Unformatted text preview: Capacitors and Inductors Overview
• Deﬁning equations
• Key concepts and important properties ic
C Symbol  vc + Capacitors and Inductors Dielectric with
permittivity e Conducting plates
each with area A Capacitors: Concept ECE 221 • Series and parallel equivalents
• Integrator
• Diﬀerentiator ic d Portland State University vc Physical Concept Ver. 1.46 • Conceptually, a capacitor consists of two conducting plates
• Charge builds up on each of the plates
• Does this violate KCL (conservation of charge)?
• Capacitors and inductors are almost as common as resistors ECE 221 Capacitors and Inductors Ver. 1.46 • The resistance of physical capacitors is usually ∼100 MΩ and can
be safely ignored
Portland State University Introduction: The Impact of Time • Up to this point we have assumed everything happens
instantaneously • We will now discuss elements that depend on time
• These elements can store energy Ver. 1.46 • At diﬀerent times they may either produce or absorb energy Capacitors and Inductors • Called energy storage elements ECE 221 i(t) C
v(t)  q = Cv charge in Coulombs (C)
capacitance in Farads (F)
voltage in Volts (V) ECE 221 Capacitors and Inductors Ver. 1.46 Capacitors: Relationship of Charge and Potential 1 Portland State University where 3 Portland State University • Capacitance usually represented by the letter C • 1 F = 1 C/V • The “strength” of capacitors is measured in Farads • The charge stored is directly proportional to the voltage q=
C=
v= + 2 4 vc ic d Dielectric with
permittivity e Conducting plates
each with area A 1
C C t0 t
−∞ +
vc
 v(t0 ) v(t) i(t) dt = C dv(t)
i(τ ) dτ = C i(τ ) dτ Capacitors and Inductors ic
C Ver. 1.46 Ver. 1.46 i(t) C
v(t)  Capacitors: Deﬁning Equation Capacitors are linear.
7 Portland State University ECE 221 = C Capacitors and Inductors Capacitors: Linearity ECE 221 v(t) = a1 v1 (t) + a2 v2 (t) Ver. 1.46 Ver. 1.46 i(t) = f (a1 v1 (t) + a2 v2 (t)) = a1 i1 (t) + a2 i2 (t) Capacitors and Inductors d (a
1 v1 (t) + a2 v2 (t))
,
i(t) = C
dt
dv1 (t)
dv2 (t)
+ a2 C
,
= a1 C
dt
dt
= a1 i1 (t) + a2 i2 (t). 1 (t)
2 (t)
If i1 (t) = C dvdt and i2 (t) = C dvdt , then a voltage
v(t) = a1 v1 (t) + a2 v2 (t) would produce i2 (t) = f (v2 (t)) i1 (t) = f (v1 (t)) A device is linear if and only if the relationship between the device
voltage and current is linear: 5 Portland State University This equation deﬁnes the behavior of capacitors (just as Ohm’s law is
the deﬁning equation for resistors). i(t) q(t) = Cv(t)
d
d
q(t) =
Cv(t)
dt
dt
dv(t)
dt + Capacitors: Physical Dimensions dv(t)
dt i(τ ) dτ + v(t0 ) v(t) = ECE 221 dv(t) Symbol Capacitors and Inductors capacitance in Farads
permittivity in C2 /N · m2
area in m2
length in m Physical Concept
eA
, where
C=
d
C=
e=
A=
d= ECE 221 i(t)
v(t)  Capacitors: Voltage versus Current Portland State University i(t) = C t0 t i(τ ) dτ = C dv(τ )
1
C Portland State University Alternatively, if we assume limt→−∞ v(t) = 0 v(t) = t We can also solve for v(t) in terms of i(t): + 6 8 v(τ )C dv(τ )
dτ
dτ i(t) 0.1 mF
v(t) 11 Portland State University ECE 221 v(t) (V)
10
1 ECE 221 2 C v(t) Capacitors and Inductors t (s) i(t) (mA) 1 Capacitors and Inductors Find an expression for i(t) and plot i(t) versus t.  2 Example 1: Capacitor Deﬁning Equations 9 Portland State University Ver. 1.46 t (s) Ver. 1.46 1
• Thus, the energy stored in a capacitor is given by w(t) = 2 Cv(t)2 • By convention, this is the 0reference • At what v can we no longer extract energy from the capacitor? • Similar to potential energy: w = mgh, where is h = 0? • We need a reference energy to answer this question • The equation above only tells us how much energy was stored in
the capacitor over a period of time • We want an absolute measure of energy so we can answer the
question: “How much energy is stored in the capacitor?”  Capacitors: Energy Stored Continued t
to
2 v(t)
1
2 Cv(τ ) v(to ) Ver. 1.46 Ver. 1.46 i(t) Capacitors: Energy Stored  =
= C
v(t)  Capacitors and Inductors i(t) Capacitors and Inductors 1
1
w(t) − w(t0 ) = 2 Cv(t)2 − 2 Cv(to )2 C
v(t) dv(t)
dt From the previous slide, the energy stored by a capacitor from time to
to time t is given by, ECE 221 + i(t) The power absorbed by a capacitor is given by,
p(t) = v(t) × i(t) = v(t) × C v(τ ) dv(τ ) p(τ ) dτ The energy stored by a capacitor from time to to time t is given by,
t
to
v(t)
v(to ) dv(t)
dt Capacitors: Key Concepts ECE 221 1
1
= 2 Cv(t)2 − 2 Cv(to )2 =C w(t) − w(t0 ) = Portland State University i(t) = C • How quickly can the voltage v change? • How quickly can the current i change? • What other circuit element has this property? • Thus, if a DC voltage is applied, i = 0 • If the voltage v(t) is constant, i(t) = 0 for all t + • How much energy does a capacitor dissipate? Portland State University + + 10 12 i(t) i(t)  i(t) (mA) 1 1 2 t (s) v(t) (V) 1 2 Example 2: Capacitor Deﬁning Equations 0.1 mF
v(t) ECE 221 ECE 221 C2
i2 i(t) Capacitors and Inductors Ceq Capacitors and Inductors +
v(t)
 Example 3: Capacitors in Parallel C1
i1 Portland State University Find i1 (t) and i2 (t). +
v(t)
 Portland State University t (s) Ver. 1.46 Ver. 1.46 Find an expression for v(t) and plot v(t) versus t. The voltage across
the capacitor at t = 0 is 5 V. + vs is
i2
C2 i3
C3 vs is Capacitors: Parallel Equivalent
i1
C1 Ceq Right circuit, = Ceq 14 dvs (t)
dt We need to ﬁnd the value of Ceq that makes the relationship between
vs (t) and is (t) the same for both circuits.
