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**Basic Electronics Ninth Edition Grob Schultz**

©2002 The McGraw-Hill Companies

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**RC and L/R Time Constants**

Basic Electronics Ninth Edition 23 CHAPTER RC and L/R Time Constants ©2003 The McGraw-Hill Companies

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**Topics Covered in Chapter 23**

High Voltage Produced by Opening an RL Circuit RC Time Constant RC Charge and Discharge Curves High Current Produced by Short-circuiting RC Circuit RC Waveshapes

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**Topics Covered in Chapter 23**

(continued) Long and Short Time Constants Charge and Discharge with Short RC Time Constant Long Time Constant for RC Coupling Circuit Universal Time Constant Graph Comparison of Reactance and Time Constant

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**Transient Response of an Inductor**

The transient response of an inductor has a time constant T = L / R. T is in seconds L is in henries R is in ohms The time (T) required for a change in current of 63 % is one time constant, T. Inductor current reaches its steady-state value in five time constants.

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**Transient Response of an Inductor (charge)**

10 L 1 H 8 10 V 6 R 1 kW iC in mA 4 2 T = L/R = 1/1 x 103 = 1 ms ISS = 10 V/1 kW = 10 mA T in ms Steady state circuit current (ISS) = 10 mA

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**Interrupting Inductor Current**

The instant an inductive circuit is opened, a high voltage is generated across the inductor. Opening the circuit causes a rapid decrease in current. In an inductive circuit, the faster the rate of change of current, the higher the amount of induced voltage: vL = L(di/dt).

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**Interrupting Current Flow in an Inductive Circuit**

Large di/dt generates a very high voltage.

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**Shorting a Charged Capacitor**

The instant a charged capacitor is shorted with a very low resistance, considerable discharge current flows through the short. Shorting the capacitor causes a rapid decrease in capacitor voltage. In a capacitor circuit, the faster the rate of change of voltage, the higher the amount of current: iC = C(dv/dt).

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**Shorting a charged capacitor**

With large capacitors, this can be dangerous!

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**Transient Response of a Capacitor**

The transient response of a capacitor has a time constant T = RC. T is the time in seconds C is the capacitance in farads R is the resistance in ohms The time required for a change in voltage of 63 % is one time constant, T. Capacitor voltage reaches its steady-state value in five time constants.

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**Transient Response of a Capacitor (charge)**

10 1 kW 8 R 10 V 6 1 mF C vC in Volts 4 Initial charge = 0 V 2 T in ms T = RC = 1 x103 x 1 x 10-6 = 1 ms

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**Transient Response of a Capacitor (discharge)**

10 8 1 kW R 6 vC in Volts 1 mF C 4 2 Initial charge = 10 V T in ms T = RC = 1 x103 x 1 x 10-6 = 1 ms

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Short Time Constants A short time constant is 1/5 or less of the pulse width of the applied voltage. RC Circuits: Produces sharp voltage spikes for vR and rounded edges for vC at the leading and trailing edges of an applied rectangular wave. L/R Circuits: Produces rounded edges for vR and sharp voltage spikes for vL.

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**Short Time Constant in an RC Circuit**

VS 1 kW R VS 100 Hz 1 mF C vC Pulse width = 5 ms T = RC = 1 ms vR

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Long Time Constants A long time constant is 5 or more times longer than the pulse width of the applied voltage. RC Circuits: vR is very nearly the same as the applied waveform, and the capacitor blocks the average dc level of the applied waveform. L/R Circuits: vR resembles the average dc level of the applied waveform.

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**Long Time Constant in an RC Circuit**

VS 1 kW R VS 2.5 kHz 1 mF C vC Pulse width = 0.2 ms T = 1 ms vR

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**Output Waveshapes With long time constants: With short time constants:**

vC and iL rise slowly to the steady-state value. The output waveform is said to be integrated. With short time constants: vC and iL rise rapidly to the steady-state value. The output waveform is said to be differentiated.

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**Integrators and Differentiators**

VS vOUT vOUT C Integrators use a relatively long time constant. C vOUT VS R vOUT Differentiators use a relatively short time constant.

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**Universal Time Constant Graphs**

Rising curve: Value increases 63% with each time constant. Can represent: Capacitor charging voltage, vC Rising inductor current, iL Falling curve: Value decrease 63% with each time constant. Can represent: Capacitor discharging voltage, vC Decaying inductor current, iL

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**Universal Rising Curve**

T 0.5 39.3 % 1.0 63.2 % 1.5 77.7 % 2.0 86.5 % 2.5 91.8 % 3.0 95.0 % 3.5 97.0 % 4.0 98.2 % 4.5 98.9 % 5.0 99.3 % 100 % 80 % 60 % 40 % 20 % 0 % T

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**Universal Falling Curve**

T 100 % 0.5 60.7 % 1.0 36.8 % 1.5 22.3 % 2.0 13.5 % 2.5 8.21 % 3.0 4.98 % 3.5 3.02 % 4.0 1.83 % 4.5 1.11 % 5.0 0.674 % 100 % 80 % 60 % 40 % 20 % 0 % T

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