Question about class A amplifiers

amplifierclass-acommon-emitter

I am trying to understand the working principle of a simple class A amplifier.

Below you can find a diagram of the inner components.

I am struggling to understand why the gain \$G\$ is given by the formula \$G = Rc/Re\$.

image

Image from:
https://en.m.wikipedia.org/wiki/Common_emitter

Best Answer

DC quiescent situation

The base will have a DC biasing voltage, \$V_\text{bq}\$, necessary to properly DC bias the bipolar transistor into active operation.

The bipolar's base-emitter voltage, a diode drop called \$V_{_\text{BE}}\$, is ideally a constant value and the emitter voltage will be less than the base by this amount: \$V_\text{eq}=V_\text{bq}-V_{_\text{BE}}\$.

\$V_\text{eq}\$ sets the quiescent operating current, \$I_{_\text{Q}}\$, due to its presence across the emitter resistor. Ideally, this quiescent current is copied exactly to the collector and travels via the collector resistor to the supply rail, leading to a quiescent collector resistor voltage drop that sets the quiescent collector voltage.

(In practice, the base recombination current robs some of the emitter current before it becomes a copied collector current. But I'm keeping this simple and ideal here.)

We know everything we need to know right now about the DC quiescent state:

  • \$V_\text{eq}=V_\text{bq}-V_{_\text{BE}}\$
  • \$I_{_\text{Q}}=\frac{V_\text{eq}}{R_{_\text{E}}}\$
  • \$V_\text{cq}=V_{_\text{CC}}-I_{_\text{Q}}\cdot R_{_\text{C}}\$

AC gain

A tiny signal voltage, \$v_\text{b}\$, is now added to the DC biasing base voltage: \$V_\text{b}=V_\text{bq}+v_\text{b}\$.

This is replicated at the emitter, with the base-emitter voltage ideally being taken as constant: \$V_\text{e}=V_\text{eq}+v_\text{b}\$.

The emitter current is \$I_\text{e}=\frac{V_\text{e}}{R_{_\text{E}}} = \frac{V_\text{eq}+v_\text{b}}{R_{_\text{E}}}=I_{_\text{Q}}+\frac{v_\text{b}}{R_{_\text{E}}}=I_{_\text{Q}}+i_\text{e}\$. Note that this is just the quiescent operating current plus a small current change, \$i_\text{e}=I_\text{e}-I_{_\text{Q}}=\frac{V_\text{e}}{R_{_\text{E}}}-\frac{V_\text{eq}}{R_{_\text{E}}}=\frac{v_\text{b}}{R_{_\text{E}}}\$, due to the signal. Also note: \$V_\text{c}=V_{_\text{CC}}-I_\text{e}\cdot R_{_\text{C}}\$.

\$i_\text{e}\$ is then copied to the collector and yields a change in voltage drop across the collector resistor:

$$\begin{align*}v_\text{c}&=V_\text{c}-V_\text{cq}\\\\&=V_{_\text{CC}}-I_\text{e}\cdot R_{_\text{C}}-\left(V_{_\text{CC}}-I_{_\text{Q}}\cdot R_{_\text{C}}\right)\\\\&=\left(I_{_\text{Q}}-I_\text{e}\right)\cdot R_{_\text{C}}\\\\&=- i_\text{e}\cdot R_{_\text{C}}\\\\&=-\frac{v_\text{b}}{R_{_\text{E}}}\cdot R_{_\text{C}}\\\\&=-v_\text{b}\cdot\frac{R_{_\text{C}}}{R_{_\text{E}}}\\\\ A_v=\frac{v_\text{c}}{v_\text{b}}&=-\frac{R_{_\text{C}}}{R_{_\text{E}}}\end{align*}$$

And there you have it for the ideal case.