Inductors, Capacitors, and Transient Response

ENGR2105_Lab4_CapcitorsInductorsTransientResponse2.pdf

Home >Engineering homework help >Inductors, Capacitors, and Transient Response

ENGR 2105 – Inductors, Capacitors, and Transient Response

1. Introduction and Goal: Exploring transient behavior due to inductors and workcapacitors in DC circuits; gaining experience with lab instruments.

2. Equipment List: Required instruments In Multisim. • Multisim

• Calculator

3. Experimental Theory: The three common passive circuit elements are resistors, capacitors, and inductors. We study DC capacitor and inductor

circuits today.

3.1 Capacitor: A capacitor collects (stores) electrical charge. It is made of two or more conductors separated by insulators.

3.1.1 Applying DC voltage causes current (charge flow) to enter a capacitor. Charge accumulates on its surfaces like water in

a reservoir (Figure 1).

3.1.2 In Figure 2, when voltage V is applied, current 𝑖(𝑡) initiall flows until the capacitor develops an equal and opposite

voltage to the DC source (VC = -V). When VC = -V, current

ceases. Thus a capacitor “eventually” blocks DC current

flow. (In most cases, this takes a very small amount of

time.)

3.1.3 With no initial charge, the capacitor has a large capacity to absorb current initially. Thus voltage across a capacitor

cannot change instantaneously. It builds from 0 as charge

collects on the capacitor plates. When VC = -V, current flow

ceases.

3.1.4 Capacitors are used in all modern electrical circuits (TV’s, mobile Phones, etc.). The unit of capacitance is the Farad,

(after Faraday, another early experimenter). The Farad is a

very large measure of capacitance. Capacitors usuall have

values between micro- (10−6) and pico- (10−12) Farads. 3.2 Inductor: An inductor is a coil of wire with the property of electrical

inertia. An analogy is the inertia of mass. A large truck accelerates

slowly due to its mass. At high speed, it is hard to stop for the same

reason. Similarly, inductors resist any change (increase or decrease)

in current (Figures 3 and 4).

3.2.1 Inductor characteristics are due to its magnetic properties. The inductive effect in a coil of wire occurs due to changes

in the current. Constant inductor current (or no current)

causes no inductive effect.

3.2.2 As voltage cannot change instantaneously on a capacitor, current cannot change instantaneously in an inductor.

3.2.3 Inductance is measured in Henry’s (for another pioneer). As one Henry is a large inductor, practical inductors are in

milli-Henry’s (mH).

3.3 Capacitors and Inductors in a DC Circuit: Capacitors and inductors cause very brief non-linear effects when a DC voltage is applied or

changed. Shortly after a DC voltage change, capacitor and inductor

circuits reach “steady state.” These extremely brief effects are called

transient behavior.

3.3.1 Exponential functions review: If𝑦 = 3𝑥, y is an exponential function of x. In many exponential functions, e (≈ 2.71828,

the “base of natural logarithms”) appears. In Figure 5, 𝑦 = 1 − 𝑒 −𝑥. At x = 0, y = 0. But as x increases y rapidly increases in the beginning, but quickly approaches the

asmptote. As x → ∞, 𝑒 −𝑥 → 0, so y →1. Mathematically, y never reaches 1, although by x = 10, 𝑒 −𝑥 = 0.000045, and the difference is negligible. Approaching a value

(“asymptote”) but never reaching it is typical of exponential

functions. Such functions describe capacitor and inductor

behavior in DC circuits.

3.3.2 The equation 𝑣𝐶 (𝑡) = 𝑉(1 − 𝑒

−(𝑡 𝜏⁄ )) describes behavior of

current in the RC circuit of Figure 6, where 𝑣𝐶 (𝑡) is the capacitor voltage after the switch is closed (at t = 0), and V

is the DC voltage. Since vC cannot change instantaneously,

𝑣𝐶 (𝑡 = 0) = 0 (assuming no charge on the capacitor at t = 0). τ is the “time constant,” the time it takes for the voltage

to change to (1 − 1 𝑒⁄ ) of its former value. Thus, τ is a measure of the duration of transient behavior. The unit of a

time constant is seconds, and the smaller it is, the quicker

transient behavior is over. Although an exponential function

never mathematically reaches its asymptote, transient

behavior is over in about five time constants. For a series

RC circuit, 𝜏=RC (R in Ohms, C in Farads). (RC has units of

seconds!) Thus, 𝑣𝐶 (𝑡) = 𝑉(1 − 𝑒 −(𝑡 𝑅𝐶⁄ )).

