Frequency response of amplifiers
A 2 pF capacitance between base and collector looks like almost 400 pF from the input. The Miller effect is what limits the bandwidth of a common emitter.
01Three frequency regions
The gain of a transistor amplifier is constant only over a band, the midband. Below it the gain falls because of the coupling and bypass capacitors, which stop being short circuits; above it, because of the internal capacitances of the transistor, which stop being open circuits. Since the two causes act in regions separated by several decades, they are analyzed separately:
| Region | External capacitors | Transistor capacitances | Method |
|---|---|---|---|
| Low frequency | Act | Open | Short-circuit time constants |
| Midband | Short circuits | Open | Resistive small-signal model |
| High frequency | Short circuits | Act | Miller effect and open-circuit time constants |
The small-signal model is that of the bipolar transistor as an amplifier, and the frequency response of linear systems is that of frequency response, in Signals and Systems.
02Low frequency: short-circuit time constants
With several capacitors, calculating the exact poles is laborious. A widely used estimate takes each capacitor separately, with the others short-circuited, calculates the resistance \( R_{is} \) it sees and adds the reciprocals of the time constants:
03High frequency: the Miller effect
In the hybrid-π model, the bipolar transistor has the diffusion and junction capacitance of the base–emitter junction \( C_\pi \), and that of the base–collector junction \( C_\mu \), much smaller. Its transition frequency, where the current gain falls to 1, is:
The problem is in \( C_\mu \): it connects the input to the output, which has a gain \( -g_m R_L \). The current through it is that of a capacitor \( 1 + g_m R_L \) times larger connected to ground. Seen from the input, the total capacitance is:
04Open-circuit time constants
The Miller approximation ignores that \( C_\mu \) also loads the output. A more complete method, by Gray and Searle, adds the time constants of each capacitor seen with the others open, and gives exactly the first-order coefficient of the denominator:
A common-emitter stage with β = 100 and \( f_T \) = 300 MHz, with the complete hybrid-π model. The blue curve is the exact response, with its two poles and the zero; the green one, that of a single pole with the open-circuit time constants, and the orange one, that of a pole with the Miller capacitance.
The second pole of the common-emitter stage is hundreds of times higher, and the zero, at \( g_m/C_\mu \), is in the right half-plane: above it the signal passes directly through \( C_\mu \). Both matter when the amplifier is fed back, as will be seen in the next unit.
05How to gain bandwidth
If the problem is the voltage gain between base and collector, the solution is not to have it. The cascode, which already appeared in multistage amplifiers, puts a common-base stage on top of the common-emitter: the collector of the first transistor sees \( 1/g_m \) and its gain is −1, so \( C_\mu \) only doubles. Other solutions are to drive the base from a follower, which presents a small \( R' \), or to use a differential pair with the output taken from the opposite side, with no voltage gain in the input transistor.
06In the lab
Measure the gain of a common-emitter stage from 10 Hz to 10 MHz and locate the lower and upper cutoff frequencies. Repeat with half the emitter bypass capacitor.
Measure the upper cutoff frequency with two different source resistances, and compare with the Miller and time-constant estimates.
Add a common-base transistor on top of the common emitter, with the same current and load, and measure the new bandwidth.
07Common mistakes
- Forgetting the source resistance. The cutoff frequency depends on \( R' \): the same amplifier has a different band with a different source.
- Applying Miller with the midband gain at any frequency. Near the cutoff the gain has already changed and the approximation worsens.
- Taking the transition frequency as the bandwidth of the amplifier. \( f_T \) belongs to the short-circuit current gain, not to the circuit.
- Making all the low-frequency capacitors equal. The emitter bypass sees a much smaller resistance and has to be much larger.
08Self-assessment
What Cπ + Cμ does a transistor with fT = 250 MHz at 2 mA have?
\( g_m \) = 77.4 mS; \( 77.4\,\mathrm{mS}/(2\pi \cdot 250\,\mathrm{MHz}) \approx 49 \) pF.
With Cμ = 3 pF and gmRL = 100, what Miller capacitance appears at the input?
303 pF.
Why does the cascode have more bandwidth than the common emitter?
Because the input transistor has a voltage gain of −1 and its \( C_\mu \) only doubles.
What term of the time constant does the Miller approximation ignore?
\( C_\mu R_L \): the load that \( C_\mu \) places on the output.
09Further reading
- Adel S. Sedra and Kenneth C. Smith. Microelectronic Circuits. 8th ed., Oxford University Press, 2020. Frequency response with the Miller effect and time constants.
- Paul R. Gray and Campbell L. Searle. Electronic Principles: Physics, Models, and Circuits. Wiley, 1969. The origin of the open-circuit time constants method.
- Behzad Razavi. Fundamentals of Microelectronics. 2nd ed., Wiley, 2013. The Miller effect and the cascode explained with a designer's intuition.