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Chapter 11: Problem 23

Three identical amplifiers having \(A_{\mathrm{voc}}=25\) \(R_{l}=2 k \Omega,\) and \(R_{c}=3 k \Omega\) are cascaded. Determine the input resistance, the open- circuit voltage gain, and the output resistance of the cascade.

Short Answer

Expert verified
Input resistance is 2 k惟, open-circuit voltage gain is 15625, and output resistance is 3 k惟.

Step by step solution

01

Understanding amplifier parameters

We are given three identical amplifiers with the following parameters for each amplifier: the open-circuit voltage gain \(A_{\mathrm{voc}} = 25\), load resistance \(R_{l} = 2 \, \mathrm{k}\Omega\), and collector resistance \(R_{c} = 3 \, \mathrm{k}\Omega\). These amplifiers are cascaded, meaning they are connected in series, where the output of one feeds into the input of the next.
02

Calculate the input resistance

For a cascade of amplifiers, the input resistance \(R_{in, \text{cascade}}\) is the same as the input resistance of the first amplifier. Since each amplifier is presumably connected in series without any modification, assuming ideal conditions, \(R_{in, \text{cascade}} = R_{l} = 2 \, \mathrm{k}\Omega\).
03

Calculate the open-circuit voltage gain

The total open-circuit voltage gain of cascaded amplifiers is the product of the voltage gains of the individual amplifiers. Given three amplifiers each with a gain of \(A_{\mathrm{voc}} = 25\), the cascade gain is calculated as:\[A_{\mathrm{voc, \text{cascade}}} = A_{\mathrm{voc}}^3 = 25^3 = 15625.\]
04

Calculate the output resistance

For the cascaded configuration, the output resistance \(R_{out, \text{cascade}}\) is essentially the output resistance of the final amplifier. Since all amplifiers are identical, the output resistance of each is given by \(R_{c} = 3 \, \mathrm{k}\Omega\), thus \(R_{out, \text{cascade}} = R_{c} = 3 \, \mathrm{k}\Omega\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Input Resistance
When dealing with cascaded amplifiers, input resistance is a fundamental concept. It sets the stage for how efficiently signals can transfer into the amplifier chain. For cascaded amplifiers, the input resistance of the entire chain is simply the input resistance of the very first amplifier. This is due to the fact that each amplifier feeds directly into the next without affecting the initial resistance.
To illustrate, consider the given values: if each amplifier shares identical characteristics, the first amplifier's load resistance, denoted as \(R_l = 2 \, \mathrm{k}\Omega\), establishes the input resistance for the complete system. This property is immensely useful for analyzing must-have-resistance values, especially in scenarios where signal integrity is paramount.
Deciphering Voltage Gain
In the world of amplifiers, voltage gain is a measure of how much an amplifier boosts an input signal. For a single amplifier, this gain, often denoted by \(A_{\mathrm{voc}}\), is straightforward. But with cascaded amplifiers, it becomes a multiplicative property.
Imagine stacking three identical amplifiers, each with a unit gain of 25. The total voltage gain of the cascaded system is achieved by multiplying each amplifier's gain. In this case, it results in:
  • Total Cascade Gain: \(A_{\mathrm{voc, \text{cascade}}} = 25^3 = 15625\).
This exponential increase illustrates how small gains in individual amplifiers can culminate in significant output, highlighting the efficiency of cascading in amplifying tasks.
Exploring Output Resistance
Output resistance is the "window" through which the amplifier system delivers its final signal. It's a crucial parameter as it affects how the system interacts with connected components downstream. For cascaded amplifiers, this property boils down to the output resistance of the last amplifier in the chain.
If each amplifier's collector resistance is \(R_c = 3 \, \mathrm{k}\Omega\), then this serves as the output resistance for the entire system. The choice of this component is pivotal for reducing power loss and ensuring effective power delivery to subsequent stages or loads. Keeping this resistance in check can maintain signal strength across connected devices, ensuring the integrity of the system's output.

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Most popular questions from this chapter

An amplifier has a short-circuit current gain of \(10 .\) When operated with a \(50 . \Omega\) load, the current gain is \(8 .\) Find the output resistance of the amplifier.

The output terminals of an ideal trans resistance amplifier are connected to the input terminals of an ideal trans conductance amplifier. What type of ideal amplifier results? Determine its gain parameter in terms of the gain parameters of the separate stages.

The transfer characteristic of an amplifier is described by the equation $$v_{o}(t)=10 v_{\mathrm{in}}(t)+0.6 v_{\mathrm{in}}^{2}(t)+0.4 v_{\mathrm{in}}^{3}(t)$$ For the input \(v_{\mathrm{in}}(t)=2 \cos (200 \pi t),\) determine the distortion factors \(D_{2}, D_{3},\) and \(D_{4}\) Also, compute the total harmonic distortion. You may find the following trigonometric identities useful: $$\begin{array}{l} \cos ^{2}(A)=\frac{1}{2}+\frac{1}{2} \cos (2 A) \\ \cos ^{3}(A)=\frac{3}{4} \cos (A)+\frac{1}{4} \cos (3 A) \end{array}$$

The output terminals of an ideal voltage amplifier are connected to the input terminals of an ideal trans conductance amplifier. What type of ideal amplifier results? Deter mine its gain parameter in terms of the gain parameters of the separate stages.

An amplifier with \(R_{i}=2 \mathrm{k} \Omega, R_{\mathrm{e}}=500 \Omega,\) and \(R_{\text {mec }}=-10^{7} \Omega\) is operated with a 1-k\Omega load. A source having a Th茅venin resistance of \(1 \mathrm{k} \Omega\) and an open-circuit voltage of \(2 \cos (200 \pi t) \mathrm{mV}\) is connected to the input terminals. Determine the output voltage as a function of time and the power gain.
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