Lesson 15 – Capacitors Transient Analysis. Learning Objectives Calculate capacitor voltage and current as a function of time. Explain Capacitor DC characteristics
Learning Objectives Calculate capacitor voltage and current as
a function of time. Explain Capacitor DC characteristics.
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TRANSIENTS IN CAPACITIVE NETWORKS: THE CHARGING PHASE The
placement of charge on the plates of a capacitor does not occur
instantaneously. Instead, it occurs over a period of time
determined by the components of the network. This period of time is
called the Transient Phase.
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Capacitor Current and Voltage Capacitor v-i relationship
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Capacitor Current and Voltage The charge on a capacitor is
given by: Current (i C ) is the rate of flow of charge: Current
through a capacitor is equal to C times the rate of change of
voltage across it.
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Circuit Analysis (for Physics Majors) Using KVL: Substituting
in using ohms law and the capacitor current relationship: Using
Calculus:
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Capacitor charging Capacitor is initially fully discharged acts
like a short circuit When switch is closed (position 1), the
current instantaneously jumps to:
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Capacitor charging As charge is stored in the capacitor, the
voltage across the capacitor starts to rise. This makes the voltage
drop across the resistor drop, so current in the circuit drops
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Capacitor Charging Equations Voltages and currents in a
charging circuit change exponentially over time
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Steady State Condition (Fully Charged) Circuit is at steady
state When voltage and current reach their final values and stop
changing Capacitor has voltage across it, but no current flows
through the circuit Capacitor looks like an open circuit
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The Time Constant Rate at which a capacitor charges and
discharges depends on R and C, which is called the TIME CONSTANT:
Transients can be considered to last for five time constants
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Example Problem 1 The capacitor in the circuit below is
initially uncharged. After the switch is shut: a. determine how
long it will take for the capacitor to reach a steady-state
condition (>99% of final voltage). b. Write the equation for v c
(t). c. Sketch the transient.
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Capacitor Discharging Capacitor is initially fully charged acts
like a open circuit When switch is moved to discharge, the current
instantaneously jumps to -E/R
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Capacitor Discharging As charge flows out of the capacitor, the
voltage across the capacitor drops. This makes the voltage drop
across the resistor drop, so current in the circuit drops until the
capacitor is fully discharged
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Capacitor Discharging Equations Voltages and currents in a
discharging circuit also change exponentially over time
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More complex circuits If the circuit does not look like the
simple charge-discharge circuit, then you will need to use
Thvenin's Equivalent to make it into the simple circuit. The
circuit below does not have the same charging equation as the
previous circuits, since the voltage drop across the capacitor is
controlled by the voltage divider circuit.
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More complex circuits Thvenin's Equivalent of charging
circuit:
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More complex circuits Now you can calculate the charging time
constant using the Thvenin Equivalent resistance. You write the
charging equation using Thvenin Voltage.
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More complex circuits The discharge portion of the circuit
operates the same as we previously analyzed. The steady-state
(fully charged) voltage across the capacitor can be determined by
the VDR (this is the Thvenin voltage found earlier).
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Example Problem 2 The capacitor in the circuit below is
initially at steady state with the switch open and capacitor fully
discharged. After the switch is shut: (CHARGING) a. determine how
long it will take for the capacitor to fully charge (>99% of
final voltage). b. Write the equation for v c (t). Sketch the
transient.
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Example Problem 2b The capacitor is now fully charged and at
steady-state condition. The switch is opened to start the discharge
cycle. After the switch is open:(DISCHARGING) a. determine how long
it will take for the capacitor to fully discharge. b. Identify the
direction of current flow. c. Write the equation for v c (t).
Sketch the transient.