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**RC and L/R Time Constants**

Chapter 22 RC and L/R Time Constants Topics Covered in Chapter 22 22-1: Response of Resistance Alone 22-2: L/R Time Constant 22-3: High Voltage Produced by Opening an RL Circuit 22-4: RC Time Constant 22-5: RC Charge and Discharge Curves 22-6: High Current Produced by Short-Circuiting RC Circuit © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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**Topics Covered in Chapter 22**

22-7: RC Waveshapes 22-8: Long and Short Time Constants 22-9: Charge and Discharge with Short RC Time Constant 22-10: Long Time Constant for RC Coupling Circuit 22-11: Advanced Time Constant Analysis 22-12: Comparison of Reactance and Time Constant McGraw-Hill © 2007 The McGraw-Hill Companies, Inc. All rights reserved.

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**22-1: Response of Resistance Alone**

Resistance has only opposition to current. There is no reaction to a change. R has no concentrated magnetic field to oppose a change in I, like inductance, and no electric field to store charge that opposes a change in V, like capacitance.

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**22-1: Response of Resistance Alone**

When the switch S is closed in Fig (a), the battery supplies 10 V across the 10-Ω R and the resultant I is 1 A. The graph in Fig (b) shows that I changes from 0 to 1 A instantly when the switch is closed. Fig. 22-1: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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22-2: L/R Time Constant The action of an RL circuit during the time current builds up to a specific value is its transient response. Transient response is a temporary condition that exists only until the steady-state current is reached. The transient response is measured in terms of the ratio L/R, which is the time constant T of an inductive circuit. T = L/R The time constant is a measure of how long it takes the current to change by 63.2%.

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22-2: L/R Time Constant When S is closed, the current changes as I increases from zero. Eventually, I will reach the steady value of 1 A, equal to the battery voltage of 10 V divided by the circuit resistance of 10 Ω. Fig. 22-2: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-3: High Voltage Produced by Opening an RL Circuit**

When an inductive circuit is opened, the time constant for current decay becomes very short because L/R becomes smaller with the high resistance of the open. Then the current drops toward zero much faster than the rise of current when the switch is closed. The result is a high value of self-induced voltage across a coil whenever an RL circuit is opened. This high voltage can be much greater than the applied voltage.

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**22-3: High Voltage Produced by Opening an RL Circuit**

In Fig. 22-3, the neon bulb requires 90 V for ionization, at which time it glows. The source is only 8 V, but when the switch is opened, the self-induced voltage is high enough to light the bulb for an instant. The sharp voltage pulse or spike is more than 90 V just after the switch is opened. Fig. 22-3: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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22-4: RC Time Constant The transient response of capacitive circuits is measured in terms of the product R x C. To calculate the time constant, T = R x C where R is in ohms, C is in farads, and T is in seconds.

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**22-4: RC Time Constant T = 3 x 106 x 1 x 10−6 = 3 s Fig. 22-4:**

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-5: RC Charge and Discharge Curves**

In Fig. 22-4, the rise is shown in the RC charge curve because the charging is fastest at the start and then tapers off as C takes on additional charge at a slower rate. As C charges, its potential difference increases. Then the difference in voltage between VT and v C is reduced. Less potential difference reduces the current that puts the charge in C. The more C charges, the more slowly it takes on additional charge. Fig. 22-4

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**22-5: RC Charge and Discharge Curves**

T in ms 2 4 6 8 vC in Volts On discharge, C loses its charge at a declining rate. At the start of discharge, vC has its highest value and can produce maximum discharge current. As the discharge continues, vC goes down and there is less discharge current. The more C discharges, the more slowly it loses the remainder of its charge. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-6: High Current Produced by Short-Circuiting RC Circuit**

A capacitor can be charged slowly by a small charging current through a high resistance and then be discharged quickly through a low resistance to obtain a momentary surge, or pulse of discharge current. This idea corresponds to the pulse of high voltage obtained by opening an inductive circuit.

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**22-6: High Current Produced by Short-Circuiting RC Circuit**

The circuit of Fig illustrates the application of a battery-capacitor (BC) unit to fire a flashbulb for a camera. The flashbulb needs 5 A to ignite, but this is too much load current for the small 15-V battery. Instead of using the bulb as a load for the battery, the 100-μF capacitor is charged. The capacitor is then discharged through the bulb in Fig (b). With large capacitors, this can be dangerous! Fig. 22-5: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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22-7: RC Waveshapes Voltage and current waveshapes in RC circuits can show when a capacitor is allowed to charge through a resistance for RC time and then discharge through the same resistance for the same amount of time. Waveshapes show some useful details about the voltage and current for charging and discharging.

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22-7: RC Waveshapes Fig. 22-6: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-8: Long and Short Time Constants**

Useful waveshapes can be obtained by using RC circuits with the required time constant. In practical applications, RC circuits are used more than RL circuits because almost any value of an RC constant can be obtained easily. Whether an RC time constant is long or short depends on the pulse width of the applied voltage. A long time constant can be arbitrarily defined as at least five times longer than the pulse width, in time. A short time constant is defined as no more than one-fifth the pulse width, in time.

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**22-8: Long and Short Time Constants**

Integrators and Differentiators VA R C vOUT Integrators use a relatively long time constant. Differentiators use a relatively short time constant. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-9: Charge and Discharge with Short RC Time Constant**

Fig illustrates the charge and discharge of an RC circuit with a short time constant. Note that the waveshape of VR in (d) has sharp voltage peaks for the leading and trailing edges of the square-wave applied voltage. Fig. 22-7

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**22-10: Long Time Constant for RC Coupling Circuit**

Fig illustrates the charge and discharge of an RC circuit with a long time constant. Note that the waveshape of VR in (d) has the same waveform as the applied voltage. Fig. 22-8: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-11: Advanced Time Constant Analysis**

Transient voltage and current values can be determined for any amount of time with a universal time-constant chart. A universal time-constant chart is a graph of curves obtained by plotting time in RC or L/R time constants versus percent of full voltage or current. An example of a universal time-constant chart for RC and RL circuits is shown in Fig (next slide).

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**22-11: Advanced Time Constant Analysis**

. Fig. 22-9: Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

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**22-12: Comparison of Reactance and Time Constant**

Table 22-2 Comparison of Reactance XC and RC Time Constant Sine-Wave Voltage Nonsinusoidal Voltage Examples are 60-Hz power line, af signal voltage, rf signal voltage Examples are dc circuit turned on and off, square waves, rectangular pulses Reactance XC = 1/(2πfC) Time constant T = RC Larger C results in smaller reactance XC Larger C results in longer time constant Higher frequency results in smaller XC Shorter pulse width corresponds to longer time constant IC = VC/XC iC = C(dv/dt) XC makes IC and VC 90° out of phase Waveshape changes between iC and vC

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