3.3.3 In Figure 7, since current cannot start instantaneously in an

inductor, the inductor voltage vL = V when the switch is

closed (i = 0, thus iR = 0). As current increases, vL falls. At

steady state, vL = 0, and current equals V/R. An equation for

inductor voltage is: 𝑣𝐿(𝑡) = 𝑉𝑒 −(𝑡 [𝐿 𝑅⁄ ]⁄ ) = 𝑉𝑒 −(𝑅 𝐿⁄ )𝑡. At t

= 0, vL = V. 𝝉 = 𝐿 𝑅⁄ is the RL circuit time constant (inductance is in Henry’s, Resistance is in Ohms.)

3.4 RLC Circuit: A capacitor, inductor, and resistor circuit can oscillate.

The signal generators in the lab make use of oscillating circuits to

produce our desired voltage waveforms.

3.4.1 In Figure 8, at t = 0, V causes current to flow in the circuit. The inductor reacts with its transient response to the change

in current, so current starts from 0 and then current

gradually increases, and charge collects on the capacitor.

When the capacitor charges to -V, (reverse polarity of the

voltage source), current flow ceases in the circuit (after

about 5 time constants).

3.4.2 The RLC circuit will resonate just as a bell that is rung, with

proper choice of R, L, and C. The oscillation is also

transient.

3.4.3 Skipping the mathematical derivation, for a resonant series RLC circuit, capacitor voltage can be expressed as:

𝑣𝐶 (𝑡) = 𝑉𝑠 (1 − [cos 𝜔𝑑 𝑡]𝑒 −𝑡(𝛼))

if the components R, L, and C, are chosen properly (for many

component values, no oscillation occurs), and V is the applied voltage.

3.5 The cosine function above describes the voltage oscillation, and the e- term clearly makes the behavior transient. 𝝎𝒅 is the radian frequency of oscillation (𝜔𝑑 = 2𝜋𝑓𝑑 , where 𝑓𝑑 is the resonant frequency of the

circuit, in Hz), defined as 𝜔𝑑 = √(1 𝐿𝐶⁄ ) − (𝑅 2𝐿⁄ ) 2. α is the

damping factor, defined as α = 𝑅 2𝐿⁄ . Like 𝜏, α determines how fast

the oscillation dies away.

4. Pre-work: We will use the equations above as we examine transient behavior.

4.1 Make sure you understand the concepts of transient behavior discussed above and complete the worksheet.

5. Experimental Procedure – RL and RC circuits: 5.1 Voltage Across a Capacitor in a Series RC Circuit: The capacitor

voltage equation is: 𝑣𝐶 (𝑡) = 𝑉(1 − 𝑒 −(𝑡 𝑅𝐶⁄ )) , where the time

constant 𝜏 = 𝑅𝐶. 5.1.1 Select 1 kΩ resistor and 0.05 μF capacitor. Measure R and C

values use an LC meter for capacitor.

5.1.2 In our RC circuit, 𝜏 = 𝑅𝐶 = (1000Ω)(0.05 × 10−6𝐹). That is, 𝜏 = 50μsec. Since transient circuit behavior lasts ~ 5τ, the behavior of the circuit lasts about 5 × 50𝜇𝑠 = 250𝜇𝑠, or ¼ millisecond.

5.1.3 Quarter-millisecond events are hard to see. If we were in the lab, we would use an oscilloscope to observe our

transients and a signal generator for “DC voltage.” Since we

are not in the lab, we will use the Multisim Live “Grapher”

feature.

5.1.4 Connect a 0.05𝜇𝐹capacitor and 1𝑘Ωresistor (located in the “Passive” menu item on the left) in series with a “Clock

Voltage” (located under “Sources”). Don’t forget to place

the ground connection.

5.1.5 The Clock Voltage can generate a “DC voltage” that cycles on/off at a designated frequency, which is a square wave.

Set the Clock Voltage to 5V and a frequency of f = 250 Hz.

Hz means “per second”, so the voltage will cycle on/off 250

times per second.

5.1.6 The period of the square wave is 𝑇 = 1

𝑓 =

250𝐻𝑧 = 0.004𝑠 =

4𝑚𝑠. This will provide an “off” (0 V) and an “on” (5 V) over a 4 ms period. Since the voltage is on half of the time

and off half of the time, the voltage is in each state for 2

msec.

5.1.7 2 msec is 10 time constants for our chosen values of R and C. Since we reach steady-state after 5 time constants, the

voltage will be “on” sufficiently long for us to see the entire

transient response.

