Four Fundamental Single BJT Configurations#
1. Common-Emitter Amplifier#
Basic characteristics#
Incomplete - Need to clean up and add diagrams / re-add all derivation steps
DC bias: $V_{DD} \to R_C \to$ collector; base driven through source resistance $R_S$; emitter grounded.
Small-signal (hybrid-π) model, with $V_\pi$ the base-emitter voltage and $g_m V_\pi$ the dependent collector current source:
$$V_o = -g_m V_\pi R_C$$If $R_S$ is small enough that $V_i \approx V_\pi$:
$$A_v = \frac{V_o}{V_i} = -g_m R_C$$Gain variation with temperature: Since $g_m = I_C/V_T$ and $V_T = kT/q$, both $g_m$ and $I_C$ vary with temperature — so the gain $A_v = -g_m R_C$ drifts with temperature.
Gain variation with input amplitude: Because $I_C$ depends exponentially on $V_{BE}$ (Shockley diode equation), the transconductance itself is signal-dependent for large input swings — this nonlinearity distorts the amplified signal.
Emitter degeneration (adding $R_E$)#
Both issues can be linearized by adding a series emitter resistor $R_E$ (undecoupled — no bypass capacitor).
KCL at the output (collector) node:
$$V_o = -g_m V_\pi R_C$$KCL at the emitter node (emitter current is base + collector current):
$$I_E = g_m V_\pi + \frac{V_\pi}{r_\pi} = V_\pi\left(g_m + \frac{1}{r_\pi}\right)$$KVL around the input loop ($I_B = V_\pi/r_\pi$, $I_E = (\beta+1)I_B$):
$$V_i = I_B R_S + V_\pi + I_E R_E = \frac{V_\pi}{r_\pi}R_S + V_\pi + (\beta+1)\frac{V_\pi}{r_\pi}R_E$$$$V_i = \frac{V_\pi}{r_\pi}\Big[R_S + r_\pi + (\beta+1)R_E\Big]$$Solving for $V_\pi$ and substituting into $V_o = -g_mV_\pi R_C$:
$$\boxed{A_v = \frac{V_o}{V_i} = \frac{-g_m r_\pi R_C}{R_S+r_\pi+(\beta+1)R_E} = \frac{-\beta R_C}{R_S+r_\pi+(\beta+1)R_E}}$$Neglecting $R_S$ (or dividing through by $r_\pi$, using $g_m r_\pi = \beta$):
$$A_v \approx \frac{-g_m R_C}{1+g_m R_E}$$Design note: Resist adding extra variables where possible — keep expressions in their lowest-entropy form.
If $R_E \to 0$: $A_v \to -g_m R_C$, as expected (recovers the basic CE result).
If $g_m R_E \gg 1$ (i.e., $R_E$ large enough):
$$\boxed{A_v \approx -\frac{R_C}{R_E}}$$This is independent of $g_m$ (and hence temperature and signal amplitude) — the nonlinearity is minimized! To desensitize the gain from $g_m$ variation, make $R_E$ large enough that $g_m R_E \gg 1$.
Naturally, with this configuration the maximum achievable gain is limited by the additional DC voltage drop across $R_E$, which eats into the available headroom for $R_C$.
Input and output impedance (with emitter degeneration)#
Finding $Z_{in}$ (zero independent sources, apply test current $i_x$ at the base):
KCL at emitter: $I_E = (\beta+1)i_x$
KVL: $V_x = i_x r_\pi + (\beta+1)i_x R_E = i_x\big[r_\pi+(\beta+1)R_E\big]$
$$\boxed{Z_{in} = \frac{V_x}{i_x} = r_\pi + (\beta+1)R_E}$$$R_E$ significantly boosts $Z_{in}$.
Finding $Z_{out}$ (looking into the collector, $V_i = 0$): with no early-effect/output resistance ($r_o$) modeled, the dependent current source $g_mV_\pi$ has infinite output resistance on its own — so the impedance looking into the collector node (excluding $R_C$ itself) is effectively infinite:
$$Z_{out}\Big|_{\text{transistor}} \to \infty \quad\Rightarrow\quad Z_{out}\big|_{\text{node}} = R_C$$2. Common-Base Amplifier#
DC characteristics and voltage gain#
Base is AC-grounded; input signal is applied at the emitter; output is taken at the collector.
