BJT CE Amplifier — Two Topologies, Two Different Circuits

Circuit Parameters (shared)

DC Operating Point (identical for both)

$$I_B = \frac{V_{BB} - V_{BE}}{R_B} = \frac{3 - 0.7}{100k} = 23,\mu A$$

$$I_C = \beta \cdot I_B = 2.3,\text{mA}, \quad V_{CE} = V_{CC} - I_C R_C = 10.8,\text{V} \quad \text{(active)}$$

$$g_m = \frac{I_C}{V_T} = 88.5,\text{mS}, \quad r_\pi = \frac{\beta}{g_m} = 1130,\Omega$$

Topology A — Original Problem (R_B in Series)

Schematic

        ┌─────[R_B 100kΩ]─────┬──── base ─── BJT ─── emitter → GND
        │                      │               │
  v_i ──┤                      │            collector
        │                      │               │
  V_BB ─┘                      │          ┌───[R_C]───┐
   3V                           │          │           V_CC
  GND ──────────────────────────┘──────── GND

Signal definition: v_i is defined at the V_BB node itself. No coupling capacitor.

AC Equivalent

When V_BB = 0 (AC ground at supply), the signal v_i drives through R_B to the base:

  v_i ──[R_s]──[R_B]──── base ──[r_π]── GND

R_s and R_B are in series. The base sees the full series resistance.

Midband Analysis

$$\frac{v_{be}}{v_i} = \frac{r_\pi}{R_s + R_B + r_\pi}$$

At R_s = 0:

$$\frac{v_{be}}{v_i} = \frac{1130}{100{,}000 + 1130} = 0.01117$$

$$A_d = \frac{v_o}{v_{be}} = -g_m R_C = -354 \quad (51.0,\text{dB})$$

$$A_v = \frac{v_o}{v_i} = A_d \cdot \frac{r_\pi}{R_B + r_\pi} = -354 \times 0.01117 = -4.0 \quad (11.9,\text{dB})$$

R_B attenuation: 39.0 dB — the 100 kΩ bias resistor wastes almost all the signal.

High-Frequency Analysis (OCTC)

Source resistance at base = R_s + R_B (series). Effective resistance for OCTC:

$$R_{eff} = r_\pi | (R_s + R_B)$$

At R_s = 0:

$$R_{eff} = r_\pi | R_B = 1130 | 100k = 1117,\Omega$$

Miller capacitance is fully activated because R_eff ≈ 1117 Ω ≫ 0:

$$C_{Miller} = C_\mu (1 + g_m R_C) = 2,\text{pF} \times 355 = 710,\text{pF}$$

OCTC time constants:

$$\tau_{C_\pi} = R_{eff} \cdot C_\pi = 1117 \times 50 \times 10^{-12} = 55.9,\text{ns}$$

$$\tau_{C_\mu} = [R_{eff}(1 + g_m R_C) + R_C] \cdot C_\mu = [1117 \times 355 + 4000] \times 2 \times 10^{-12} = 801,\text{ns}$$

$$f_H(A_v) = \frac{1}{2\pi(\tau_{C_\pi} + \tau_{C_\mu})} = \frac{1}{2\pi \times 857 \times 10^{-9}} = 185,\text{kHz}$$

For A_d (v_o/v_be), only the collector-node pole matters:

$$f_H(A_d) = \frac{1}{2\pi R_C C_\mu} = \frac{1}{2\pi \times 4000 \times 2 \times 10^{-12}} = 19.9,\text{MHz}$$

Effect of R_s

Key behavior: R_s barely matters because R_B = 100 kΩ already dominates. Adding R_s actually increases f_H because R_eff decreases (parallel with larger denominator).

