Authors: Denard Lynch Date: Aug 30 - Sep 28, 2012 Sep 23, 2013: revisions-djl Description: This laboratory explores the behaviour of resistive, capacitive and inductive elements in alternating current (AC) circuits. The student will observe and measure the phase difference between voltage and current in AC circuits with combinations of resistance, capacitance or inductance. Learning Objectives: In this laboratory, the student will: Measure voltage and current in AC circuits, Understand the difference between RMS, peak-to-peak and zero-to-peak values, Measure phase difference between voltage and current in AC circuits, Calculate reactance and impedance based on AC circuit measurements, Verify phasor addition of voltages in an AC circuit Reporting- Use your lab notebook (logbook) to document the key objectives of this laboratory, your theoretical calculations/expectations, Parts List: your equipment and circuit components used any measured values of components your measurements verifying your theoretical expectations (you can paste in screen shots from your ADM where appropriate), use Power Triangles and Phasor Diagrams to help illustrate results (e.g. the relationship between voltages or currents or powers). your observations and comments about how closely your observations matched your expectations, related comments on practical limitations for your observations and comments on possible sources of error Safety Considerations: In addition to general electrical safety considerations, the student should also be aware of the following considerations specific to this laboratory exercise: Denard Lynch Page 1 of 9 Sep 2013
Resistors carrying current will generate heat energy and can be overheated in AC circuits. Power is based on RMS voltages and currents. In all other respects, the same considerations as in DC circuits apply. Unlike DC circuits, capacitors and inductors under AC excitation will conduct current and can have potentials across them even in steady-state (not only transient) conditions. The reactance (opposition to flow), current and voltage present are also a function of the frequency of the excitation source. Always consider these parameters when anticipating potential safety hazards and the required ratings of components. Measurement of AC circuit parameters requires suitable test gear or the selection of the appropriate scale and range. Use of DC instruments or incorrect scales can result in equipment damage or safety risks. Background and Preparation: Continue to familiarize yourself with the capabilities and operation of your Analog Discovery Module, particularly the Measurements feature, which will be very useful in this lab. You will also make use of the differential oscilloscope inputs and Math features to make several observations and measurements in this lab. Review the instructive material given in Lab #1, and refer to any included Help documentation. Please refer to the class notes on AC circuits. A summary of key points is provided in Appendix A at the end of this procedure. Terms: AC Reactance, X Impedance, Z RMS φ Phasor Resonant frequency Alternating Current. Usually used generally to refer to sinusoidally alternating voltage and current Opposition to flow in the AC world. This is a scalar value, measured in Ωs. Opposition to flow in the phasor world. Z is a complexvalued number whose components are the scalar values of resistance (real part) and reactance (imaginary part) Root-Mean-Square. This is the effective value of an AC waveform, or the equivalent DC voltage or current that would provide the same heating effect. Phase angle of a waveform. This is with reference to another waveform, or an arbitrary phase or reference point A representation of an AC waveform that can be used to easily add and subtract AC signals of the same frequency. It is a complex-valued number representing the magnitude and phase angle of the waveform. The frequency at which X L = X C in an AC circuit Denard Lynch Page 2 of 9 Sep 2013
Theoretical calculations of various circuit parameters should be performed as part of the lab preparation (i.e. prior to your lab period). Calculating the expected currents and voltages will also allow you to determine the required ratings for your components (i.e. how much power they must dissipate, how much voltage they must withstand etc.) Please refer to the Class Notes (4) for background theory on impedance, reactance and complex power. ` Procedure: The procedure will involve constructing several different circuits with combinations of resistance, capacitance and inductance, and using your Analog Discovery Module (ADM) to verify the AC voltage and current in each element as well as the phase difference between the voltage and current. You will also measure parameters required to calculate real, reactive and apparent power and compare those to theoretical values. Modeling- (determining what you would expect to see) This will involve using AC circuit theory to predict the circuit parameters for each circuit. The required parameters are described in the detailed procedure below. Please read the entire procedure over carefully before the lab and calculate the expected values of the various parameters using AC circuit theory. Note that you are again using a practical inductor, which has some internal resistance. You should account for this fact in your theoretical calculations, and adjust your observable expectations accordingly. Measurements- I. R-C Circuit Impedance In this part of the procedure, you will observe and measure various parameters of an AC waveform. You also want to verify that a capacitor (unlike in a DC circuit) offers an opposition to the flow of AC current somewhat like a resistor, and that this opposition (Reactance, X) is inversely proportional to the capacitance and the frequency. Also, you will want to verify that the phase difference between the voltage and current in this element is as expected (i.e. which leads which, and by how much?). Use your solderless breadboard and set up the circuit shown in Figure 1 below. A good first step is to examine the circuit, make a list of the parts you will need, obtain the necessary parts and construct the circuit. (The parts for this lab are the same as for the previous lab, except that the resistors are a different value). Refer You may also need to measure the actual value of your components (e.g. using the RLC tester or equivalent) versus their nominal value. For example: Denard Lynch Page 3 of 9 Sep 2013
