Question

The amplification or gain of a transistor amplifier may depend on the temperature. The gain for a certain amplifier at room temperature $\left({20.0}^{\circ }\mathrm{C}\right)$ is $30.0$, whereas at ${55.0}^{\circ }\mathrm{C}$ it is 35.2. What would the gain be at ${28.0}^{\circ }\mathrm{C}$ if the gain depends linearly on temperature over this limited range?

Short answer

The gain of the amplifier at 28.0°C is 31.17.

Step-by-step solution

Find the rate of Change

We first find the rate of change or slope of the gain with respect to temperature. This can be done by calculating the difference in gains divided by the difference in corresponding temperatures. The formula to find the rate of change ($m$) is $m=\frac{y2-y1}{x2-x1}$ where x refers to the temperature and y refers to the gain. Substituting the given values, we get $m=\frac{35.2-30.0}{55.0-20.0}$

Calculate the Rate of Change

Carrying out the above operation, we find the rate of change $m=\frac{5.2}{35.0}=0.14857$ gain/°C. This means that for each degree Celsius change in temperature, the gain changes by 0.14857.

Use the Rate of Change to Find the Gain at 28.0°C

Now using the rate of change, we can determine the gain at 28.0°C. We can use the linear equation y = mx + c to calculate this. Here, y is the gain we want to find, m is the rate of change we calculated in step 2, x is the new temperature i.e 28.0°C, and c is the y-intercept. The y-intercept can be calculated by substituting the values of one of the points into the equation. Let's use the point (20.0°C, 30.0). So, c = 30 - 0.14857*20.0. Then we substitute m, c and x into the equation to get the gain at 28.0°C.

Calculate the Y-Intercept and the Gain at 28.0°C

The y-intercept c is calculated as 30 - 0.14857*20.0 = 27.0286. Substituting the values into the equation, we get $y=0.14857\ast 28.0+27.0286=31.17$

Key concepts

Physics of Transistor Amplifier Gain

The transistor amplifiers are essential components in modern electronic devices, where they function to strengthen or 'amplify' electrical signals. The physics underlying their operation involves the control of electronic currents by altering the voltage applied across the device. Importantly, the gain or amplification provided by these transistors can vary with external conditions, such as temperature.

As temperature affects the mobility of charge carriers (electrons or holes), the transistor's ability to amplify signals also changes. A higher temperature typically increases the energy of these carriers, potentially leading to a higher amplification as they move more quickly through the semiconductor material. Yet, this relationship isn't limitless and can be constrained by various factors intrinsic to the transistor's design and the material properties.

Understanding the temperature dependence of amplifier gain is crucial not only for designing stable amplifiers but also for troubleshooting and predicting performance issues in various climatic conditions.

Linear Relationship in Amplifier Gain

In many situations, the relationship between two variables can be approximated as a line, especially over a limited range—this is what we refer to as a linear relationship. When we discuss the gain of a transistor amplifier versus temperature, we're examining how a change in one (temperature) affects the other (gain).

A linear relationship is beneficial for its simplicity, allowing for straightforward predictions about one variable based on the other. The formula describing a straight line, often known as the linear equation, is given as y = mx + c, where y represents the dependent variable (gain, in our case), m is the slope of the line (rate of change), x is the independent variable (temperature), and c is the y-intercept. In the context of our exercise, assuming linearity between gain and temperature enables us to use these constants to describe the behavior of the amplifier across the specified temperature range.

Rate of Change in Amplifier Gain

The rate of change is a concept that is pivotal in understanding how a change in one variable impacts another in a system exemplified by its slope in a linear relationship. In our exercise, the rate of change represents how much the gain of an amplifier varies with every unit increase in temperature. This can be quantified as a numerical value, calculated by dividing the difference in gain by the corresponding difference in temperature. For the given problem, the calculated rate of change of the gain with respect to the temperature is approximately 0.14857 gain/°C. This means that for every degree Celsius increase in temperature, the gain increases by approximately 0.14857. Using this rate, we can make predictions about the gain at any temperature within the range observed, provided that the linear relationship holds true. This concept not only applies to transistor amplifiers but also to a wide range of physical phenomena where changes in one quantity directly influence another.