5.1.8 Start the simulation for several seconds and then stop it. Refer to the lecture video for more detail on the following

steps.

5.1.9 Place Cursor 1 at the start of the transient response. Place Cursor 2 on the trace 1 time constant to the right.

5.1.10 Move the time cursor horizontally until it intersects the signal at 2τ (i.e. 2 vertical lines to the right) and take another

reading. Take a screen shot of the 2τ measurement, paste

it in a word doc and include it in your lab report. I want

to be able to see all of the circuit components as well as

the oscilloscope. Continue measurements at 3τ, 4τ, 5τ, 6τ,

8τ, and 10τ and record these in the data sheet.

5.2 Inductor Voltage in a Series RL Circuit: 5.2.1 Replace the capacitor with a 10 mH inductor. Use the same

resistor and Clock Voltage settings.

5.2.2 For an inductor circuit, 𝜏 = 𝐿 𝑅⁄ . Calculate the time constant for this new circuit and record this in your data sheet.

5.2.3 Place Cursor 1 at the start of the transient response. Place Cursor 2 on the trace 1 time constant to the right

5.2.4 Move the time cursor horizontally until it intersects the signal at 2τ and take another reading. Take a screen shot of

the 2τ measurement, and include it in your lab report. I

want to be able to see all of the circuit components as

well as the oscilloscope.

5.2.5 Note that inductor voltage falls off exponentially. Also, the inductor voltage spikes negatively on voltage turn-off. This

means that the inductor opposes any change in current.

Using the cursors, measure and record the time until

transient behavior ends. Convert this value to a number of

time constants on your data sheet.

5.3 RLC Circuit: Use a 1000Ω resistor, 10 mH Inductor, and 0.05 μF. 5.3.1 In your lab report, show the calculation that proves that this

circuit will not oscillate.

5.3.2 RLC Circuit: Connect the 1𝑘Ω resistor, 10 mH inductor, and 0.05 μF capacitor as shown below (Figure 13).

Figure 13

5.3.3 Verify that the capacitor voltage does not oscillate: Measure the voltage across the capacitor. To measure only the

capacitor voltage, the ground needs to connected to one side

of the capacitor, and the voltage probe needs to be connected

to the other side. Verify in the grapher that the circuit does

not oscillate. (An example of RLC oscillation is shown in

Figure 14.)

5.3.4 From the worksheet, you know that for oscillation to occur, there must be the right combination of R, L, and C.

5.3.5 Replace 1000 Ω resistor with 51 Ω resistor. In your lab report, show the calculation that proves that this circuit will

oscillate.

5.3.6 In Multisim, you should now see the capacitor voltage oscillate as in Figure 14. Note that you may need to change

the frequency of the clock source in order to see the entire

transient response.

5.3.7 The capacitor voltage “overshoots” the 5V level (up to ~ 7-8

volts), oscillates several times, then settles to 5 V after 5 or 6

cycles.

5.3.8 Calculate 𝜔𝑑 = √(1 𝐿𝐶⁄ ) − (𝑅 2𝐿⁄ ) 2. Using 𝜔𝑑 = 2𝜋𝑓, or

𝑓 = 𝜔𝑑

2𝜋 . Calculate the oscillation frequency, f. Calculate the

period of the oscillation, 𝑇 = 1

𝑓 .

5.3.9 Use vertical cursors to measure the period of the waveform (the distance between identical points on the wave, e.g., two

maxima) and record. Take a screen shot of this

measurement and include it in your lab report. Also

record the amount of time for the oscillation to die out

(transient time).

5.3.10 Change the 0.01 μF capacitor to 0.05 μF capacitor. On applying the square-wave, you should see a different

frequency. Using vertical cursors, measure and record the

period of the new waveform. Take a screen shot of this

measurement and label it Figure 4. Also, record the

elapsed time to end. This should be the same as that

measured above, since the damping term α depends only on

R and L.

6. Writing Laboratory Report: Include the following: 6.1 You need to do a formal lab report for this lab. Lab reporting is an

essential Engineering function. Lab reports serve two primary

purposes. 1) They instruct those that come after you how you got

your data and arrived at your conclusion. 2) They provide evidence

of discoveries that are very important if a patent lawsuit later arises.

For this lab, your lab report should have the following sections.

• Objective

• Schematics (You may use Multisim Screen Shots)

• Equipment/Supplies

• Procedure

• Data Tables

• Measurements (These are the screen shots you were instructed to take. You should include a description for each screen shot)

• Calculations

• Graphs

• Conclusions 6.2 Submit the screenshots as described above.