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Chapter 11: Problem 28

A diff-amp is biased with a constant-current source \(I_{Q}=0.25 \mathrm{~mA}\) that has an output resistance of \(R_{o}=8 \mathrm{M} \Omega\). The bipolar transistor parameters are \(\beta=120\) and \(V_{A}=\infty\). Determine (a) the differential-mode input resistance and (b) the common-mode input resistance.

Short Answer

Expert verified
(a) Differential-mode input resistance is 48 k惟; (b) Common-mode input resistance is 16 M惟.

Step by step solution

01

Understanding the Parameters

We have a differential amplifier with a constant current source \(I_Q = 0.25 \text{ mA}\), and an output resistance \(R_o = 8 \text{ M}\Omega\). The bipolar transistor parameters are \(\beta = 120\) and \(V_A = \infty\), which means the Early voltage is very high, indicating the output resistance is not affected by \(V_A\).
02

Determine the Differential-Mode Input Resistance

The differential-mode input resistance \(R_{id}\), for a bipolar junction transistor (BJT) differential amplifier, can be approximated by:\[ R_{id} \approx 2 \times r_{\pi} \]where \(r_{\pi} = \frac{\beta}{g_m}\), and \(g_m = \frac{I_C}{V_T}\) with \(I_C \approx \frac{I_Q}{2} = 0.125 \text{ mA}\). Assuming \(V_T \approx 25 \text{ mV}\),\[ g_m = \frac{0.125 \text{ mA}}{25 \text{ mV}} = 5 \text{ mS}\]\[ r_{\pi} = \frac{120}{5} = 24 \text{ k}\Omega\]Thus, the differential-mode input resistance is:\[ R_{id} \approx 2 \times 24 \text{ k}\Omega = 48 \text{ k}\Omega\].
03

Determine the Common-Mode Input Resistance

The common-mode input resistance \(R_{ic}\) is determined by the output resistance of the current source since the transistors share the current path. Hence,\[ R_{ic} \approx 2 \times R_{o} = 2 \times 8 \text{ M}\Omega = 16 \text{ M}\Omega\].

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Key Concepts

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

Bipolar Junction Transistor (BJT)
The Bipolar Junction Transistor, often abbreviated as BJT, is a fundamental semiconductor device used in amplifiers and switches. The BJT has three layers of doped semiconductor material, forming two pn junctions. The three layers are called the emitter, base, and collector. It functions by using small input currents to control larger output currents. This makes BJTs highly efficient for amplification applications.
  • The two main types of BJTs are NPN and PNP transistors, which differ in the polarity of their power supply.
  • BJTs are current-controlled devices. This means that the output current is a function of the input current applied to the base terminal.
  • The beta (\( \beta \)) parameter of a BJT is the current gain, which represents the ratio of the collector current (\( I_C \)) to the base current (\( I_B \)). A higher beta implies more efficiency in amplifying the signal.
Understanding BJTs is crucial for grasping the concept of differential amplifiers, where they can serve as the primary amplifying component.
Differential-mode Input Resistance
In a differential amplifier, the differential-mode input resistance (\( R_{id} \)) is a measure of how much the amplifier resists incoming signals that are equal and opposite across its input terminals. It's vital because it shows how effectively the amplifier can handle and process these signals.
The calculation of \( R_{id} \) involves using the transistor's parameters, particularly its \( g_m \) (transconductance) and \( \beta \) (current gain).
  • The transconductance (\( g_m \)) is calculated using the formula: \( g_m = \frac{I_C}{V_T} \), where \( I_C \) is the collector current and \( V_T \) is the thermal voltage. This tells us how effectively the input voltage changes control the output current.
  • Armed with \( g_m \), the base resistance (\( r_{\pi} \)) can be derived as: \( r_{\pi} = \frac{\beta}{g_m} \).
For a differential amplifier using BJTs, \( R_{id} \) is approximated by \( 2 \times r_{\pi} \). This reflects the resistance faced by differential signals and is a key determinant of the amplifier's performance.
Common-mode Input Resistance
Common-mode input resistance (\( R_{ic} \)) in a differential amplifier is the resistance against identical signals at both input terminals. It's crucial as it affects how the amplifier suppresses common-mode signals while amplifying differential signals.
In a BJT diff-amp, \( R_{ic} \) primarily depends on the output resistance of the constant current source driving the circuit. Often, these currents share the same path through both transistors, thus, common-mode input resistance is represented with:
  • The formula \( R_{ic} = 2 \times R_{o} \) shows how the output resistance (\( R_o \)) of the current source impacts common-mode resistance.
  • This aspect is significant for determining the common-mode rejection ratio, which is a measure of how well the amplifier rejects signals common to both inputs.
Understanding \( R_{ic} \) is essential for optimizing the balance between common-mode rejection and circuit stability.
Constant Current Source
A constant current source is a crucial element in the design of a differential amplifier. It provides a stable current over a wide range of output voltages, ensuring that the amplifier operates optimally. The steady current prevents variations in biasing conditions, improving both stability and performance of the amplifier circuit.
  • The current source is characterized by its output resistance (\( R_o \)), which affects the common-mode input resistance (\( R_{ic} \)) directly.
  • By maintaining a constant current, the transistor parameters remain stable, leading to predictable and reliable amplification.
  • When designing a current source, factors like temperature drift and component tolerance need to be accounted for to ensure continual operation within the desired parameters.
Incorporating a constant current source in differential amplifiers leverages its ability to supply consistent current, thus aiding in maintaining uniform amplifier operation.

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Most popular questions from this chapter

(a) A differential-amplifier has a differential-mode gain of \(A_{d}=250\) and a common-mode rejection ratio of \(\mathrm{CMRR}_{\mathrm{dB}}=\infty\). A differential-mode input signal of \(v_{d}=1.5 \sin \omega t \mathrm{mV}\) is applied along with a common-mode input signal of \(v_{c m}=3 \sin \omega t \mathrm{~V}\). Assuming the common-mode gain is positive, determine the output voltage. (b) Repeat part (a) if the common-mode rejection ratio is \(C M R R_{\mathrm{dB}}=80 \mathrm{~dB}\). (c) Repeat part (a) if the commonmode rejection ratio is \(C M R R_{d B}=50 \mathrm{~dB}\).

A BJT diff-amp is biased with a current source \(I_{Q}=2 \mathrm{~mA}\), and the circuit parameters are \(R_{C}=10 \mathrm{k} \Omega\) and \(R_{B}=1 \mathrm{k} \Omega\). The transistor parameters are: \(\beta=120, f_{T}=800 \mathrm{MHz}\), and \(C_{\mu}=1 \mathrm{pF}\). (a) Determine the upper \(3 \mathrm{~dB}\) frequency of the differential-mode gain. (b) If the current source impedance parameters are \(R_{o}=10 \mathrm{M} \Omega\) and \(C_{o}=1 \mathrm{pF}\), find the frequency of the zero in the common-mode gain.
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