**Voltage Amplification Calculation of a Multi-Stage Amplifier Circuit**
When calculating the voltage gain of a multi-stage amplifier circuit composed of discrete components, there are two main approaches. The first method involves considering the input resistance of the subsequent stage as the load for the previous stage. This means that the input resistance of the second stage is connected in parallel with the collector load resistance of the first stage. This approach is commonly referred to as the **input resistance method**.
The second method is based on the **open-circuit voltage method**, where the output of the previous stage is considered as an independent signal source. In this case, we calculate the open-circuit voltage gain and output resistance of the first stage, treating it as the internal resistance of the signal source. This signal then drives the input of the next stage. This method is especially useful when analyzing the overall gain of the system.
To illustrate these methods, let’s consider the two-stage amplifier circuit shown in Figure 1.
*Figure 1: Two-Stage Amplification Circuit*
In this example, both transistors have β = 100, and the base-emitter voltage VBE = 0.7 V for both. We will calculate the total voltage gain using both the input resistance method and the open-circuit voltage method.
**Solution (1): Using the Input Resistance Method**
**Step 1: Determine the DC Operating Point**
To find the static operating point, we analyze the biasing network and determine the quiescent current and voltage levels. This step ensures that the transistor operates in the active region, allowing proper amplification.
**Step 2: Calculate the Voltage Gain**
First, we compute the input resistance of the transistor. This is essential because it affects how the signal is coupled from one stage to the next.
$$
R_{in} = r_{be} + (1 + \beta) R_E
$$
Next, we calculate the voltage gain of each stage. For a common-emitter configuration, the voltage gain is approximately:
$$
A_v = -\frac{R_C}{r_e}
$$
Where $ R_C $ is the collector resistor and $ r_e $ is the small-signal emitter resistance.
Finally, the total voltage gain is the product of the gains of each stage, taking into account the loading effect between stages.
If we are interested in the **source voltage gain** (from the input signal source to the output), we must also consider the input resistance of the first stage and the source resistance.
**Solution (2): Using the Open-Circuit Voltage Method**
This method involves separating the stages and calculating the open-circuit voltage gain of the first stage. Then, we treat this as the signal source for the second stage, taking into account its input impedance.
For the first stage, the open-circuit voltage gain is calculated as:
$$
A_{v1} = -\frac{R_{C1}}{r_{e1}}
$$
Then, we combine this with the input impedance of the second stage to find the overall gain.
**Voltage Gain of a Single-Stage Amplifier**
The general formula for the voltage gain of a single-stage BJT amplifier is:
$$
A_v = \frac{U_{out}}{U_{in}}
$$
Where $ U_{out} $ is the change in output voltage, and $ U_{in} $ is the change in input voltage. The negative sign indicates a 180° phase shift between input and output.
For a common-collector amplifier (emitter follower), the voltage gain is typically less than 1 but close to 1, due to the following formula:
$$
A_v = \frac{(1 + \beta) R_E // R_L}{r_{be} + (1 + \beta) R_E // R_L}
$$
Since $ (1 + \beta) R_E // R_L $ is generally much larger than $ r_{be} $, the voltage gain is slightly less than 1, which is why this configuration is often used as a **buffer** or **emitter follower**.
By applying these methods, engineers can accurately determine the performance of multi-stage amplifier circuits and optimize their design for specific applications.
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