Inductors and Capacitors
• Inductor is a Coil of wire wrapped around a supporting (mag or non mag) core
• Inductor behavior related to magnetic field
• Current (movement of charge) is source of the magnetic field
• Time varying current sets up a time varying magnetic field
• Time varying magnetic field induces a voltage in any conductor linked by the field
• Inductance relates the induced voltage to the current
• Capacitor is two conductors separated by a dielectric insulator
• Capacitor behavior related to electric field
• Separation of charge (or voltage) is the source of the electric field
• Time varying voltage sets up a time varying electric field
• Time varying electric field generates a displacement current in the space of field
• Capacitance relates the displacement current to the voltage
• Displacement current is equal to the conduction current at the terminals of capacitor
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Inductors and Capacitors (contd)
• Both inductors and capacitors can store energy (since both magnetic fields and
electric fields can store energy)
• Ex, energy stored in an inductor is released to fire a spark plug
• Ex, Energy stored in a capacitor is released to fire a flash bulb
• L and C are passive elements since they do not generate energy
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Inductor
• Inductance symbol L and measured in Henrys (H)
• Coil is a reminder that inductance is due to conductor
linking a magnetic field
• First, if current is constant, v = 0
• Thus inductor behaves as a short with dc current
• Next, current cannot change instantaneously in L i.e.
current cannot change by a finite amount in 0 time since an
infinite (i.e. impossible) voltage is required
• In practice, when a switch on an inductive circuit is opened,
current will continue to flow in air across the switch (arcing)
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Inductor: Voltage behavior
• Why does the inductor voltage
change sign even though the
current is positive? (slope)
• Can the voltage across an
inductor change
instantaneously? (yes)
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Inductor: Current, power and energy
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Inductor: Current behavior
• Why does the current approach a constant
value (2A here) even though the voltage across
the L is being reduced? (lossless element)
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Inductor: Example 6.3, I source
• In this example, the excitation comes from a
current source
• Initially increasing current up to 0.2s is storing
energy in the inductor, decreasing current after
0.2 s is extracting energy from the inductor
• Note the positive and negative areas under the
power curve are equal. When power is positive,
energy is stored in L. When power is negative,
energy is extracted from L
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Inductor: Example 6.3, V source
• In this example, the excitation comes from a
voltage source
• Application of positive voltage pulse stores
energy in inductor
• Ideal inductor cannot dissipate energy – thus a
sustained current is left in the circuit even after
the voltage goes to zero (lossless inductor)
• In this case energy is never extracted
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Capacitor
• Capacitance symbol C and measured in Farads (F)
• Air gap in symbol is a reminder that capacitance occurs
whenever conductors are separated by a dielectric
• Although putting a V across a capacitor cannot move
electric charge through the dielectric, it can displace a
charge within the dielectric displacement current
proportional to v(t)
• At the terminals, displacement current is similar to
conduction current
• As per above eqn, voltage cannot change
instantaneously across the terminals of a capacitor i.e.
voltage cannot change by a finite amount in 0 time since an
infinite (i.e. impossible) current would be produced
• Next, for DC voltage, capacitor current is 0 since
conduction cannot happen through a dielectric (need a time
varying voltage v(t) to create a displacement current).
Thus, a capacitor is open circuit for DC voltages.
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Capacitor: voltage, power and energy
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Capacitor: Example 6.4, V source
• In this example, the excitation comes from a
voltage source
• Energy is being stored in the capacitor
whenever the power is positive and delivered
when the power is negative
• Voltage applied to capacitor returns to zero with
increasing time. Thus, energy stored initially (up
to 1 s) is returned over time as well
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Capacitor: Example 6.5, I source
• In this example, the excitation comes from a
current source
• Energy is being stored in the capacitor
whenever the power is positive
• Here since power is always positive, energy is
continually stored in capacitor. When current
returns to zero, the stored energy is trapped
since ideal capacitor. Thus a voltage remains on
the capacitor permanently (ideal lossless
capacitor)
• Concept used extensively in memory and
12 imaging circuits
Series-Parallel Combination (L)
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Series Combination (C)
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Parallel Combination (C)
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First Order RL and RC circuits
• Class of circuits that are analyzed using first order
ordinary differential equations
• To determine circuit behavior when energy is
released or acquired by L and C due to an abrupt
change in dc voltage or current.
