Root-Mean-Square (RMS), Peak, and Peak-to-Peak Values, Measurements with Oscilloscope

Transcription

1 Salman bin Abdulaziz University College of Engineering Electrical Engineering Department EE 2050 Electrical Circuit Laboratory Root-Mean-Square (RMS), Peak, and Peak-to-Peak Values, Measurements with Oscilloscope Objectives: 1. To understand the relationship between rms and peak-to-peak values of a sinusoidal wave. 2. Familiarization with Oscilloscope in measuring different parameters and evaluation of Phase difference, Angle, Time Period and Frequency under sinusoidal excitation Equipments: 1. DC power supply and function generator, 2. Digital multi-meters and Oscilloscope, 3. Resistors: 100 Ω, 470 Ω and decade box, 4. Inductors: 17.5 mh with 1000 tuns air core. 205 Electrical Circuit Laboratory Experiment # 2 Note: Please record any damage, if exists any while performing the experiment. You can also write the difficulties you are confronted with when using the equipment, and your suggestions and critics related with the equipment you used. Background: I. RMS, Peak, And peak-to Peak values Alternating current (AC) circuits have sources that change in value with time. The most common AC source is a sine wave generator. Two common ways to describe a sine wave voltage are: Peak -to- peak value. The oscilloscope can be used and the distance from the top of the wave to the bottom will be the peak-to-peak value of the signal. RMS (root means square) value. A digital multi-meter meter (DMM) can be use to obtain the RMS value of an AC signal. A digital multi-meter meter measures the AC voltage as if it were a DC voltage. What DC voltage would cause the same effect (power dissipation in a resistor) as the AC voltage being measured? This value is called the effective voltage or RMS value. Mathematically the RMS value can be calculated by squaring the voltage, finding the average (mean), then taking the square root. Experiment # 2 Page 1

2 Therefore, RMS value of an ac voltage or current x(t) is defined by the relation: x 1 2 RMS ) T = x ( t dt T where T is the period of the ac wave o It is known for a sine wave that: 1 x RMS = X PP where X PP is the peak to peak value 2 2 Or 1 X RMS = X P where X P is the peak value 2 This mean that X P = Note: The multi-meters used in this lab are, generally, rectifying-type meter does not indicate correctly the RMS value for non-sinusoidal wave forms. II. Cathode-Ray Oscilloscope (CRO) CRO (cathode ray oscilloscope), often referred to as a scope, is the most powerful tool available for measuring electrical quantities associated with electronic circuitry. It is such an important instrument that a thorough understanding of its operations is absolutely essential for any electrical engineer. The display on the CRO screen is created by an electron beam exciting a phosphor coating on the inside face of the CRT (cathode ray tube) shown in Fig. 1. The electrons generated by the cathode are accelerated and focused into an electron beam. When the beam strikes the fluorescent screen, it emits a tiny spot of visible light. The position of the electron beam (hence the spot on the screen) generated by the electron gun is controlled by the vertical and horizontal deflection plates. The spot on the screen changes its position depending on the voltages applied to the vertical and horizontal deflection plates. Continuously changing signals on vertical and horizontal deflection plates cause the beam to trace out a path on the screen. Even though only one spot is highlighted on the screen by the electron beam at any given moment, the persistence of the phosphor makes the path appear to be continuous. The path must be retraced frequently in order to render a steady display. Based on the operations of the CRT discussed above, there are four major electronic subsystems needed to generate control signals for the CRT: Display control for controlling the intensity and focus of the electron beam. Vertical control for controlling the up-down deflection of the electron beam by external signals. These signals are amplified through a series of electronic circuits controlled by the front panel knob whose scale is calibrated in VOLTS/DIV (called sensitivity). Experiment # 2 Page 2

3 Horizontal control for controlling the right-left deflection of the electron beam by either internal or external signals. If the external input is selected for the horizontal deflection system, the scope is operating in the x-y mode and the horizontal scale is calibrated in VOLTS/DIV. If the internal source is chosen, the scope is operating in the y-t (or timebase) mode and the horizontal scale is given in SEC/DIV. There is an internal circuit (known as the sweep circuit) producing a time-calibrated signal (sweep signal) for the horizontal deflection plates. Trigger control for controlling the starting point of each trace when operating in the y-t mode. In order to obtain a steady display pattern, each trace (and there could be thousands of traces per second) must duplicate exactly the previous trace. Thus, each trace must begin from precisely the same point of a periodic waveform in order for traces to fall on top of each other. The trigger subsystem permits one to select the exact starting point on each trace. (a) Single-beam tube (a) Dual-beam tube Fig. 1. Cathode Ray Tube (CRT) Experiment # 2 Page 3

