplied Cµ.
• Miller capacitance: CM = Cµ*(1 + gm*RC).
• fT = gm / (2π*(Cπ+Cµ)): listed in datasheet.
• Beta falls at fβ = fT/β: above fβ, current gain decreases at -20 dB/decade.
• Cascode eliminates Miller multiplication; used for wideband amplifiers.
• Exam trap: Students compute fH = 1/(2π*rπ*Cπ) and forget the Miller component Cµ*(1+gm*RC). This gives fH many times too high, missing the dominant pole completely.