Component Nominal Value Measured Value 1/4W resistor 220Ω 216.4Ω Capacitor 0.1µF 0.096µF Etc. Your lab instructor will indicate where to obtain the necessary parts if they are not already in your parts kit, and how to measure their actual value. You can use your Analog Discovery Module (ADM) as the AC source for these procedures. (Note: there is a very limited amount of current available from the ADM; ~10mA maximum from the WaveGen. Check your theoretical calculation to make sure this supply isn t overloaded whenever you use it. Also, check the actual waveform across your circuit to detect any variations from you nominal setting.) Digilent ADM WaveGen (W1) Amplitude: 5V Offset: 0V Symmetry: 50% Figure 1 Check your breadboarded circuit for correctness. Start a Waveform Generator window in your ADM and set the output to a 7.20kHz sine wave, amplitude of 5 VAC, zero offset, and connect it to the W1(AC) input shown in Figure 1 (remember to make good ground connections from the ADM to your circuit). In this circuit, you can conveniently determine the current by measuring the voltage across the 220Ω resistor, and dividing by the actual R value; you do not need the 10Ω current sense (CS) resistor in this case. (In cases where this isn t convenient, for instance in a circuit with parallel elements, you can add a small sense resistor, like R1, and connect the differential oscilloscope inputs, 1+ and 1-, to the test points CS+ and CS- respectively. Note: the differential ± inputs are not connected to ground, which allows you to measure voltages across any point in the circuit without inadvertently grounding the wrong point in the circuit!). Using Ohm s Law, the series current in the circuit is simply the voltage measured divided by the value of the sense resistor. You can display this directly on your oscilloscope display by selecting Add a Mathematical Channel from the +C Add Channel button above the traces pane. Select Custom from the pull-down menu and enter C1/10 in the Enter Function dialog box. The settings window for the additional trace will appear in the parameters pane, and you can use the icon in the settings window for that channel (immediately left of the X ) and select A (amperes) from the Units: menu and the display will show ma for units for that channel. You can then add a measurement to display the RMS current from that channel (In the Measurements pane, +Add/ M1/ +Vertical/ ACRMS then +Add Denard Lynch Page 4 of 9 Sep 2013
Selected Measurement ). Alternatively, you can use a Digital Multimeter (DMM) on the ma current scale in series with the circuit to measure AC current separately. (Note: DMMs usually read in RMS values on the AC scales.) Next connect your other oscilloscope channel, 2+ and 2-, across the R-C circuit of interest (use the points V+ and V- shown in Figure 1) to display the AC voltage across the elements of interest ( load ). Again, you can add a measurement to display the RMS voltage impressed across the circuit. Now compare the phase of the two traces, C1 representing the current and C2 representing the voltage across the R-C impedance. Pick similar points on both traces and use the X cursors (X1, X2) to determine the time differential between the voltage and current in this R-C impedance. (You may need to select a smaller horizontal time base to spread the trace and make it easier to adjust the cursors accurately.) Convert this time difference to an angle, which is the phase angle of the complex valued impedance of the R-C element under test.) Remember, the sign of the angle depends on the whether the voltage is ahead of the current (+ve) or behind (-ve). The magnitude of the impedance, Z, is simply the ratio of V/I. Finally, move your C2 scope probes and measure the voltage across the resistor, R2, and then the capacitor, C1, again noting the phase difference relative to the current in the circuit. (Remember: in a series circuit, the current is common to all elements and is a good reference.) you can draw a Phasor Diagram to represent the current and all the component voltage in your circuit. You may want to use a table similar to Table 1 to record your measurements and results in your laboratory notebook (logbook). Table 1 Circuit Parameter I RMS V P-P V 0-P V RMS Δt period Expected Result* Measured Value Δφ Δt/T*360 0 Z φ Z V R2 φ VR2 V C1 φ VC1 * where applicable Comments Between voltage and current T=f -1 (can add a measurement for this on ADM too) Denard Lynch Page 5 of 9 Sep 2013
