DELFT UNIVERSITY OF TECHNOLOGY Faculty of Electrical Engineering, Mathematics and Computer Science EXAM Amplifiers and Instrumentation (EE1C31) April 18, 2017, 9.00-12.00 hr This exam consists of four questions, each with a number of sub-questions. The number of points per question is indicated. The total number of points in the exam amounts to 40. During this exam you are allowed to use P.P.L. Regtien, Electronic Instrumentation and the reader J. Hoekstra, Versterkerschakelingen only in their original form: no print-out, photo-copy or any other material is allowed. The use of a (graphical) calculator is allowed. Examiners / reviewers: Dr. ir. Michiel Pertijs, Dr. Jaap Hoekstra Good luck! For each sub-question, please provide a short motivation for your answer. Start EVERY QUESTION on a NEW ANSWER SHEET. Write on each answer sheet your name and student number. Page 1 of 6
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Question 1 Op amp as a nullor implementation With an operational amplifier (op amp) various amplifiers can be built. In these amplifiers, the op amp is an implementation of the circuit element nullor. a) Write down the chain matrix of the nullor. (1 pt) b) Is it possible to define the impedance matrix for the nullor? Give a brief motivation of your answer. (2 pt) c) Now, we consider the non-inverting voltage amplifier circuit with an op amp. The op amp can be considered ideal; meaning: it has an infinite amplification, a zero input conductance, and a zero output resistance. Give an expression for the closed-loop gain. (1 pt) d) Again, we consider the non-inverting voltage amplifier circuit. As soon as the amplification is not infinite, but A, the op amp is non-ideal. Give an expression for the open-loop gain in case of a non-ideal op amp with amplification A, a zero input conductance, and a zero output resistance. (2 pt) e) In reality, the op amp will have amplification A, an input conductance G in, and an output resistance R o. Determine the K-matrix for this non-ideal op amp. (4 pt) Page 3 of 6
Question 2 Op amp built using bipolar transistors (new answer sheet please) The figure below shows a very simple op amp amplifier circuit. T1 and T2 are NPN transistors having both a h FE of 100; T3 is a PNP transistor also having a h FE of 100. All three transistors are biased in such a way that the absolute value of the current through them is 1 ma in the operating point. Answer the following subquestions: a) What kind of amplifier circuit is shown in the above figure? (1 pt) b) What is the amplification factor, assuming that the nullor is ideal? (1 pt) c) Calculate the g m of transistor T2. (2 pt) d) Calculate the voltage at the base of T3 in the operating point. (2 pt) e) The output of the differential pair is taken as a single-ended signal from T1. Because of symmetry the output signal is the input signal multiplied by g m R C /2. Calculate the total amplification of the op amp nullor, where you can neglect the effect of the load of the nullor. (If you do not have an answer for the value of g m, take g m =1). (4 pt) Page 4 of 6
Question 3 Battery current measurement (new answer sheet please) R S V s As shown in the circuit diagram on the right, the current I L provided by a battery to a load is measured by measuring the voltage drop V S across a resistor R S in series with load. V bat I L Given: 2.7 V V bat 3.3 V I L 1.0 ma R S = 100 Ω load a) The voltage V S is measured using a multi-meter. Determine the maximum error in the measured load current if one calculates the load current based on the reading of the multi-meter without taking into account that the multi-meter has an internal resistance of 1 MΩ. (2 pt) As shown below, the voltage V S is now measured using an instrumentation amplifier with a voltage gain of 100. The output of this amplifier is fed into an ADC to digitize the measurement result. R S V s instr. amplifier ADC digital output signal V bat I L load V out b) Assume that the instrumentation amplifier is ideal, except that it has offset. Calculate the maximum equivalent input offset voltage for which the resulting error in the measured current is less than ±1.0 µa. (1 pt) c) Assume that the instrumentation amplifier is ideal, except that it has finite CMRR. The finite CMRR causes V out to change as V bat drops from 3.3 V (fully charged) to 2.7 V (almost empty), even if I L is constant. Assume for simplicity that I L = 0. Determine the minimum CMRR for which the drop in V out is equivalent to an error in the measured current of less than ±1.0 µa. (3 pt) d) The instrumentation amplifier has a bandwidth of 1.0 MHz and an equivalent input noise voltage spectral density of 3 nv/ Hz. Calculate the rms noise voltage at the output of the instrumentation amplifier due to the combination of the thermal noise of the resistor R S and the noise added by the instrumentation amplifier. Assume 4kT = 1.6 10-20. (3 pt) e) De reference voltage of the ADC has been chosen such that the full scale of the digital output signal corresponds to the maximum load current of 1.0 ma. The ADC has a resolution of 12 bits. What is the maximum error in the measured current (in µa) due to the quantization in the ADC? (1 pt) Page 5 of 6
Question 4 Capacitive displacement sensor (new answer sheet please) movable plate A capacitive displacement sensor consists of a fixed bottom plate and a movable top plate. At a distance x = 1.0 mm, the capacitance C x is 1.0 nf. x fixed plate C x Together with a resistor R of 10 kω, the sensor forms an RC filter that is driven by a sinusoidal voltage source U b with a peak amplitude of 1.0 V and a frequency of 1.0 MHz. When C x changes due to a displacement of the movable plate, the filter s corner frequency changes, and hence the displacement can be detected as a change in the amplitude of the output voltage U o. + U b -- R C x U o a) Determine the sensitivity of U o (in V/mm) for small displacement around x = 1.0 mm. (2 pt) b) The output is connected to an oscilloscope using a coax cable with a capacitance of 50 pf. The oscilloscope has an input impedance of 1 MΩ // 20 pf. Determine for x = 1.0 mm the relative error (in %) in the amplitude of the output signal due to the loading by the cable and the oscilloscope. (2 pt) Next, the circuit shown below is used to read out the sensor. The voltage source U b still has an amplitude of 1.0 V and a frequency of 1.0 MHz. Assume for the remaining questions that C x has a nominal value of 1.0 nf and a sensitivity of 100 pf/mm. C F + U b -- C x U o c) Determine the value of the feedback capacitance C F for which the sensitivity of the output voltage U o equals 10 mv/mm. Assume that the opamp is ideal. (2 pt) d) The feedback capacitor has a temperature coefficient of 5.0 10-4 K -1. The other components can be assumed to be temperature independent. The sensor is calibrated at 25 C by measuring the output amplitude at x = 1.0 mm. Determine the maximum error in the measurement of x (in µm) for x = 1.0 mm if the sensor is used in the temperature range from -50 C to 125 C. (2 pt) e) In practice, the circuit above will not work well due to the opamp s bias current of 10 na. Propose a change to the circuit that will solve the problem caused by the bias current. If you add any components, indicate how you would choose their value. (2 pt) --- end of the exam --- Page 6 of 6