Topology B — Realistic Circuit (R_B as Shunt)

Schematic

                                   V_BB (AC ground)
                                    │
                                   [R_B 100kΩ]  ← shunt
                                    │
  v_i ──[R_s]──[C_in]──────── base node ──[r_π]── GND (emitter)
                                    │
                                   BJT
                                    │
                               collector ──[R_C]── V_CC

Signal definition: v_i is a separate AC source, coupled through C_in to the base. V_BB biases through R_B independently.

AC Equivalent

With C_in short and V_BB = 0, both R_B and r_π connect from base to ground:

  v_i ──[R_s]──── base ─┬─ [R_B] ─ GND   (shunt!)
                         └─ [r_π] ─ GND   (shunt!)

R_B and R_s are in parallel as seen from the base looking back toward the source.

Midband Analysis

$$\frac{v_{be}}{v_i} = \frac{R_B | r_\pi}{R_s + R_B | r_\pi}$$

$$R_B | r_\pi = \frac{100{,}000 \times 1130}{100{,}000 + 1130} = 1117,\Omega$$

At R_s = 0:

$$\frac{v_{be}}{v_i} = \frac{1117}{0 + 1117} = 1.000$$

$$A_v = A_d = -354 \quad (51.0,\text{dB})$$

No attenuation at all! The source connects directly to the base; R_B shunts current to ground but cannot drop any voltage because R_s = 0.

High-Frequency Analysis (OCTC)

Source resistance at base = R_s ∥ R_B (both paths to AC ground):

$$R_s | R_B = \frac{R_s \cdot R_B}{R_s + R_B}$$

Effective resistance:

$$R_{eff} = r_\pi | (R_s | R_B)$$

At R_s = 0:

$$R_s | R_B = 0 \quad \Rightarrow \quad R_{eff} = 0$$

No Miller effect! When the source has zero impedance, the base voltage is rigidly held by the source. C_π and C_μ draw current from the source but cannot change v_be. The only HF pole is the collector-node RC:

$$f_H(A_v) = f_H(A_d) = \frac{1}{2\pi R_C C_\mu} = 19.9,\text{MHz}$$

Effect of R_s

Key behavior: Extremely sensitive to R_s. Even 50 Ω crashes f_H from 19.9 MHz to 3.6 MHz (5.5× reduction). This is because R_s ∥ R_B ≈ R_s when R_s ≪ R_B, so the full source impedance drives Miller multiplication.

Side-by-Side Comparison

Why the Difference Matters

The fundamental question: "What does Miller see?"

In Topology A, the source impedance at the base is R_s + R_B. Since R_B = 100 kΩ is already huge, Miller is fully activated regardless of R_s. The bandwidth is always ~185 kHz, determined almost entirely by R_B.

In Topology B, the source impedance is R_s ∥ R_B. When R_s = 0, this is zero — the ideal source clamps the base voltage perfectly, and no amount of feedback current through C_μ can change v_be. Miller has nothing to grab onto. But even a tiny R_s provides a finite impedance that Miller amplifies by (1 + g_m R_C) ≈ 355×.

Design implications

Topology B is what you actually build, and it reveals why CE amplifier bandwidth is so sensitive to source impedance. A signal generator with 50 Ω output impedance would limit this amplifier to ~3.6 MHz — far below the intrinsic 19.9 MHz collector pole. This is why techniques like cascode (removing the Miller effect) or emitter degeneration (reducing effective g_m) are essential in wideband design.

The Error in the Original Document

The document's JSX code drew Topology B (R_B as shunt from base to V_BB ground, with C_in coupling) but computed the math for Topology A (R_s + R_B in series, A_v = −4.0). This mismatch was exposed by the question "isn't Miller seeing R_B and R_s in parallel?" — which is correct for the drawn circuit but wrong for the intended analysis.

The corrected interactive visualization now uses Topology B's math consistently:

• R_B ∥ r_π for midband voltage divider
• R_s ∥ R_B for Miller source impedance
• A_v = A_d = −354 when R_s = 0
• f_H(A_v) = f_H(A_d) = 19.9 MHz when R_s = 0