Circuit Response to Time Varying Signals Lab Write Up

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Lab Write Up - Circuit Response to Time Varying Signals

Laboratory Procedures and Lab Objectives Section located at,
Back - Circuit Response to Time Varying Signals Lab

Procedure and Data:
Part A: Meters

1. Section one of Part A is to determine the output for a given input waveform with no DC offset. Three waveforms, a sine wave, square wave, and triangle wave were used for the input waveforms at a frequency of 100Hz and a 10volt peek to peek voltage. The input waveforms and peek to peek voltage were set using a function generator and confirmed by the use of an oscilloscope. First before any measurements were taken, hand calculations were performed to predict the output of each waveform using the following equations.

RMS = V_P/√2 (E.1)

(V_P/√2) / (2V_P/π) = 1.11 (sin wave only) (E.2)

The sine wave calculations were straight forward based upon the equations are derived specifically for sine waves. For a triangle wave another equations was given as follows.

RMS = V_P/√3 (triangle wave only) (E.3)

Each of these two wave forms, the triangle and sine wave, and there given equations help determine the predicted AC RMS values and the True RMS values. In this case the AC RMS and True RMS values are equal for triangle and sine waves separately because our DC offset is set equal to zero.

No equations were given for the square wave however on close examination of the characteristics of a square wave and knowing generally what a meter does to an input signal the predictions became quite apparent. For starters a meter first takes an input signal and rectifies it.

Thus making all negative components of the waveform positive. Then the meter takes the average value or mean of the input signal, and finally multiplies 1.11 to the output signal. This final step of multiplying 1.11 will be explained shortly. By examining a square wave we are essentially determining the area under the curve and because a square wave is exactly that, 'square' our wave form is turned into a flat rectangle sitting on the x‑axis at zero (see Figure 1.1).

Figure 1.1: Squarewave

By close observation of our reflected square wave our predicted AC RMS and True RMS would equal 5volts.

As mentioned the meter multiples a variable equal to 1.11 to the output signal. This factor of 1.11 comes from our equation (E.2) where it is found that our multimeter is calibrated for a sin wave. Any input wave be it a sin wave or not will always be effected by this variable. Looking back to our triangle wave our predicted meter read must be the mean of our reflected waveform multiplied by our variable of 1.11. To determine the mean of our reflected triangle wave the area under the curve for one period must be found. This was done by splitting the wave into four right triangles, finding the area of one right triangle, multiplying it by four, and divided by the period to get the total area over one period.

Area = 1/2bh (E.4)

This resulted in determining the mean which was multiplied by our variable, 1.11 to predict the output meter reading. Finally our predictions were tested and confirmed by actually testing the true output in a laboratory setting. Data from these tests and predicted values are listed in Table 1.

Table 1: DMM Readings for 10Vpp AC Waveforms

2. In section two of part A a DC offset of 10 volts is now considered in our procedures for the sin, square, and triangle waveforms. This procedure is identical to section one part A, except our input waveforms are now offset by an average DC value equaling 5 volts. Again frequency is 100Hz and the input signals were confirmed using an oscilloscope. Our predictions of True RMS value do take into account a DC signal, our calculations show this characteristics.

Table 2: DMM Readings of 0 - 10V DC Offset Waveforms

Part B: C and L Response to Time Varying Waveforms

Figure 1.2: Triangle Wave Circuit

1. For this section of the experiment a 0.01micro farad capacitor was used driven by a triangle wave at 100Hz and 1000Hz. To view the current passed by the capacitor a 100 ohm resistor is placed in series to the capacitor (see Figure 2). Inputting a triangle wave simplifies hand calculations do to a triangle wave has a constant slope. Therefore, the following equations were used in calculating the characteristics of the circuit.

slope = rise / run = ∆V_C/∆t ≈ dv_C/dt (E.5)

i_C = Cdv_C/dt (E.6)

Following the hand calculations (see Table 3) measurements needed to be taken using the following techniques. By measuring the voltage across the resistor and applying ohms law the current for the circuit was calculated. After measurements were taken the output waveform was drawn over the input waveform as a dotted line providing an opportunity to graphically determine the characteristics of the circuit (see Graph 1.1).

Table 3: Current Through a Capacitor Driven by a Triangle Wave

2. In this section a 100mH inductor will be examined much like the capacitor in Part B,1. Although the input waveform will be that of a 10Vpp square wave at frequencies of 5,000Hz and 50,000Hz. Like the capacitor a 100 ohm resistor will be used to indirectly measure the current through the inductor. Like the capacitor the inductor has specific equations associated with them.

slope = ∆i_L/dt = di_L/dt (E.7)

V_L = Ldi_L/dt (E.8)

After completion of our hand calculations (see Table 4) measurements needed to be taken. This procedure is exactly the same as our last section using the capacitor where by the voltage is measured across the resistor and using ohms law to determine the current through the inductor. The input and output data taken was then transposed to our graphs (see Graph 2.1) in order too produce a graphical repression of this circuits characteristics.

