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
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
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. Transient Response of an Inductor
L R Transient Response of an Inductor (charge) 10 V T in ms i C in mA 1 k 1 H T = L/R = 1/1 x 10 3 = 1 ms Steady state circuit current (I SS ) = 10 mA I SS = 10 V/1 k = 10 mA
Interrupting Inductor Current 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: v L = L(di/dt). The instant an inductive circuit is opened, a high voltage is generated across the inductor.
10 V Interrupting Current Flow in an Inductive Circuit Large di/dt generates a very high voltage.
Shorting a Charged Capacitor 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: i C = C(dv/dt). The instant a charged capacitor is shorted with a very low resistance, considerable discharge current flows through the short.
Shorting a charged capacitor With large capacitors, this can be dangerous!
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. Transient Response of a Capacitor
T in ms R C 1 k 1 F 10 V v C in Volts Transient Response of a Capacitor (charge) T = RC = 1 x10 3 x 1 x = 1 ms Initial charge = 0 V
T in ms R C 1 k 1 F v C in Volts Transient Response of a Capacitor (discharge) T = RC = 1 x10 3 x 1 x = 1 ms Initial charge = 10 V
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 v R and rounded edges for v C at the leading and trailing edges of an applied rectangular wave. L/R Circuits: Produces rounded edges for v R and sharp voltage spikes for v L.
VSVS vCvC VSVS vRvR R C Short Time Constant in an RC Circuit 100 Hz 1 k 1 F Pulse width = 5 ms T = RC = 1 ms
Long Time Constants A long time constant is 5 or more times longer than the pulse width of the applied voltage. RC Circuits: v R is very nearly the same as the applied waveform, and the capacitor blocks the average dc level of the applied waveform. L/R Circuits: v R resembles the average dc level of the applied waveform.
VSVS vCvC VSVS vRvR R C Long Time Constant in an RC Circuit 2.5 kHz 1 k 1 F Pulse width = 0.2 ms T = 1 ms
Output Waveshapes With long time constants: v C and i L rise slowly to the steady-state value. The output waveform is said to be integrated. With short time constants: v C and i L rise rapidly to the steady-state value. The output waveform is said to be differentiated.
VSVS R C Integrators and Differentiators v OUT Integrators use a relatively long time constant. VSVS R C v OUT Differentiators use a relatively short time constant.
Universal Time Constant Graphs Rising curve: Value increases 63% with each time constant. Can represent: Capacitor charging voltage, v C Rising inductor current, i L Falling curve: Value decrease 63% with each time constant. Can represent: Capacitor discharging voltage, v C Decaying inductor current, i L