Small Signal Analysis Tutorial

Small signal analysis is a method used to analyze the behavior of electronic circuits, particularly those containing non-linear devices like MOSFETs (Metal-Oxide-Semiconductor Field-Effect Transistors) and BJTs (Bipolar Junction Transistors). This technique involves linearizing the circuit's behavior around a specific operating point, allowing for the application of linear circuit analysis methods.

Small Signal Model of a MOSFET

When a MOSFET operates in the saturation region, it functions as a voltage-controlled current source. The relationship between the gate-source voltage (V_GS) and the drain current (I_D) is crucial for understanding its behavior.

Transconductance (g_m):

Transconductance (g_m) is a key parameter that represents the slope of the I_D vs. V_GS relationship. It is defined as the partial derivative of I_D with respect to V_GS:

In practice, the I_D vs. V_GS relationship is non-linear. Therefore, we analyze the behavior for small signals around a specific operating point Q. When the signal amplitude is very small, this change in I_D relative to V_GS can be approximated as linear (meaning g_m is treated as a constant).

In the small-signal model, we use lowercase i_d and v_gs to represent these “small changes” (AC signals). Thus, the relationship from Equation 1 can be expressed in the small-signal model as:

i_d = gm * v_gs

This is precisely why this analysis is called the “small-signal model.”

Channel Length Modulation (CLM):

Channel Length Modulation (CLM) is another important effect to consider. Ideally, once the drain-source voltage (V_DS) exceeds a certain value, the current stops increasing with V_DS. However, in reality, the current does slightly increase due to the effective shortening of the channel length.

To incorporate the effect of CLM into our model, we modify the current formula using the channel length modulation parameter, lambda (λ):

Complete Small-Signal Equivalent Circuit for an NMOS

The complete small-signal equivalent circuit for an NMOS includes the transconductance (g_m) and the output resistance (r_o) in parallel. The gate is an open circuit to the drain and source, and current flows only between the drain and source.

MOSFET Small Signal Model

Small Signal Analysis of a BJT

A bipolar junction transistor, or BJT, is a type of transistor. They are commonly found in electronic amplifier circuits such as those used to transmit data wirelessly, and in radios.

The internal characteristics of a BJT can vary with temperature, voltage, and current. Because of this, before putting a transistor into your project or device you may want to measure the internal characteristics of the transistor at the voltage and current you will have it operating.

To measure these values, you need to take the following steps:

Create a circuit to test your transistor in.
Measure the voltage or current going into the transistor.
Calculate the Beta value (current gain at low frequencies) and the transconductance.
Using the values calculated for current gain and transconductance, find R-π, the internal resistance.

Important Safety Note: If the current through the transistor is too high, it may melt or explode. For most small transistors, anything close to an amp of current is high.

Step 1: Set Up a Test Circuit

To test the collector and base currents you will first need to set up a circuit to test them in. Because BJTs usually operate at very low currents, we attach 1000 ohm resistors on each node of the transistor to ensure that the transistors are not damaged.

Step 2: Measure the Voltage Across the Resistors

Because BJTs operate at such low current, instrument noise can make it difficult to accurately measure the current. Instrument noise is small, random variations in the measured values. In most multimeters, this is a very small amount, but it is similar in magnitude to the currents in the test circuit. In order to compensate for this issue, you may instead want to measure the voltage across the resistors and then use Ohm's law to calculate the current.

Ohm's law states the the voltage across a component equals the current through it multiplied by its resistance: V = I * R. To find the current through the resistors divide the voltage by the resistance: I = V / R.

Step 3: Calculate the Internal Characteristics Based on Input Currents

Now that you've calculated the internal currents you can calculate the small signal values of the BJT. These are as follows:

β₀: Current Gain - This value represents the ratio between the current into the collector and the current into the base.
g_m: Transconductance - This value represents the ratio between the current out of the transistor and the voltage into it.
r_π: Internal Resistance - This value is the resistance between the base and emitter terminals of the transistor.

Step 4: Calculate Current Gain

To calculate current gain, divide the value of the current going into the base terminal by the value of the current going into the collector terminal.

Step 5: Calculate Transconductance

Transconductance is just the ratio between the voltage into the transistor and current out of the transistor. To calculate it, divide the current into the collector terminal by VT, thermal voltage. Doing so gives you a value in inverse ohms, written as mhos.

Step 6: Calculate Internal Resistance

The last step is to calculate the internal resistance of the BJT. This value represents how much of the voltage placed on the collector terminal makes it to the emitter terminal. To calculate it, divide the value of current gain by the transconductance, both calculated in previous steps. This calculation gives you a result in ohms.

Step 7: Conclusions

After having gone through these steps, you will have calculated the main internal characteristics of whatever BJT you were testing. Keep in mind that these values are only valid for lower frequencies, and will change at high frequencies.

Table of BJT Small-Signal Parameters

Parameter	Description	Formula
β₀	Current Gain	I_C / I_B
g_m	Transconductance	I_C / V_T
r_π	Internal Resistance	β₀ / g_m