II. R L Circuit Impedance In this part of the procedure, you will verify that an inductor (unlike in a DC circuit) offers an opposition to the flow of AC current somewhat like a resistor, and that this Reactance, X, is proportional to the frequency and the value of the inductor. Also there the phase difference between the voltage and current in this element is as expected, and opposite to that of the capacitor. Use your solderless breadboard and set up a similar circuit using an inductor as shown in Figure 2 below. Again, gather and check your components and construct the circuit. Continue to use your Analog Discovery Module (ADM) as the AC source for these procedures. (Be careful not to draw too much current.) Digilent ADM WaveGen (W1) Amplitude: 5V Offset: 0V Symmetry: 50% Figure 2 Check your breadboard circuit for correctness. Adjust the Waveform Generator output from your ADM to a 4.50kHz sine wave output of 5 VAC, zero offset and connect it to the W1(AC) input shown in Figure 2 (remember to make good ground connections from the ADM to your circuit). You can use the same connections and setup as in Part I using both input channels to observe and measure the desired circuit parameters. Compare the phase of the two traces, C1 representing the current and C2 representing the voltage across the R-L circuit (or the individual elements as required). Remember, with an inductive circuit you will expect the sign of the impedance s angular argument to be opposite that for the capacitive circuit you measured in Part I. (Also remember, the is phase difference is generally considered to be the smaller of the two choices; i.e. <180 0.) Finally, move your C2 scope probes and measure the voltage across the resistor, R2, and then the Inductor, L1, again noting the phase difference relative to the current in the circuit. You may again want to use a table similar to Table 1 to record your measurements and results in your laboratory notebook (logbook). (The individual voltage and phase measurements will be across the inductor this time.) III. Verifying Phasor summation in an R L C Circuit In this part of the procedure, you will again verify Ohm s Law for AC circuits as well as KVL for AC circuits. This will also allow you to verify phasor addition in a series AC circuit. You will also be able to observe a phenomenon in AC circuits known as resonance. Denard Lynch Page 6 of 9 Sep 2013
Adding the capacitor to the circuit from Part II, you can determine the phasor current and voltages in this circuit (Figure 3). This time, set the frequency of the WaveGen to 6.0 khz. (Note: If you try this circuit with the 220Ω resistor first, you will see the effect of the current limits of the WaveGen output.) Digilent ADM WaveGen (W1) Amplitude: 5V Offset: 0V Symmetry: 50% Figure 3: Phasor Addition R-L-C Again using the current as reference, measure the RMS magnitude and phase of each of the voltages in the circuit, and the overall voltage and phase. Compare this to your theoretical calculations for the individual voltages and the sum. Record relevant values, such as those shown in Table 2 and any others you may need in your lab notebook. Show that the values you observed are substantially what you would expect form your theoretical predictions. Remember that the inductor has some internal resistance which cannot be separated from the true inductance. How does this affect your magnitude and phase measurements? Table 2 Circuit Parameter I RMS V R φ R V C φ C V L φ L V T φ Z Expected Result* Measured Value Comments Common to all series elements Again, a Phasor Diagram makes a good graphical representation of your measurements. Denard Lynch Page 7 of 9 Sep 2013
Optional: Vary the WaveGen frequency while observing the overall voltage and current phase difference, and observe where the total impedance phase angle is 0 0. This is know as the resonant frequency of the circuit; a frequency where V L and V C are equal and opposite directions (i.e. cancel each other out completely) on the phasor diagram! You will also note that at this point, the voltage across the resistor is ~ E, the source voltage. This phenomenon also has some relevance for power, as we ll see in the next lab. Denard Lynch Page 8 of 9 Sep 2013
APPENDIX A: Background Theory Opposition to flow of one of the basic elements in AC circuits is a scalar value measured in Ohms (Ω) and called reactance, X, for capacitors and inductors. Resistance is generally the same value in AC circuits (R AC = R DC ), but the reactance of capacitors and inductors is given by: XC = 1, (C in Farads), (L in Henrys), ωc Ω XL = ω LΩ where ω = 2π frequency( Radians second ) Use of these scalar values will provide some useful information about the individual elements, but for an AC circuit with a combination of R, L and C, we find is very useful to represent the AC voltage and current as phasor quantities (which are represented as complex-valued numbers). Phasors are a transformation of AC voltages and currents form the time domain into a phasor domain: a two-dimensional complex plane, where the operations of adding and subtracting sinusoidal waveforms of the same frequency is greatly simplified. Phasors are generated from the time domain expression by using only the magnitude and phase angle of the wave form: Time Domain to Phasor Domain Conversion Time: v(t) = Asin( ωt +ϕ)v Phasor: V = A V (polar form), (rectangular form) 2 ϕ = A 2 cos(ϕ ) + j A 2 sin(ϕ ) The reverse conversion is also straight forward, except that the actual frequency cannot be determined from the phasor quantity alone. Phasor Domain to Time Domain Conversion Phasor: V = B θv (polar form) or V = (X +jy)v (rectangular form) Time: v( t ) = 2Bsin(ωt + θ )V, or v( t ) = 2 X 2 + Y 2 sin ωt + tan 1 Y X V The opposition to flow in the phasor domain is called the impedance, Z, and is also measured in Ohms and is defined as: phasor _ voltage, or Z = V, where we use the standard units of measure (Ohms, phasor _ current I Volts, Amperes), but each quantity is a complex valued number (usually represented as a magnitude and angle, if the polar form is used, or an x and y component if the rectangular form is used, in a two-dimensional complex plane). In rectangular form, the complex impedance can be written in terms of the scalar resistance and reactances: ( ) Z = R + j XL XC, and can be shown graphically on a Impedance Diagram (again, on a two-dimensional complex plane). Denard Lynch Page 9 of 9 Sep 2013