4 The magnitude of unknown voltage using an oscilloscope is given by: Volts (unknown) = Voltage sensitivity (volts/division) x Deflection (vertical divisions) The current measurement is done by measuring the voltage across a resistor and dividing by the value of that used resistor. The used resistor should be much less than the total resistance of the circuit. III. AC Circuit analysis Analyzing AC circuits in time domain (real analysis) is a difficult task. So different methods have been introduced; among these is what is called phasor algebra (complex analysis). The different methods of circuits analysis as KVL, KCL, voltage division rule, current division rule, superposition, Thevenin s theorem, Norton s theorem, maximum power transfer theorem, and others, are still applied to analyze AC circuits using the phasor vector analysis. Remember that: Z L = jωl = ωl 90 (impedance of an inductor in phasor form) = (impedance of a capacitor in phasor form) Z R = R = R 0 o (impedance of a resistor in phasor form) Z = Z 1 + Z 2 + (total impedance in series AC circuit) PF = cos θ = P / (V I) = R / Z From impedance diagram: 2 2 T L C PF : Power Factor P : average power delivered (active) θ : the angle of impedance vector V : magnitude of the supply voltage I : magnitude of current Z = R + ( X X ) = magnitude of the impedance of RLC circuit. θ = tan -1 (X L - X C ) / R= angle of the impedance ZT Where X L = ω L = reactance of the inductance L X C = 1/ ω C = reactance of the capacitor C V = V 2 R + ( V V ) C L 2 Where V R = magnitude of the voltage across the resistance R V L = magnitude of the voltage across the inductance L V C = magnitude of the voltage across the capacitance C Experiment # 2 Page 4

5 Procedure I: Familiarization with the CRO 1. Turn on the oscilloscope and observe the function of the following control switches. a) Focus b) Intensity c) Horizontal position, and d) vertical position 2. Using, variable DC supply, connect the circuit shown in Fig. 2. Table 1: Fig. 2. Measurements of DC voltage by Oscilloscope a) Locate the zero line on the scope screen. b) Use ch-1 switch to be at dc mode c) Vary the supply voltage to obtain the values recorded in Table1, and fill up the table with the missing data V dc in volts Vertical scope deflection (divs.) x vertical sensitivity (V/div) = measured voltage by CRO 1.. =. 5.. = =. In case of V dc = 10 volt, plot what you observe from the CRO on the following axes: Experiment # 2 Page 5

6 3. Turn the dc voltage supply off and keep your circuit connected. 4. Switch the display mode switch to ch 2 and using a function generator as an AC supply, wire the circuit shown in Fig.3. Fig. 3. Measurements of AC voltage by Oscilloscope a. Use ch-2 switch to be at ac mode b. Display a sinusoidal signal on the CRO c. Calculate the peak value and period and record them in Table 2 Table 2: Ac signal Sinusoidal Waveform Peak voltage (V P ) = vertical Deflection (divs.) x vertical Sensitivity (V/div) V P = = Period (T) = Horizontal deflection (divs) x Horizontal sensitivity (ms/div) T = = d. Calculate the waveform frequency (f) and RMS value (V RMS ) f =. (Hz), V RMS Calculated =. (V) e. Using a DMM measure the RMS value of the waveform V RMS Measured =. (V) f. Regarding RMS value, does your measurement agrees with the calculated value g. plot what you observe from the CRO on the following axes: Experiment # 2 Page 6

7 5. Return the dc supply ON for the circuit of Fig.2. Connected to ch 1 and switch the display mode switch to DUAL (which means that the two displayed signals will be traced simultaneously on the scope). plot what you observe from the CRO on the following axes: 6. Then switch the mode to (ADD) mode (which means that the displayed signal will be the addition of signals entering channels 1 and 2 simultaneously). plot what you observe from the CRO on the following axes: Procedure II: Series RL Circuits 1. Construct the circuit shown in Fig. 4 and measure the values of E, V L, V R and I using a DMM. E = V, V L = V, V R = V, I = A Notice: The measured values obtained are the rms values. 2. Display V R and E on the CRO. Plot the waveforms on the following axes: Experiment # 2 Page 7

8 Fig. 4. Circuit used in procedure II. 3. Measure the phase angle θ, and indicate the leading waveform. Explain why? Lissajous pattern on the oscilloscope can be used to measure the phase angle between two equal-amplitude deflection signals defined as: θ = sin -1 (L 1 /L 2 ) ; see Fig. 3 (b) Note: You would display the above pattern by using X-Y mode of the oscilloscope. θ = rad = degree, 4. Calculate the circuit impedance Z using the relation: Z = E / I. 5. Draw the impedance diagram used in your calculation. 6. Using phasor analysis prove that E = V R + V L PF =... Impedance diagram Voltage diagram Experiment # 2 Page 8