Transistor Amplifiers
Large Signal (DC Analysis)
We only consider the values from the circuit that do not change with time. We call this the Q-Point.
- AC Volt sources: \(\int V\ dt\) has no constants -> short circuit
- AC Current sources: \(\int I\ dt\) has no constants -> open circuit
Consider the impedance of capacitors and inductors:
| Capacitor | Inductor |
| \(\Z_C = \frac{1}{j\omega C}\) | \(\Z_L = j\omega L\) |
With only DC sources, \(\omega = 0\):
- Capacitor: \(\Z_C = \infty\) -> open circuit
- Inductor: \(\Z_L = 0\) -> short circuit
Assume the input is a sine wave. A large signal analysis gives the middle point (Q-point) where the sine wave moves.
With the simplified circuit, analyze as usual (step by step here).
Small Signal (AC Analysis)
Small Signal is like a derivative of the circuit with respect to time.
- DC Volt sources: \(\frac{dV}{dt} = 0\) -> short circuit
- DC Current sources: \(\frac{dI}{dt} = 0\) -> open circuit
Consider the impedance of capacitors and inductors:
| Capacitor | Inductor |
| \(\Z_C = \frac{1}{j\omega C}\) | \(\Z_L = j\omega L\) |
Assuming \(\omega\) has a high value:
- Capacitor: \(\Z_C = 0\) -> short circuit
- Inductor: \(\Z_L = \infty\) -> open circuit
This assumption works only for high frequencies. Read more about frequency in amplifiers here.
The transistors require an extra step, shown for FETs and BJTs:
FET Small-Signal Model
Original circuit:
The transistor becomes:
- A dependent current source \(g_m V_{gs}\) from Drain to Source, parallel to a resistor \(r_o\)
- An open circuit at the Gate
Where:
\[g_m = 2 K_n (V_{GSQ} - V_{TN})\]Is the transconductance,
\[r_o = \frac{1}{\lambda I_{DQ}}\]Is the finite output resistance.
Here, the output voltage is
\[V_o = -g_m V_gs(r_o \| R_D)\]And the input voltage is
\[V_i = V_{gs}\]The small-signal voltage gain is defined as
\[A_v = \frac{V_o}{V_i} = -g_m (r_o \| R_D)\]BJT Small-Signal Model
Original circuit:
The transistor becomes:
- A dependent current source \(\beta i_b\) from Collector to Emitter, parallel to a resistor \(r_o\)
- A resistor \(r_{\pi}\) from Base to Emitter
Where:
\[r_{\pi} = \frac{V_T}{I_{BQ}} = \frac{\beta V_T}{I_{CQ}}\]Is the hybrid \(\pi\) model resistance.
\[r_o = \frac{V_A}{I_{CQ}}\]Is the transistor output resistance, and \(V_A\) is the early voltage.
Here, the output voltage is
\[V_o = -R_C \beta i_b\]And the input voltage is
\[V_i = i_b r_{\pi}\]The small-signal voltage gain is defined as
\[A_v = \frac{V_o}{V_i} = -\frac{\beta R_C}{r_{\pi}}\]The dependent current source may also be expressed as
\[g_m v_{be}\]Where
\[g_m = \frac{I_{CQ}}{V_T}\]Is the transconductance.
There is also a model with transconductance \(g_m\) but given that \(i_c = \beta i_b\) we can use this simpler approach.
For PNPs, switch the direction of the currents.
Examples
When in doubt, KVL is your best friend.
FET Amplifiers
Basic NMOS amplifier:
Solving Steps
Assume NMOS is in saturation, and \(V_{GSQ},\ V_{DD},\ V_{TN},\ K_n,\ R_D\) are given.
- DC Analysis: Q Point (Large Signal)
- AC: Short Volt, Open Current Sources
- Short Inductors, Open Capacitors
- Get \(I_{DQ},\ V_{DSQ}\)
- Verify \(V_{DSQ} > V_{DSQ}(sat)\) (otherwise, non-linear)
- AC Analysis (small signal)
- DC: Short Volt, Open Current Sources
- Open Inductors, Short Capacitors
- Replace NMOS by small-signal model
- Get \(v_{out},\ v_{in}\)
- Voltage gain: \(A_v = \frac{v_{out}}{v_{in}}\)
FET Amplifier Comparison
| Configuration | \(A_v\) | Input R | Output R |
| Common source | \(A_v > 1\) | \(R_{TH}\) | \(R_D\) |
| Common drain | \(A_v \approx 1\) | \(R_{TH}\) | Low |
| Common source | \(A_v \geq 1\) | \(\frac{1}{g_m}\) | \(R_D\) |
BJT Amplifier Comparison
| Configuration | \(A_v\) | Input R | Output R |
| Common collector | \(A_v > 1\) | \(R_1 | R_2 | R_{ib}\) | \(R_C | r_o\) |
| Common base | \(A_v \approx 1\) | \(R_1 | R_2 | R_{ib}\) | \(\frac{r_{\pi}}{\beta + 1} | R_E | r_o\) |
| Common emitter | \(A_v > 1\) | \(\frac{r_{\pi}}{\beta + 1}\) | \(r_o | R_C\) |
Frequency
See transistors frequency here.