• Natural response: i(t) and v(t) when energy is
released into a resistive network (i.e. when L or C is
disconnected from its DC source)
• Step response: i(t) and v(t) when energy is
acquired by L or C (due to the sudden application of a
DC i or v)
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Natural response: RL circuit
• Assume all currents and voltages in circuit
have reached steady state (constant, dc) values
Prior to switch opening,
• L is acting as short circuit (i.e. since at DC)
• So all Is is in L and none in R
• We want to find v(t) and i(t) for t>0
• Since current cannot change
instantly in L, i(0-) = i(0+) = I0
• v(0-) = 0 but v(0+) = I0R
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Natural response time constant
• Both i(t) and v(t) have a term
• Time constant τ is defined as
• Think of τ as an integral parameter
• i.e. after 1 τ, the inductor current has been reduced to e-1 (or
0.37) of its initial value. After 5 τ, the current is less than 1%
of its original value (i.e. steady state is achieved)
• The existence of current in the RL circuit is momentary –
transient response. After 5τ, cct has steady state response
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Extracting τ
• If R and L are unknown
• τ can be determined from a plot of the natural
response of the circuit
• For example,
• If i starts at I0 and decreases at I0/τ, i becomes
• Then, drawing a tangent at t = 0 would yield τ at the x-axis intercept
• And if I0 is known, natural response can be written as,
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Example 7.1
• To find iL(t) for t ≥0, note that since cct is in
steady state before switch is opened, L is a short
and all current is in it, i.e. IL(0+) = IL(0-) = 20A
• Simplify resistors with Req = 2+40||10 = 10Ω
• Then τ = L/R = 0.2s,
• With switch open,
• voltage across 40 Ω and 10 Ω,
• power dissipated in 10 Ω
• Energy dissipated in 10 Ω
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Example 7.2
• Initial I in L1 and L2 already
established by “hidden sources”
• To get i1, i2 and i3, find v(t) (since
parallel cct) with simplified circuit
• Note inductor current i1 and i2 are
valid from t ≥ 0 since current in
inductor cannot change
instantaneously
• However, resistor current i3 is valid
only from t ≥ 0+ since there is 0
current in resistor at t = 0 (all I is
shorted through inductors in steady
21 state)
Example 7.2 (contd)
• Initial energy stored in inductors
• Note wR + wfinal = winit
• wR indicates energy dissipated in
resistors after switch opens
• wfinal is energy retained by inductors
due to the current circulating between
the two inductors (+1.6A and -1.6A)
when they become short circuits at
22 steady state again
Natural Response of RC circuit
• Similar to that of an RL circuit
• Assume all currents and voltages in circuit
have reached steady state (constant, dc) values
Prior to switch moving from a to b,
• C is acting as open circuit (i.e. since at DC)
• So all of Vg appears across C since I = 0
• We want to find v(t) for t>0
• Note that since voltage across capacitor
cannot change instantaneously, Vg = V0, the
initial voltage on capacitor
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Example 7.3
• To find vC(t) for t ≥0, note that since cct is in
steady state before switch moves from x to y, C is
charged to 100V. The resistor network can be
simplified with a equivalent 80k resistor.
• Simplify resistors with Req = 32+240||60 = 80kΩ
• Then τ = RC = (0.5µF)(80kΩ)=40 ms,
• voltage across 240 kΩ and 60 kΩ,
• current in 60 kΩ resistor
• power dissipated in 60 kΩ
• Energy dissipated in 60 kΩ
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Example 7.4: Series capacitors
• Initial voltages
established by
“hidden” sources
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Step response of RL circuits
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Example 7.5: RL step response
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Step response of RC circuits
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Example 7.6: RC Step Response
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