**Voltage Amplification Calculation of a Multi-Stage Amplifier Circuit**
When calculating the voltage gain of a multi-stage amplifier circuit using discrete components, there are two main approaches. The first method involves considering the input resistance of the subsequent stage as the load of 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 technique is known as the **input resistance method**.
The second approach is to open the connection between the stages and calculate the open-circuit voltage gain and output resistance of the first stage. This is then treated as the internal resistance of the signal source, which works together with the input of the second stage. This method is referred to as the **open-circuit voltage method**.
To illustrate these methods, let’s take the two-stage amplifier circuit shown in Figure 1 as an example.
**Figure 1: Two-Stage Amplification Circuit**
In this example, both transistors have β = 100, and the base-emitter voltage VBE = 0.7 V for each. We will calculate the total voltage gain using both the input resistance method and the open-circuit voltage method.
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**Solution (1): Using the Input Resistance Method**
**Step 1: Determine the DC Operating Point**
We begin by finding the DC bias point of each transistor. This involves calculating the base, emitter, and collector voltages and currents to ensure proper operation in the active region.
**Step 2: Calculate the Voltage Gain**
First, we determine the input resistance of the first transistor. Then, we calculate the voltage gain of the first stage, taking into account the loading effect of the second stage's input resistance.
The voltage gain of the first stage is calculated as:
$$
A_{v1} = \frac{R_C || R_{in2}}{r_e + R_E}
$$
Where:
- $ R_C $ is the collector resistor,
- $ R_{in2} $ is the input resistance of the second stage,
- $ r_e $ is the small-signal emitter resistance,
- $ R_E $ is the emitter resistor.
Next, we compute the voltage gain of the second stage, considering the output resistance of the first stage and the input resistance of the second stage.
Finally, the overall voltage gain is the product of the gains from both stages.
If we want to find the **source voltage gain** (from the source voltage VS), we must also consider the input resistance of the entire circuit and how it interacts with the source impedance.
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**Solution (2): Using the Open-Circuit Voltage Method**
In this method, we first calculate the open-circuit voltage gain of the first stage, assuming no load is connected. Then, we determine the output resistance of the first stage, treating it as the internal resistance of the signal source.
This effective source is then applied to the input of the second stage, allowing us to calculate the total voltage gain.
For a single-stage BJT amplifier, the voltage gain can be approximated as:
$$
A_v = -\frac{R_C}{r_e + R_E}
$$
Where:
- $ R_C $ is the collector resistor,
- $ r_e $ is the small-signal emitter resistance,
- $ R_E $ is the emitter resistor.
The negative sign indicates a phase inversion between the input and output signals. However, if only the magnitude of the gain is needed, the phase difference can be ignored.
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**Voltage Gain of a Common Collector Amplifier (Emitter Follower)**
The voltage gain of a common collector amplifier is given by:
$$
A_v = \frac{(1+\beta) R_E || R_L}{r_{be} + (1+\beta) R_E || R_L}
$$
Since $ (1+\beta) R_E || R_L $ is typically much larger than $ r_{be} $, the voltage gain is less than 1 but close to 1. This is why the common collector configuration is often called an **emitter follower**, as it follows the input voltage with minimal attenuation.
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