Ten problems to solve before 6 December, 2010

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Hope Golden
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1 Ten problems to solve before 6 December, 2010 From Bengtsson: 1. (11.1) a) Calculate the total capacitance Ctot between the centre conductors of two coaxial cables, each 1 m long, separated by 10 cm. The two shields are not connected to ground. Help: Any signal must pass through three capacitances in series; one of these is the shield-to-shield capacitance. Some further information needed: This type of coaxial cable has a capacitance between the centre conductor and the shield of 100 pf per m cable, and an outer radius of 0.1 inch = 2.54 mm. The capacitance between two parallel conductors, each with a diameter d, at a distance of D from each other, is C = πεεo/ln(2d/d) [Fm -1 ]. b) Which of the three capacitances in a) has the largest effect on the total Ctot? c) The two cables in a) are connected to two signal sources as shown on the right, and the right cable is connected to an amplifier with input impedance Zin = R = 100 kω. The left cable is only connected to a source. If we assume that the signal source impedances are infinitely large, what is the input voltage to the amplifier arising from the left signal source? (Include the frequency dependence, please!) 2. (2.2) We use a Pt-100 sensor to measure temperature in a control circuit. The distance between the sensor and the regulator is 5 m and the sensor is connected using copper wires with effective diameters of 0.5 mm. What is the temperature error (in K) due to the resistance of the connecting wires? (You can assume that the temperature is not very far from 0 o C. The resistivity of copper is taken as Ωm.)

2 From Problems in Circuit Theory: 3. The circuit below illustrates a common industrial application where the signal is transformed into a current and sent by the current source (left) through a double wire (10 Ω wire resistances) to a measurement instrument (represented by the resistance right). An interfering voltage, capacitively or inductively coupled to the signal wires, is represented by a voltage source in series with the signal. a. What is the output voltage UAB between points A and B due to the current source alone? b. What is the output voltage UAB due to the interfering voltage source alone? I = 10 ma R1 = 100 Ω R3 = R4 = 10 Ω U = 10 V R2 = 1 M Ω 4. Calculate the gain G = Uout/Uin as a function of frequency for the circuit on the right. (This can be considered as a very simple active low-pass filter). 5. The differential amplifier (solved last time) gives an output voltage relative to ground which is proportional to the difference between the two input voltages. The circuit below is similar, but gives an output current I to ground. Show that the current I is proportional to the difference between the input voltages and that it does not depend on the load resistance RL attached! (This circuit is one possible way to realize a practical, almost ideal current source.)

3 From Bentley: 6. (6.4) A thermocouple giving a 10 mv DC output is connected to a high impedance digital voltmeter some distance away. A difference in potential exists between the earth points at the thermocouple and the earth point at the voltmeter. An equivalent circuit is shown below. a. Calculate the r.m.s. value of the series mode and common mode interference voltages at the voltmeter input. b. Given that the DVM has a common mode rejection ratio of 100 db, find the minimum and maximum possible measured voltages. (Help: Assume the voltmeter measures instantaneous values, and remember the difference between amplitude and r.m.s. value!) 7. (9.16) a The figure below shows a four-lead bridge circuit, which is a British standard circuit for high-accuracy measurements of temperature using resistance sensors. This circuit uses special sensors with an extra pair of connecting wires, not connected to the sensor itself but included together with the sensor wires inside the connection cable. Rc is the resistance of the leads connecting the sensor to the bridge circuit. Show that Eth VS(R0/R3) αt, i.e. the bridge output voltage is unaffected by changes in Rc. b. There is a related German standard circuit intended for similar applications, but using a sensor with three identical connecting wires. The circuit is shown on next page; show again that the bridge output voltage is unaffected by changes in Rc.

4 Other problems: 8. We need a very-low-noise amplifier for a special application, and we buy such an amplifier with a noise bandwidth of 2.4 MHz and an amplification of To test it, we take a set of "perfectly" shielded standard resistors, connect each if these to the amplifier input, and measure the rms output noise voltage in each case. The data are given in the Table below. Calculate the following noise parameters of the amplifier: Equivalent noise input voltage, equivalent noise input current, optimum source resistance Ro, and noise factor F at Ro. Table of output noise voltages: R (Ω) Urms (mv) R (Ω) Urms (mv) You have on your bench a Thermistor (NTC resistor) and you want to measure temperature. According to its data sheet, this thermistor has a resistance that varies as R = A exp (B/T), with B = 4000 K. With your ohmmeter you measure that it has a resistance of 10 kω at +20 o C. a. What is the resistance of this sensor at 100 o C? b. What is the temperature coefficient of the resistance at +20 o C?

5 10. A very common source of resistively (galvanically) coupled interference is that we make several connections in a circuit, using a common "ground" (return lead), as shown on the right. (This circuit is very good from an economic point of view to connect N sensors we need only N+1 wires, instead of the 2N ones needed for a circuit with separate return leads for every sensor!) Suppose we connect a small heater and a temperature sensor as shown. The heater has a resistance of 49.8 Ω, and the sensor will give a DC voltage of 200 mv at the desired heater temperature. If the three wires have resistances R1 = R2 = R3 = 0.1 Ω, the amplification A is 5, and the voltmeter has a very high input impedance, what is the amplifier output voltage when the heater power is a) switched off, b) switched on? (Note sensor polarity!) The circuit shown is a typical simple temperature controller the amplifier output is used to change the switch, such that the heater switches on when the temperature (and the amplifier output voltage) falls below a pre-determined desired value.) Answers: 1. a. 6.6 pf b. Inter-shield capacitance c. U = 10 5 U1/(10 5 j( /ω) K 3. a. 1 V b. 1 mv 4. -5/( jω) 6. a. Series voltage 3.14 mvrms, common mode voltage 100 Vrms. b mv and mv. 8. uen = 2 µvrms, ien = 0.8 narms, Ro = 2.5 kω, Fmin = 1.08 (= 0.7 db) 9. a. 535 Ω b K a V b V

Differential Amplifier : input. resistance. Differential amplifiers are widely used in engineering instrumentation

Differential Amplifier : input resistance Differential amplifiers are widely used in engineering instrumentation Differential Amplifier : input resistance v 2 v 1 ir 1 ir 1 2iR 1 R in v 2 i v 1 2R 1 Differential