Small-signal model: since the base is grounded, $V_\pi = V_B - V_E = -V_i$, and:
$$V_o = -g_m V_\pi R_C$$$$\boxed{A_v = \frac{V_o}{V_i} = \frac{-g_m R_C(-V_i)}{V_i} = g_m R_C}$$Note the gain is positive (no phase inversion) — a hallmark of the common-base stage.
Input impedance#
Apply test source $V_x$ at the emitter (base grounded, so $V_\pi = -V_x$). The current pulled from the test source is the full emitter current:
$$i_x = g_m V_\pi + \frac{V_\pi}{r_\pi} = V_x\left(g_m+\frac{1}{r_\pi}\right)$$$$Z_{in} = \frac{V_x}{i_x} = \frac{1}{g_m+\dfrac{1}{r_\pi}} = \frac{r_\pi}{g_m r_\pi + 1} = \frac{r_\pi}{\beta+1}$$$$\boxed{Z_{in} = \frac{r_\pi}{\beta+1} \approx \frac{1}{g_m}}$$This is characteristically small — a signature of the common-base topology.
Output impedance#
With no $r_o$ modeled, the ideal dependent current source gives:
$$Z_{out}\big|_{\text{transistor}} \to \infty \quad\Rightarrow\quad Z_{out}\big|_{\text{node}} = R_C$$Current gain#
$$A_i = \frac{I_C}{I_E} = \frac{\beta}{\beta+1} = \alpha \approx 1$$(slightly less than unity — current gain is essentially unity for CB, unlike voltage gain).
Effect of base bias network#
If the base is biased through resistors $R_1 | R_2$ rather than being ideally AC-grounded, that impedance appears in the emitter path as well — but reduced by a factor of $(\beta+1)$:
$$\boxed{Z_{in} = \frac{r_\pi + (R_1\|R_2)}{\beta+1}}$$General rule: any impedance placed in series with the base always appears $(\beta+1)$ times smaller when reflected into the emitter.
For proper small-signal CB operation, place a bypass capacitor from the base to ground so that the base is a true AC ground at the signal frequency (i.e., $X_C = \dfrac{1}{2\pi f C}$ is small compared to the bias network impedance at the frequencies of interest).
3. Emitter Follower (Common-Collector Buffer)#
Input applied to the base; output taken at the emitter, across $R_E$; collector tied to $V_{DD}$ (AC ground).
Voltage gain#
KCL at the emitter node:
$$\frac{V_o}{R_E} = g_m V_\pi + \frac{V_\pi}{r_\pi}$$where $V_\pi = V_i - V_o$ (base-emitter voltage is the difference between input and output).
Substituting and solving:
$$\frac{V_o}{R_E} = (V_i-V_o)\left(g_m+\frac{1}{r_\pi}\right)$$$$V_o\left[\frac{1}{R_E}+g_m+\frac{1}{r_\pi}\right] = V_i\left(g_m+\frac{1}{r_\pi}\right)$$$$\boxed{A_v = \frac{V_o}{V_i} = \frac{(\beta+1)R_E}{r_\pi+(\beta+1)R_E} = \frac{g_m R_E}{1+g_m R_E}}$$As expected, the gain is a voltage divider between $R_E$ and the intrinsic emitter resistance $r_e = 1/g_m$ — for $g_m R_E \gg 1$, $A_v \to 1$ (a unity-gain buffer, hence “emitter follower”).
Input impedance#
Applying a test current $i_x$ at the base (same topology as the degenerated CE stage):
$$V_x = i_x r_\pi + (\beta+1)i_x R_E$$$$\boxed{Z_{in} = r_\pi + (\beta+1)R_E}$$(Identical form to the CE-with-degeneration input impedance — same physical topology, just probed differently.)
Output impedance#
Looking into the emitter (test source $V_x$ applied at the emitter node, $V_i$ set to zero so the base sees only $r_\pi$ to AC ground):
$$i_x = g_m V_\pi + \frac{V_\pi}{r_\pi}, \qquad V_\pi = -V_x$$$$\frac{V_x}{i_x}\bigg|_{\text{transistor}} = \frac{1}{g_m+\dfrac{1}{r_\pi}} = \frac{r_\pi}{\beta+1} \approx \frac{1}{g_m} = r_e$$$R_E$ sits directly in parallel with this looking into the emitter node, so:
$$\boxed{Z_{out} = R_E \,\|\, \frac{r_\pi}{\beta+1} \approx R_E \,\|\, \frac{1}{g_m}}$$This is the classic emitter-follower result: low output impedance, dominated by $r_e = 1/g_m$ when $R_E \gg r_e$ — which is what makes the emitter follower useful as a buffer stage.