Berkeley ELENG 40  Guide 4 (8 pages)
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Guide 4
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 Eleng 40  Introduction to Microelectronic Circuits
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EECS40 RLC Lab guide Introduction Second Order Circuits Second order circuits have both inductor and capacitor components which produce one or more resonant frequencies 0 In general a differential equation for the circuit can be written in the form d 2 v t dv t 2 02 v t f t 2 dt dt Eq 1 Where and 0 depend on values for R L and C present in the circuit and the forcing function f t depends on these values as well as voltage sources present This differential equation follows the same form as that for a damped harmonic oscillator ie A mass oscillating due to a spring resonance but experiencing a damping force friction and possibly a driving force on the mass to counteract the spring In the case of our second order circuit the damping is produced by a resistor burning energy resonance occurs between the inductive and capacitive components and driving force would be a voltage source to force the circuit against the damping by the resistor For a series RLC circuit the damping factor R and resonant frequency 0 2L 1 LC Figure 1 Series RLC circuit The solution to the differential equation Eq 1 depends on these parameters and 0 in that if 0 The circuit is overdamped and the solution is v t K1e s t K 2 e s t 1 0 2 The circuit is critically damped and the solution is v t K1e s t K 2 te s t 1 0 2 The circuit is underdamped and the solution is Lab 4 5 guide 1 v t e t K1 cos n t K 2 sin n t The roots s1 and s 2 of the over and critically damped solutions are given by s1 2 02 The natural frequency s 2 2 02 n of the underdamped case is given by n 02 2 In this lab we will observe the step response of a series RLC circuit The step response is how the circuit behaves in response to a forcing function that is a step function f t 0 for t 0 and f t 1 if t 0 which we apply periodically to our circuit with a square wave The solution is shown in Figure 2 6 where the values are normalized amplitude of the forcing function divided out Figure 2 Normalized Step Response of 2nd order circuit Quality Factor and Bandwidth Lab 4 5 guide 2 Figure 3 Series RLC Bandpass Filter with AC source The circuit shown in Figure 3 has a resonance frequency 0 which has the same value as shown on page 1 At resonance the reactance of the capacitor cancels out the reactance of the inductor so they must be equal in magnitude The quality factor Q of a series RLC filter is defined as the ratio of the inductive reactance to the resistance at the resonant frequency Q 0 L R 2 f 0 L R Figure 4 Plots of Transfer Function magnitude for different values of Qs the quality factor of a series RLC circuit Note that this plot is not on a log log scale which Bode plots feature Lab 4 5 guide 3 The bandwidth of a bandpass filter is the region between which the output is above half the maximum power This is also the 3dB point because in decibels 10 log 0 5 3 2 where 0 5 comes from the power ratio or H The bandwidth B of a series bandpass filter is related to quality factor Q by the equation f B 0 Q The voltage measured at the half power frequency should be 0 707 or 0 5 of the maximum voltage because power is proportional to the square of voltage Phasors and Complex Impedance Phasors can be used to represent the complex amplitude of a sinusoidal function For example Acos t A Impedance measures the opposition of a circuit to time varying current The units of impedance are ohms A resistor s impedance is simply it s resistance Capacitors and inductors have complex impedances Capacitor impedance 1 j C Inductor impedance j L An RLC circuit that is driven by a sinusoidal voltage source can be analyzed using KVL KCL and Ohm s law The rules that apply to resistors apply to the complex impedances of the elements of the circuit For example Vout in Figure 3 can be found easily using the voltage divider equation with the circuit element s impedances Doing this is part of a prelab problem Lab 4 5 guide 4 Hands On LC Circuit For this part of the lab you will need to set your function generator to generate pulses at regular intervals First create a square wave with a frequency of 100 Hz 3 Vpp and a Voltage offset of 1 5V the offset makes the square wave range between 0V and 6V Hit the shift key then the key with burst written in small blue letters above it Use the oscilloscope to make sure the output your function generator looks something like this Figure 5 pulses created by the function generator Now Attach a 10 F capacitor as shown in figure 2 make sure to check if the capacitor is polarized A1 Scope A1 Ground Figure 6 Capacitor attached to a pulse generator Question 1 Sketch the voltage across the capacitor Now add a 1 mH inductor in parallel with the capacitor as shown in figure 3 A1 Scope A1 Ground Figure 7 inductor and capacitor attached to a pulse generator Lab 4 5 guide 5 You may have to move the trigger level up or down to get a still picture If that doesn t work you can always hit the stop button to freeze the image DON T rescale the image you will need to graph the new output on the same graph as question 1 Question 2 Sketch the voltage across the capacitor and inductor in parallel and explain what you see Use the same graph as question 1 Increase the frequency of the function generator to 2 KHz Question 3 Sketch what you see when the inductor is removed like Figure 6 Sketch what you see when the inductor is present like Figure 7 Use the two time cursers hit the curser key then use the time softkey to estimate the period of the oscillation you see on your oscilloscope You may have to extrapolate the period from a trough to peak 1 4th of a period to get the most accurate answer Question 4 Use the information from the time cursors to estimate the frequency of the oscillation Remove the 1 mH inductor and replace it with a 10 mH inductor Question 5 Sketch the new waveform Series RLC Circuit Construct the circuit below with a 1 nF Capacitor 100 mH inductor and a 2 2 K resistor Use a sinusoidal input with 6 Vpp and 0V offset Determine the resonance frequency f0 and set the function generator to that frequency NOTE make sure you turn off burst before continuing A1 Scope A1 Ground Figure 8 Series RLC circuit Lab 4 5 guide 6 Have the oscilloscope display …
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