Basic BJT Differential Amplifier Construction and Analysis

Dual Input Balanced Output differential amplifier circuit diagram

There are four main types of differential amplifiers:

1. Dual Input, Balanced Output
2. Dual Input, Unbalanced Output
3. Single Input, Balanced Output
4. Single Input, Unbalanced Output

The circuit shown is a dual input, balanced output configuration, where two input signals are applied to the bases of the BJTs (Q1 and Q2), and the outputs are taken from the collectors of these transistors. This setup is also known as an emitter-coupled differential amplifier.

The following diagrams illustrate the input waveforms $V_1$ and $V_2$, along with the resulting output waveforms $V_{out1}$ and $V_{out2}$.

The BJT differential amplifier operates on the principle that the current through the two BJTs is proportional to the difference between the input voltages, $V_1$ and $V_2$. When the input signals are equal, the current through both transistors is balanced, and the output voltage is zero. However, when the input signals differ, the current through each BJT becomes unequal, creating a voltage drop across the load resistor and producing an output voltage.

The amplifier’s performance can be analyzed through both DC and AC analysis. This analysis helps determine the operating point ($V_{ceq}$ and $I_{cq}$), input and output resistances, and both differential and common mode gains.

DC Analysis

To find the Q-point collector current ($I_{cq}$), we note that it is equal to the emitter current ($I_e$). Thus, deriving an expression for the emitter current ($I_e$) is equivalent to finding the collector current ($I_{cq}$).

Under DC conditions, the input voltage $V_1$ is grounded, and we analyze the base-emitter loop to derive an expression for the emitter current $I_e$.

Using KVL around the base-emitter loop, we have:

Since $I_B = \frac{I_E}{\beta}$(since $I_C = I_E$), we can express the equation as:

Rearranging:

Or:

Thus:

Because $2R_E\gg \frac{R_1}{\beta }$, we can approximate:

Thus, the equation for the quiescent point collector current ($I_{CQ}$) is:

The emitter current ($I_E$) or the $Q$-point collector current ($I_{CQ}$) in a differential amplifier depends on the emitter resistor ($R_E$) and is independent of the collector resistor ($R_C$).

For transistor $Q_1$, we have:

Since $V_{CC}=V_C+I_C{R}_C$, we can write:

Now, $V_{BE}=V_B-V_E$. Neglecting the voltage drop across $R_1$, $V_B=0$, so:

Thus, we can rewrite:$$V_{CE}=V_{CC}-I_C{R}_C-(-V_{BE})$$

Or:$$V_{CE}=V_{CC}-I_C{R}_C+V_{BE}$$

At the $Q$-point, we have:

For the AC analysis of a differential amplifier, we can use one of the following transistor models:

• (a) h-parameter model
• (b) re model
• (c) hybrid-pi model

In this case, we will use the h-parameter model for the AC analysis of the BJT differential amplifier. For AC analysis, the two input signals should be of equal magnitude and 180 degrees out of phase with respect to each other. We assume that $V_1 = V_2 = V_{in}/2$. The AC signal across the emitter resistor ($R_e$) is zero, so it is effectively short-circuited in the AC equivalent circuit and connected to ground as shown below.

Using the h-parameter circuit model, we will perform AC analysis to determine the following:

• (a) Differential gain ($A_d$)
• (b) Common mode gain ($A_{cm}$)
• (c) Input resistance ($R_i$)
• (d) Output resistance ($R_o$)

Input Loop Analysis

In the circuit above, applying KVL around the input loop, we get:

That is:

where $R_S = R_1 = R_B$ for $Q_1$.

Output Loop Analysis

Applying KVL in the output loop, we have:

Using $I_B$ from equation (4), we get:

Or:

The negative sign in (5) shows that $V_{\text{out}}$ and $V_{\text{in}}$ are 180° out of phase.

Differential Mode Analysis

In differential mode, the two input signals are equal in magnitude but opposite in phase. The differential input voltage $V_d$ is:

Hence, the differential gain is given by:

So:

This differential gain given by equation (6) is valid when the output is taken at the collector of BJT transistor $Q_1$ or $Q_2$ with respect to ground. However, if the output is taken between the collectors of the two transistors, the differential gain will be twice the differential gain of equation (6).

Therefore, for balanced output, the differential gain is:

(b) Common Mode Gain ($A_{cm}$)

In common-mode operation of a differential amplifier, the two input signals are of the same magnitude and in phase. Thus, we have:

The common-mode signal $V_{cm}$ is the average of the two input signals:

The output voltage in the common-mode is given by:

Thus, the common-mode gain $A_{cm}$ is:

In common-mode operation, the emitter resistance is $2R_e$. The common-mode gain $A_{cm}$ is defined as:

The output voltage $V_{\text{out}}$ is the voltage across $R_c$ due to the collector current $I_c$:

Applying KVL in the input loop:

Since $I_c = h_{fe} I_b$, we have:

Simplifying:

Thus:

Substituting $I_b$ from equation (12) into equation (10), we obtain:

Therefore, the common-mode gain is:

Differential Input Resistance ($R_i$)

The differential input resistance ($R_i$) is the resistance between one of the input terminals and ground while the other input terminal is grounded. The following circuit can be used to determine the differential input resistance ($R_i$).

 

Applying KVL around the input loop:

Rearranging:

Thus, the differential input resistance $R_i$ is:

Output Resistance ($R_o$)

To determine the output resistance $R_o$, consider the following conditions:

1. $V_{\text{in}} = 0$
2. $I_b = 0$

The output resistance is defined as the resistance measured between the output terminal and ground.

From the circuit:

Applications of BJT Differential Amplifier

BJT differential amplifiers are widely used in electronic circuits due to their ability to amplify differential signals and reject common-mode noise. Key applications include:

1. Audio Amplifiers: Amplify differences between left and right audio signals for stereo output.
2. Instrumentation Amplifiers: Enhance small differential signals from sensors while rejecting noise.
3. Operational Amplifiers: Provide high input impedance and low output impedance, suitable for diverse applications.

Summary

BJT differential amplifiers can be built using discrete components or integrated circuits (ICs) in various packages, such as DIP or SMT. Their versatility makes them essential in applications ranging from audio to instrumentation. They amplify input signal differences while suppressing common-mode signals, ensuring reliable performance across many circuits.

Key Analysis Parameters

DC Analysis (Q-point):

1. $I_{cq} = \frac{V_{ee} - V_{be}}{2R_e}$
2. $V_{ceq} = V_{cc} + V_{be} - I_{cq} R_c$

AC Analysis:

1. Differential Mode Gain: $A_d = -\frac{h_{fe} R_c}{h_{ie} + R_s}$
2. Common Mode Gain: $A_{cm} = -\frac{h_{fe} R_c}{R_s + h_{ie} + 2R_e(1 + h_{fe})}$
3. Input Resistance: $R_i = 2(R_s + h_{ie})$
4. Output Resistance: $R_o = R_c$