Midterm Exam 3 -- E84, Fall 2008

• Due 5pm Tuesday 5/13.
• Take home, open everything except discussion.
• Mark your start and end times. Don't spend more than 3 hours.
• Mark your name and question number clearly on top of each page. Indicate the total number of pages submitted.
• When solving a problem, list all the steps. In each step, describe what you are doing in English, then show the calculation and the result of the step. A final answer, even if correct, without evidence of the steps leading to the answer will not receive credit.

1. Problem 1. (33 points)

An inductive load is connected to a AC voltage source of 220 Volts (RMS) and 50 Hz. It is known that the average power is 1.1 kW and the power factor is . If the power factor is to be improved to 0.866 by including a capacitor in paralle with the load, what is the its capacitance?

Solution: Given the power factor , the phase angle of the inductive impedance is . Also, given the average power

we can get the current:

The impedance is therefore

As

we have

Solving these equations we get

With a capacitor in parallel with the inductive load , the total impedance becomes:

In order to satisfy the required power factor , the phase angle of the total impedance has to be , i.e.,

where is the phase of the numerator , known to be , therefore the angle of the denominator has to be as well:

i.e.,

Substituting and and solving for we get

2. Problem 2. (33 points)

Find the DC operating point of the transistor circuit given below, where , , , , and . If you find the DC operating point is not in the middle of the linear region of the output characteristic plot, modify so that the DC operating point is in the middle of the linear region (to maximize the dynamic range of the AC output).

Solution:

Find the load line: When , , when ,

The DC operating point is in saturation region.

Now we modify to move Q-point to the middel point where .

We then plug these into

and solve it for to get .

3. Problem 3. (34 points)

The circuit shown below is called Darlington transistor amplifier which is composed of two transistors and with their collectors connected and the emitter of connected to the base of . Assume and both transistors have .

1. Give the expressions of , , , all in terms of and .
2. Given , find so that the DC operating point is in the middle of the linear region of the output V-I characteristic plot.
3. Given and the same as before, find the proper DC voltage so that the Q point is in the middle of the linear region.
For simplicity, assume for both transistors .

Solution:

1. , , , ,

2. In order for , we need to have , i.e., .

3. Solving this we get