Table 4: Inductor Response to a Square Wave of Voltage

3. Our experiment follows now with the examination of a sine wave using a 100mH inductor, a 10Vpp sine wave, and frequencies of 2kHz, 20kHz. Looking back to our triangle waveform our calculations were based upon the slope of the wave being constant. Although by examination our sin wave does not have constant slope, so to calculate the slope at any given point on our sin wave calculus must be applied. Taking into account that by taking the derivative of the sine wave gives us a positive cosine wave thus the sine waves derivative leads it by 90deg. In forming our calculated predictions the following equations were used.

ω = 2πf (E.9)

V_P / I_P = ωLI_P / I_P = ωL = X_L (E.10)

I_P = V_P/X_L (E.11)

In examination of our circuit an oscilloscope was used to confirm our predicted calculations as well as measure the phase angle of our input and output waveforms. It was discovered that in an inductor the output sine wave lead our input sin wave by almost 90deg., thereby confirming our predictions. The data predictions and measured values are represented on table 5.A graphical representation was plotted to show the following concepts (see Graph 3).

Table 5: Sine Wave Response of Inductor

4. The response for a capacitor can also be examined for a sine wave much like was done for the inductor above. However our initial conditions for the capacitor are as follows. Frequency to be set at 500Hz, 5kHz, a 10Vpp input sine wave using a 0.01E‑6F capacitor. Similarly to the inductor in Section 3 our sine wave predictions required the use of calculus to examine the slope of our waveform. The equations used in determining our predicted calculations are as follows.

ω = 2πf (E.12)

V_P / I_P = V_P/ωCV_P = 1/ωC = X_C (E.13)

Again the use on an oscilloscope was needed to determine our measured voltages and phase shifts for each frequency. Data was taken to examine the characteristics of the circuit (see Table 6) and a graphical plot was drawn to visually represent the activity of the circuit (see Graph 4).

Table 6: Sin Wave Response of Capacitor

Conclusion:

Part A: Data Analysis & Discoveries

Multimeters are calibrated for sin waves only. Any waveform inputted into a DMM will be effected by variable of 1.11 on its output reading and must be taken into consideration.

True RMS values always take into account all AC and DC signals. Thus any noise present in a signal will have an effect on the True RMS value.

Multimeters take only the average positive area under the curve for one period of any given input waveform.

Part B: Data Analysis & Discoveries

During setup of the circuit and equipment it was discovered that a slight DC offset was present in the output waveforms. Using the DC offset set to on, this initial DC offset was calibrated to zero before any measurements were made.

For any given waveform equations (E.10), (E.11), (E.13) always are true and can be applied to the given circuits.

True measurements of current can not be measured directly. Measuring the voltage across a component and applying ohms law can determine the current indirectly.

It was confirmed that for any current to be present in a capacitor the voltage must be changing with respect to time. As for a inductor it to was confirmed that for a voltage to be present in an inductor the current through it must be changing with time.

In closing the effects of multimeters and understanding how they operate provides vital information for experimental projects. By knowing the equipment and what effects that equipment may have on the data allows the user to take those variables into consideration on future projects. Examination of the effects on capacitors and inductors and proving the relationship of there prospective equations provides the user to have a better understanding of generally knowing the effects of those components and how they may react in a circuit.

For part B sections 1 through 4, Pspice simulations were ran to provide a third check to both the hand calculations as well as the measured and graphical analysis for each of the circuits. See Pspice graphs (P.1) ‑ (P.4) attached.

Figures and Graphics
Figure 1.1: 100Hz Triangle Wave
Figure 2.1: Squarewave R-L Circuit
Figure 3.1: Sinewave R-L Circuit
Figure 4.1: Simple R-C Circuit
Figure 1B: PSpice Part B, Section 1
Figure 2B: PSpice Part B, Section 2

Lab Notes
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Electrical Engineering lab key words: RLC series circuit, RLC parallel circuit, circuit behavior, resistor, inductor, capacitor, series-parallel, voltage, current, amplitude measurement, phase measurement, PSpice analysis, discrete frequency, computer simulations, voltage, current, phase, high frequency, resistance, power, work and efficiency; Ohm’s and Kirchhoff’s laws; series and parallel circuit principles. RLC circuits, equivalent circuits, series-parallel DC resistive networks, sinusoidal AC voltage, phasors, average and effective values, impedance, AC series parallel circuits, AC power, AC network analysis, AC network theorems, dependent sources, transformers. Network theorems applied to DC circuits: source conversions, Thevenin, Norton, superposition; capacitance; magnetic circuits; inductance; transient analysis of RC and RL circuits; sinusoidal waveforms; reactance and impedance; series, parallel, and series-parallel AC circuits.

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