Resistance reflection rule

The resistance reflection rule is the piece of intuition that makes BJT amplifier analysis fast: a resistance moved across the base–emitter “boundary” of a BJT changes by a factor of $(\beta +1)$.

• A resistance $R$ in the emitter is seen from the base as $(\beta +1)R$ — bigger.
• A resistance $R$ in the base is seen from the emitter as $R/(\beta +1)$ — smaller.

A resistance in the emitter looks $\beta +1$ times larger when viewed from the base.

Why the factor is (β+1)

It comes entirely from the current ratio between the two terminals. By KCL and the definition of β, $i_E=(\beta +1)i_B$ — the emitter carries $(\beta +1)$ times the base current.

Put a resistor $R$ in the emitter lead. The voltage across it is set by the emitter current: $v=i_{E}R$. But viewed from the base, that same voltage is produced by the much smaller base current $i_B=i_E/(\beta +1)$. The apparent resistance is voltage divided by the base current:

$R_{\text{seen from base}}=\frac{v}{i_B}=\frac{i_{E}R}{i_E/(\beta +1)}=(\beta +1)R$

Same voltage drop, but measured against a $(\beta +1)\times$ smaller current, so it looks $(\beta +1)\times$ more resistive. Going the other way (a base resistor seen from the emitter) the larger emitter current measures the same voltage, so the resistance looks $(\beta +1)$ times smaller. The cleanest special case is the device’s own [[Emitter resistance|$r_e$]]: $r_{\pi }=(\beta +1)r_e$ — the BJT input resistance is just the intrinsic emitter resistance reflected to the base. The BJT T-model makes this visually obvious.

For large $\beta$ the factor $\beta +1\approx \beta$, so some textbooks state the rule with just $\beta$. At this level the exact factor is $\beta +1$; use it.

Where it shows up

This rule is everywhere in BJT amplifier work:

• Emitter degeneration: an un-bypassed $R_E$ in the emitter raises the input resistance of a Common-emitter amplifier to $r_{\pi }+(\beta +1)R_E$ — the $R_E$ reflected to the base.
• Emitter follower: the high input resistance $R_{in}=r_{\pi }+(\beta +1)R_E$ is the load resistance on the emitter reflected up to the base, which is why the emitter follower is such a good Buffer amplifier.
• Working down from base to emitter explains the emitter follower’s low output resistance: the source resistance at the base reflects down by $1/(\beta +1)$.