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### CSE 323a: Measurements &Testing (1)a

2013-2014

References:

- William Dunn, Introduction to Instrumentation, Sensors, and Process Control, Artech House, 2006.
- William Bolton, Instrumentation and Control Systems, Elsevier, 2004.
- Curtis Johnson, Process control instrumentation technology, Prentice-Hall, 6th edition, 2000.

Resistors, capacitors, and inductors

The three basic passive elements used in electrical circuits.

Let us have a look at some useful slides from the course 6.091 offered at the Department of Electrical and Computer Science, MIT, available at:

http://ocw.mit.edu/courses/electricalengineering-and-computer-science/

Resistors

- Used as loads in electrical circuits.
- Resistor parameter: resistance, tolerance, and power rating.
- Standard values:

10 12 15 18 20 22 27 33 39 47 56 68 82.

- Common tolerances: ±5%, ±1%.
- Resistor are color coded.

Capacitors

Used as dc blocking devices, in level shifting, integrating, differentiating, filters, and delay circuits.

Capacitors

- Capacitors range from 1 pF (10-12) to 100,000 µF (10-1).
- Typically, capacitors larger than 1 µF are polarized.
- All capacitors have maximum voltage ratings.

How to read puff capacitor codes

- Source: http://drakedev.com/pic/capacitors.php

Inductors

- Used as current limiting devices.
- Found in relays, audio to electrical conversions, electromagnetic devices, light dimmers, and tuned circuits.
- They are also the basis for transformers and motors.

3.2 Circuits with R, L, and C

3.2.1 Voltage Step Input

When the current in the resistor is maximum, the voltage across it is maximum, given by E = IR. i.e. the voltage is said to be in phase with the current.

- For the capacitor, the voltage is zero when the current is maximum, and the voltage is a maximum when the current is zero. In this case, the voltage lags the current, or there is a phase shift between the voltage and the current of 90°.
- The voltage across the capacitor builds up exponentially, at a rate determined by the values of R and C.

Similarly, the voltage and current in the resistor are in phase, but in the inductor are out of phase. The voltage leads the current by of 90°.

- The voltage across the resistor increases exponentially, at a rate determined by the value of L and R.

Time constant

- In RC circuit, the voltage across the capacitor, while charging, is given by:

where E is the source voltage.

- and while discharging, is given by:

Time constant

- It is the time taken by the response to reach 63.2% of its full change.
- The time constant of RC circuit is given by RC, while for RL circuit, it is L/R.
- Practically, the response will complete its full change in 4 to 5 time constants.

Time constant

- Applies not only to electrical circuits, but also to sensor outputs when there is a change in the measured variable.
- The output signal from the sensor changes exponentially, so that there is a delay before the sensor output reaches its final value.

3.2.3 Sine-wave inputs

- Assuming that the circuit is capacitive.

In series RLC circuit, the same current will flow through all three devices.

- When an ac sine wave is applied to RLC circuits, the same phase shift between voltage and current occurs as when a step voltage is applied:
- IR and VRare in phase;
- ICleads VCby 90°;
- ILlags VLby 90°;
- That is,
- VCand VLare 180° out of phase; and
- VCand VLare 90° out of phase with VR

Vector addition

- Since the voltages and currents in capacitors and inductors are not in phase, they have impedance and not resistance.
- Impedance and resistance cannot be directly added.
- However, they can be combined using vectors.

E: supply voltage

- VR, VL , VC are voltage across resistor, inductor, capacitor

Example

- What is the current flowing in the series RLC if R = 27 kΩ, C = 2.2 nF, L = 33 mH, E = 20V and the input frequency = 35 kHz?

Resonance

- XLand XCare frequency dependent.
- As the frequency increases, XL and XC.
- A frequency can be reached where XL= XC, and the voltage across these components are equal, opposite, and cancel.
- At this frequency, Z=R, E=IR, and the current is maximum.
- This frequency is called the resonant frequency of the circuit.
- At resonance:

When the input frequency is below the resonant frequency XC> XL, the circuit is capacitive.

- Above the resonant frequency XC< XL, the circuit is inductive.
- Plotting the input current against the input frequency shows a peak at the resonant frequency as shown:

Example

What is the resonant frequency of the series RLC if R = 27 kΩ, C = 2.2 nF, and L = 33 mH? What is the current at this frequency?

The current can be obtained as (at resonance Z = R)

I = E/R = 20/(27 x 103) = 0.740 mA

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