ECE2006 LABORATORY 9 FREQUENCY RESPONSE OF RLC CIRCUITS OBJECTIVES The learning objectives for this laboratory are to give the student the ability to: • Experimentally determine the resonant frequency • Use the oscilloscope to display the difference of two AC signals • Use the oscilloscope to measure phase angle differences between AC signals • Use the digital multimeter to measure AC voltage magnitudes • Calculate impedances from phase and magnitude measurements • Use the PSPICE AC analysis tool to analyze AC circuits REFERENCES • Alexander/Sadiku, “Fundamentals of Electric Circuits – 2nd Edition”, 2003, McGraw-Hill • See above reference, Appendix D.4: pp.A-38 thru A-46, for PSPICE analysis of RLC transient response. BACKGROUND Phasors In sinusoidal AC systems, it is often tedious to calculate circuit voltages and currents using the standard tools of KVL, KCL and Ohm’s Law because the signals are described as sinusoidal functions of time. It is easier computationally to convert the sinusoids into Phasor form prior to applying circuit analysis techniques. To convert an AC sinusoid into its corresponding phasor, it is necessary only to know its magnitude and phase angle. The magnitude of a pure sinusoidal AC signal will be it’s peak value (with respect to ground) divided by the square-root of 2. This is called its Root-Mean-Square value. Example: v(t) = 141.4 cos (377 t + /6) Volts (a voltage in Time Domain) Voltage Magnitude = 141.4/2 = 100 Volts RMS Voltage Phasor , V = 100 /_ +30˚ Volts The phase of a pure sinusoidal AC signal must be measured with respect to a reference, another AC signal. If no other AC signal is of interest, one can assume that the signal being studied is the reference and therefore its phase angle is zero. In the example above, reference information is given as [ + /6]. Therefore, the phase angle will be + /6 Radians or +30 degrees.Example: v(t) = 141.4 cos (377 t - /4) Volts Voltage Phase Angle = (- /4)*(360/2) = -45 Degrees. Voltage Phasor , V = 100 /_ -45˚ Volts Viewing series component voltages with the oscilloscope The oscilloscope can be used to view separate component voltages with the two probes. However, both probes must share a common ground (never float the black lead). Thus, the voltages of two components in series can not be viewed directly with each of the two probes. Should one desire to view the voltage present across the capacitor and the resistor (at the same time) in the figure to the right, the Math function of the oscilloscope must be used. To do this connect the oscilloscope as shown in the figure, press the Math button and use the buttons on the right of the oscilloscope screen to select (Ch1 – Ch2). A red waveform should appear on the screen, this is voltage across the capacitor. Measuring phase angle differences between series components If components in a circuit are being excited at the same frequency but peak at different times, there exists a phase angle between the components. The phase angle is the difference ( in degrees ) between and two points in the component waveforms. Usually this is the positive or negative peak or the x-axis crossing, either with a positive or negative slope. To measure the phase angle between two series components, the Math function must be used to used as described in “Viewing series component voltages with the oscilloscope.” The menu prompted by the ‘Measure’ button allows the phase angle between two waveforms to be automatically calculated. The red ‘Math’ waveform is available as an option if it is present on the screen. Resonance The resonant frequency ( fo ) for a RLC series circuit is fo = 1 /2π√LC. At fo, ωoL = 1/ωoC. Thus, the circuit's total reactance ( jωL - j/ωC ) is zero; the current ( I ) is in phase with the circuit's source voltage ( Vs ); the circuit's impedance is a minimum, and the circuit's admittance is a maximum.EQUIPMENT • Oscilloscope • Function generator • Digital multimeter (DMM) • Resistor, 1 kΩ • Inductor, 220 mH • Capacitor, 1.0,µF PROCEDURE • Set the Digital Multimeter (DMM) to read AC Volts and connect it across the output of the Function Generator. • Select a Sinusoidal output on the Function Generator and adjust its output to read 5 Volts RMS on the DMM. • Connect the RLC circuit in Figure 1. • Display Vs and Vr on the Oscilloscope with Channel 1 and 2 respectively. Set the Scope to trigger on a positive slope for Channel 1. Make sure that both channels have Zero Offset (in other words, ground potential will be in the middle of the screen). • Adjust the frequency of the Function Generator until Vs and Vr are in phase (exactly the same zero crossings). • Record this frequency as fo in the middle row of Table 1. Measured Calculated Frequency Vs ∠Vs Vr ∠Vr Vt ∠Vt Is Zs Vr+Vt Rs+jXs xfo Hz volts degrees volts degrees volts degrees mA ohms volts ohms 0.5fo 0 fo 0 2fo 0 Figure 1: RLC Circuit Table 1: Experimental and Calculated Data• Using the Math function of the Oscilloscope, display Ch1 – Ch2 (that is, Vt = Vs – Vr). • Set the oscilloscope to perform the following measurements and record them in the middle row of Table 1: - measure the voltage magnitude, Vr, in RMS - measure the phase angle of Vr with respect to Vs - measure the voltage magnitude, Vt, in RMS - measure the phase angle of Vt with respect to Vs - • Use experimental data to make the following calculations and record them in the middle row of Table 1: - calculate the current in this series circuit by Ohm’s Law: Ir = Vr/(1 K-Ohm) mA - calculate the total impedance of the circuit by Voltage Divider: Zs = (1 K-Ohm)*Vs/Vr K-Ohm - verify KVL by adding the two phasors, Vr + Vt = Vs: **VECTOR ADDITION** [Vr cos (angVr) + Vt cos (angVt)] + j [Vr sin (angVr) + Vt sin (angVt)] - calculate the rectangular form of Zs, Rs + jXs: Rs = Zs cos(angZr) where angZr = -angVr Xs = Zs sin(angZr) • Take a screen capture from the oscilloscope showing the three waveforms, Vs, Vr, Vt. • Change the frequency of the Function Generator to be 0.5*fo. Repeat the measurements and calculations above to fill in the first line of Table 1. • Change the frequency of the Function