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Circuits, electronics and noise

Common-emitter gain with an unbypassed emitter resistor

Midband voltage gain from the base to the collector of a common-emitter stage with an unbypassed emitter resistor.

Ref: ELN-20
Rev: v1
Field: Circuits, electronics and noise
Result: A_v [V/V]

A_{v} = \frac{- g_{m} (R_{c} ∥ R_{L})}{1 + (g_{m} + 1 / r_{π}) R_{e}}

With an unbypassed emitter resistor Re, the gain is the transconductance times the load Rc∥RL, divided by 1 + (gm + 1/rπ)Re. For a large gm Re the gain approaches −(Rc∥RL)/Re, set by resistor ratios and not by the transistor. The negative sign means the output is inverted. The price of the stability is a lower gain and a much larger input resistance.

Midband voltage gain from the base to the collector of a common-emitter stage with an unbypassed emitter resistor. The emitter resistor reduces and stabilises the gain. Negative means the signal is inverted.

Calculator

Solve for

A_vVoltage gainComputed

V/V

g_mTransconductance

0.0010.4

R_cCollector resistor

100100,000

R_LLoad resistance

100100,000

r_πBase input resistance

100100,000

R_eUnbypassed emitter resistor

01,000

Gain with the emitter bypassed (Re = 0), −gm(Rc∥RL)−80 V/V
Approximate gain for large Re, −(Rc∥RL)/Re−20 V/V
Feedback factor, 1 + (gm + 1/rπ)Re5.04

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Voltage gain (V/V) current value −15.87 V/V

R_eUnbypassed emitter resistorKey driver

−80−1.93

R_cCollector resistor

−30.5−0.774

R_LLoad resistance

−30.5−0.774

g_mTransconductance

−19.5−1.75

r_πBase input resistance

−16−13.3

With 100 Ω in the emitter the gain is −15.87, against −80 with the emitter bypassed.
Re cuts the gain by a factor of 5.04, in return for less dependence on the transistor’s gm and so less distortion.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (V/V)	Local step	Change in result
R_e Unbypassed emitter resistor	0 to 1,000 Ω	−80	−1.932	78.07	+10%	−7.42 %
R_c Collector resistor	100 to 100,000 Ω	−30.53	−0.7743	29.75	+10%	+4.76 %
R_L Load resistance	100 to 100,000 Ω	−30.53	−0.7743	29.75	+10%	+4.76 %
g_m Transconductance	0.001 to 0.4 S	−19.49	−1.754	17.74	+10%	+1.91 %
r_π Base input resistance	100 to 100,000 Ω	−16	−13.33	2.663	+10%	+0.0722 %

Sweep of Unbypassed emitter resistor

Unbypassed emitter resistor (Ω)	Voltage gain (V/V)
0	−80
125	−13.22
250	−7.207
375	−4.954
500	−3.774
625	−3.048
750	−2.556
875	−2.201
1,000	−1.932

Sweep one variable

Vary

Voltage gain (V/V)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

A_v

Voltage gain

Result · V/V

What the equation gives you. Everything else feeds it.

g_m

Transconductance

S · typical 0.001 to 0.4

The transconductance, in S. A larger gm raises the size of the gain, but with less than direct proportion when Re is large. Raising it by 10 % (to 0.044) moves the result by +1.91 %, from −15.87 to −16.18 V/V.

R_c

Collector resistor

Ω · typical 100 to 100,000

The collector resistor, in Ω. A larger Rc raises the load and so the size of the gain. Raising it by 10 % (to 4,400) moves the result by +4.76 %, from −15.87 to −16.63 V/V.

R_L

Load resistance

Ω · typical 100 to 100,000

The load resistance connected to the collector, in Ω. A larger load loads the stage less, so the size of the gain rises. Raising it by 10 % (to 4,400) moves the result by +4.76 %, from −15.87 to −16.63 V/V.

r_π

Base input resistance

Ω · typical 100 to 100,000

The base input resistance rπ, in Ω. A larger rπ removes a little of the extra feedback, so the size of the gain rises a little. Raising it by 10 % (to 2,750) moves the result by +0.0722 %, from −15.87 to −15.88 V/V.

R_e

Unbypassed emitter resistor

Ω · typical 0 to 1,000

The unbypassed emitter resistor, in Ω. A larger Re gives more feedback, so the size of the gain falls. Raising it by 10 % (to 110) moves the result by −7.42 %, from −15.87 to −14.7 V/V.

Worked example with the values above

Load: Rc∥RL = 4,000 ∥ 4,000 = 2,000 Ω.
Numerator: −gm × load = −0.04 × 2,000 = −80; feedback factor: 1 + (0.04 + 1/2,500) × 100 = 5.04.
Av = −80 / 5.04 = −15.873.

Assumptions and limits

An NPN transistor in the forward-active region, with a fixed base–emitter drop of about 0.7 V.
Midband small signals, an AC-grounded supply and negligible transistor output conductance.

Common mistakes

Forgetting to check that the collector–base junction is still reverse biased.
Using the DC current gain where the small-signal gain is needed, or the reverse.
Forgetting that an unbypassed emitter resistor adds (β + 1) times its value to the input resistance.

Statistics and uncertainty

Transistor current gain β varies by a factor of two or more between parts and with temperature, so a stable design depends on emitter feedback, not on β. Thermal voltage changes by about 0.33 % per kelvin.

Where it is used

Setting the operating point of transistor amplifiers and switches.
Estimating small-signal gain, input resistance and the effect of emitter degeneration.

Maths you need

Thevenin equivalent: a source and a resistance in series
Transistor current gain: β = Ic / Ib
Thermal voltage: kT/q, about 25 mV at room temperature

Related formulas

Practise this formula