RC and RL Circuits – Page 1 RC and RL Circuits RC Circuits In this lab we study a simple circuit with a resistor and a capacitor from two points of view, one in time and the other in frequency. The viewpoint in time is based on a differential equation. The equation shows that the RC circuit is an approximate integrator or approximate differentiator. The viewpoint in frequency sees the RC circuit as a filter, either low-pass or high-pass. For each experiment starting with 2, make a copy of the screen showing input and output waveforms and place it in your lab notebook. Experiment 1, A capacitor stores charge: Set up the circuit below to charge the capacitor to 5 volts. Disconnect the power supply and watch the trace decay on the ‘scope screen. Estimate the decay time. It will be shown that this decay time, τ = RC, where R is the resistance in ohms and C is the capacitance in farads. From this estimate calculate an approximate value for the effective resistance in parallel with the capacitor. (This resistance is the parallel combination of the intrinsic leakage resistance within the capacitor and the input impedance of the ’scope.) [Ans.: about 1 s] Next, replace the 0.047 µF capacitor by a 1000µF electrolytic capacitor [Pay attention to the capacitor polarity!] and watch the voltage across it after you disconnect the power supply. While you are waiting for something to happen, calculate the expected decay time. Come to a decision about whether you want to wait for something to happen. Act according to that decision. scope 0.047 μF 5V Figure 1: Capacitor charging circuit.RC and RL Circuits – Page 2 Experiment 2, The RC integrator in time: Consider the RC circuit in Figure 2 below: In lecture you learned that this circuit can be described by a differential equation for q(t), the charge on the capacitor as a function of time. It was shown that a solution for the voltage on the capacitor, VC = q(t)/C, consistent with no initial charge on the capacitor, is: VC= V0(1 − e−t /τ) where τ = RC and V0 is the initial voltage. Now build the circuit, replacing the battery and switch by a square wave generator. (Note: The square wave generator has positive and negative outputs, but this is the same as switching the battery with an added constant offset and a scale factor.) Set the square wave frequency to 200 Hz, and observe the capacitor voltage. 10k Scope A Scope B 0.047 μF 1V Figure 2: RC Circuit.RC and RL Circuits – Page 3 Use the vertical cursors of your DPO to measure the time required to rise to a value of (V+ -V-)(1-e-1). Accuracy in this measurement is improved if the pattern nearly fills the screen. This rise time must be equal to τ. Compare with the calculated value of τ. Increase the square wave frequency to 900 Hz. Is the RC circuit a better approximation to a true integrator at this frequency? Sketch the response of a true integrator to a square-wave input. Experiment 3, The RC differentiator in time: Consider the RC circuit in Figure 4 below: VC t t τ V+ 0 V- Figure 3: Square Wave and Integrator Output. Scope A 0.047 1V 10K Scope B = VR Figure 4: RC Differentiator.RC and RL Circuits – Page 4 The output is the voltage across the resistor, which is the current, or dq/dt multiplied by the resistance R. Show that the solution for this voltage, consistent with no initial charge on the capacitor, is VR = V0e-t/τ, where τ =RC. Now build the differentiator circuit, replacing the battery and switch by a square wave generator. Set the square wave frequency to 200 Hz, and observe the resistor voltage. Use the DPO to measure the time required to fall (or rise) by a factor of e-1. This rise time must be equal to τ. Compare with the calculated value of τ. Sketch the derivative of a square wave. How does the output of the differentiator circuit compare? Experiment 4, The RC low-pass filter: The low-pass filter is simply the integrator circuit above, but we replace the source by a sine oscillator so that we can measure the response at a single frequency. (The sine wave is the only waveform that has only a single frequency.) We define the transfer function for a filter as the ratio H(ω) = Vout/Vin, the ratio of output to input voltages, where ω is the angular frequency of the input sinusoidal voltage. The transfer function in this case is given by: H(ω)=1/(1+jωτ). where, j is the imaginary number √-1. Calculate the frequency [f = ω/(2π)] of the so called “half-power point.” This is simply the frequency where the output voltage amplitude is equal to the input voltage amplitude divided by √2. Calculate the phase shift at this frequency Φ = (tan-1(Im(H(ω)/Re(H(ω)). Build the circuit and find the frequency for half power. Use the “measure” utility of your DPO to find the phase shift at that frequency and compare with calculations. (Note: The phase shift is related to the time, Δt, between equivalent zero crossings of the input and output through the formula Δ Ø = 2 π f ∆ t, or Δ Ø = 360 f ∆ t, Figure 5: Input and Output of Differentiator. V+ 0 V- VR 0 tRC and RL Circuits – Page 5 for radians or degrees respectively. We will use the convention that the phase shift is positive if the output leads the input. Does the output lead the input for this filter?) Next, instead of the half-power point we consider the half-voltage point, where the output amplitude is half the input amplitude. Show that this point occurs when ωτ = √3. Calculate the frequency for the low-pass filter above. Show that the phase shift at this point is -60 degrees. Change the oscillator frequency to find the half-voltage point. Compare frequencies and phase shifts with your calculations. Experiment 5, The RC high-pass filter: The high-pass filter is simply the differentiator circuit above, but we replace the source by a sine oscillator so that we can measure the response at a single frequency. The transfer function is H(ω) = jωτ/(1+jωτ). Show that in the limit of high frequency H = 1. Calculate the frequency [f =ω/(2π)] of the “half-power point.” Calculate the phase shift at this frequency. Build the circuit and find the frequency for half power. Use the DPO to find the phase shift at that frequency and compare with calculations. Next, do the “half-voltage” calculations and measurements, as for the RC low-pass filter. RL
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