Worked example: voltage-divider bias — Q-point and β-stability

A voltage-divider bias has $V_{CC} = 12\,\text{V}$, $R_1 = 47\,\text{k}\Omega$, $R_2 = 10\,\text{k}\Omega$, $R_C = 1\,\text{k}\Omega$, $R_E = 220\,\Omega$, $\beta = 150$, $V_{BE} = 0.7\,\text{V}$. Find the Q-point ($V_B$, $I_E$, $I_C$, $V_{CE}$). Then drop $\beta$ to 75 and quantify how much the Q-point shifts.

1. Thevenize the input divider.
   $$V_{TH} = V_{CC} \cdot \frac{R_2}{R_1 + R_2} = 12 \cdot \frac{10}{57} \approx 2.105\,\text{V}$$
   $$R_{TH} = R_1 \| R_2 = \frac{47 \cdot 10}{57} \approx 8.246\,\text{k}\Omega$$
2. Solve the base-emitter loop. KVL through the Thevenin source, the base–emitter junction, and $R_E$:
   $$I_E = \frac{V_{TH} - V_{BE}}{R_E + R_{TH} / (\beta + 1)}$$
   $$I_E = \frac{2.105 - 0.7}{220 + 8246/151} = \frac{1.405}{220 + 54.6} \approx 5.12\,\text{mA}$$
3. $I_C$ and $V_B$.
   $$I_C = \frac{\beta}{\beta + 1} \cdot I_E \approx 5.08\,\text{mA}$$
   $$V_B = V_{TH} - I_B \cdot R_{TH} \approx 2.105 - 0.034 \cdot 8.246 \approx 1.82\,\text{V}$$
   (Stiff divider — small base-loading correction.)
4. Output loop for $V_{CE}$.
   $$V_{CE} = V_{CC} - I_C \cdot R_C - I_E \cdot R_E$$
   $$V_{CE} \approx 12 - 5.08 - 1.13 \approx 5.79\,\text{V}$$
   Mid-load-line: the load line runs from $V_{CC} = 12\,\text{V}$ to $V_{CC}/(R_C+R_E) \approx 9.84\,\text{mA}$, so $V_{CEQ} \approx V_{CC}/2$ is exactly where you want it.
5. Now sweep $\beta$ from 150 to 75.
   $$I_E(\beta=75) = \frac{1.405}{220 + 8246/76} = \frac{1.405}{328.5} \approx 4.28\,\text{mA}$$
   $$I_C(\beta=75) \approx 4.22\,\text{mA}$$
   $$V_{CE}(\beta=75) \approx 12 - 4.22 - 0.94 \approx 6.84\,\text{V}$$
6. Quantify the drift. Halving $\beta$ moved the Q-point from $I_C \approx 5.08\,\text{mA}$ to $I_C \approx 4.22\,\text{mA}$ — about a 17 % change.
   Compare to fixed bias. If we picked an $R_B$ giving the same nominal $I_C \approx 5\,\text{mA}$ at $\beta = 150$, halving $\beta$ halves $I_C$ — a 50 % change. Voltage-divider bias is roughly 3× more β-stable here, all thanks to $R_E$.