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Chapter 29: Problem 47

Solve the given problems. The mutual conductance (in \(1 / \Omega\) ) of a certain electronic device is defined as \(g_{m}=\partial i_{b} / \partial e_{c} .\) Under certain circumstances, the current \(i_{b}\) (in \(\mu \mathrm{A}\) ) is given by \(i_{b}=50\left(e_{b}+5 e_{c}\right)^{1.5} .\) Find \(g_{m}\) when \(e_{b}=200 \mathrm{V}\) and \(e_{c}=-20 \mathrm{V}\).

Short Answer

Expert verified
The mutual conductance \( g_{m} \) is \( 3750 \, \mu \Omega^{-1} \).

Step by step solution

01

Identify the Function

The mutual conductance \( g_{m} \) is defined as \( \frac{\partial i_{b}}{\partial e_{c}} \). The given function is \( i_{b} = 50(e_{b} + 5e_{c})^{1.5} \). To proceed, we need to calculate the partial derivative of this function with respect to \( e_{c} \).
02

Apply the Chain Rule

To find \( \frac{\partial i_{b}}{\partial e_{c}} \), use the chain rule. The inner function is \( u = e_{b} + 5e_{c} \), so \( i_{b} = 50u^{1.5} \). The derivative \( \frac{di_{b}}{du} = 75u^{0.5} \) using the power rule.
03

Differentiate with Respect to \( e_{c} \)

Now differentiate \( u = e_{b} + 5e_{c} \) with respect to \( e_{c} \): \( \frac{du}{de_{c}} = 5 \). The partial derivative is \( \frac{\partial i_{b}}{\partial e_{c}} = 75u^{0.5} \cdot 5 = 375u^{0.5} \).
04

Substitute the Known Values

Substitute the values \( e_{b} = 200 \) V and \( e_{c} = -20 \) V into \( u = e_{b} + 5e_{c} = 200 + 5(-20) = 100 \).
05

Compute \( g_{m} \)

Substitute \( u = 100 \) into \( \frac{\partial i_{b}}{\partial e_{c}} = 375u^{0.5} \), which gives \( 375(100)^{0.5} = 375 \times 10 = 3750 \).
06

Conclude the Calculation

Thus, the mutual conductance \( g_{m} \) is \( 3750 \, \mu S \), which means \( g_{m} = 3750 \, \mu \Omega^{-1} \).

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Key Concepts

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

Partial Derivatives
Partial derivatives are used to measure how a multivariable function changes as one of the variables changes, while the others are kept constant. In this exercise, we have a function for the current \( i_b \) given by \( i_b = 50(e_b + 5e_c)^{1.5} \). Here, \( e_b \) and \( e_c \) are variables representing voltages.
  • The partial derivative \( \frac{\partial i_b}{\partial e_c} \) tells us how the current \( i_b \) changes as the voltage \( e_c \) changes, while \( e_b \) is held constant.
  • This is crucial in understanding how small changes in input affect output, especially in electrical systems.
By isolating the impact of just one variable on the function, partial derivatives become powerful tools in mathematical modeling and analysis.
Chain Rule
The chain rule is a fundamental calculus concept that allows us to differentiate composite functions. It's especially useful when dealing with functions within functions, as seen in our current equation.
  • Here, \( i_b = 50(e_b + 5e_c)^{1.5} \) includes an inner function \( u = e_b + 5e_c \).
  • We first differentiate \( i_b \) with respect to \( u \), which gives \( \frac{di_b}{du} = 75u^{0.5} \).
  • Then differentiate \( u \) with respect to \( e_c \), resulting in \( \frac{du}{de_c} = 5 \).
  • By multiplying these results, we apply the chain rule: \( \frac{\partial i_b}{\partial e_c} = 75u^{0.5} \cdot 5 = 375u^{0.5} \).
This method effectively deals with nested functions, allowing for accurate derivative calculations in complex scenarios.
Electrical Engineering
In the realm of electrical engineering, understanding how circuits and devices respond to changes is essential. This often involves calculating how current or voltage changes with respect to others, quantified by derivatives.
  • In this problem, the mutual conductance \( g_m \) gives insight into how small changes in one voltage \( e_c \) affect the current \( i_b \).
  • It is a form of gain analysis, helping engineers design circuits better by understanding sensitivity to input changes.
  • This specific mutual conductance measure, when calculated, results in \( g_m = 3750 \, \mu \Omega^{-1} \), indicating a significant sensitivity.
Such calculations are integral in device characterization and performance optimization in a myriad of electronic devices.
Mathematical Modeling
Mathematical modeling in engineering involves using equations and mathematical techniques to describe and predict real-world behaviors. This requires understanding how systems respond to various inputs and conditions.
  • The given expression \( i_b = 50(e_b + 5e_c)^{1.5} \) models the behavior of an electronic device's current in terms of voltages \( e_b \) and \( e_c \).
  • Modeling such relationships allows engineers to simulate conditions without physical prototyping, saving cost and time.
  • Using partial derivatives and techniques like the chain rule, engineers can further analyze how changes in system inputs influence outputs.
  • In this problem, \( g_m \) showcases the practical application of these calculations in predicting device behavior under changing conditions.
Mathematical modeling thus combines mathematical principles with practical engineering needs to design and optimize real-world applications effectively.

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