Left circuit: is (t) Ver. 1.46 16 = Capacitors and Inductors Ver. 1.46 Ceq Capacitors and Inductors is (t) = i1 (t) + i2 (t) + i3 (t)
dvs (t)
dvs (t)
dvs (t)
= C1
+ C2
+ C3
dt
dt
dt
dvs (t)
dt ECE 221 Example 3: Workspace ECE 221 = (C1 + C2 + C3 ) 13 Portland State University 15 Portland State University +
is C1 = C2
v2  = =
1
Ceq 1
1
1
+
+
C1
C2
C3
−∞ t = ECE 221 C3 C2
t −∞ −∞ +
vs
 is (τ ) dτ + is (τ ) dτ is 1
C3 Capacitors and Inductors Capacitors and Inductors Example 4: Workspace ECE 221 is (τ ) dτ Ceq Portland State University Portland State University t Capacitors: Series Equivalent v1 v3  vs
 + = v1 (t) + v2 (t) + v3 (t)
t
1
1
is (τ ) dτ +
C1 −∞ + vs (t) vs (t) + t C is (τ ) dτ eq −∞ Ver. 1.46 Ver. 1.46 v(t) i(t) C1
+ v1 +
v2
 i(t) Capacitors and Inductors Ceq Example 4: Capacitors in Series v(t) ECE 221 C2 Find v1 (t) and v2 (t). 17 Portland State University Capacitors: Series & Parallel Summary Ver. 1.46 • The equivalent capacitance of N capacitors connected in parallel
is the sum of the individual capacitances Ceq = C1 + C2 + · · · + CN • The equivalent capacitance of N capacitors connected in series is
the reciprocal of the sum of reciprocals of the individual
capacitances 1
1
1
1
=
+
+ ··· +
Ceq
C1
C2
CN Ceq = C1 C2
C1 + C2 Capacitors and Inductors Ver. 1.46 • Two capacitors in series have an equivalent capacitance given by ECE 221 • Will not discuss delta and wye conﬁgurations 19 Portland State University 18 20 C
eq 10 µF 40 µF 50 µF Example 5: Capacitor Network Equivalent
60 µF 30 µF
20 µF  v(t) +
L i(t) Capacitors and Inductors Number of
Turns, N Inductors: Concept ECE 221 Find the equivalent capacitance Ceq . Portland State University Core
Material
i(t)
+
v(t)
Crosssecitonal
area, A Ver. 1.46 Length, l N 2 µA , where  v(t) + Core
Material
i(t) Capacitors and Inductors Example 5: Workspace ECE 221 L=
N=
µ=
A=
= ECE 221 + v(t)  i(t) L Capacitors and Inductors Ver. 1.46 Ver. 1.46 inductance in Henrys (H)
number of turns
magnetic permeability of the core
crosssectional area (m2 )
length (m) Crosssecitonal
area, A Number of
Turns, N Inductor: Relationship Physical Dimensions 21 Portland State University L= 23 Portland State University Length, l • Conceptually, an inductor can be thought of a coil of conducting
wire wrapped around some magnetic material Capacitors and Inductors Ver. 1.46 • Energy is stored in the magnetic ﬁeld created by current ﬂowing
through the wire ECE 221 • Inductance (strength) measured in henrys (H) Portland State University 22 24 Faraday’s Law:
v
φ
v +
v
 where
v=
N=
φ=
t=
P=
i=
L= φ Ver. 1.46 voltage in volts (V)
number of turns
magnetic ﬂux in webers (Wb)
time in seconds (s)
permeance of the ﬂux space
current in amperes (A)
inductance in henrys (H) Capacitors and Inductors L General Inductor Physics i ECE 221 dφ
= N
dt
= N Pi
dφ di
= N
di dt
di
dt
di
dt = N 2P = L Portland State University L
 Inductors: Comments
i(t)
+ v(t) • Like capacitors, inductors are linear devices (proof left as exercise) 1
L t
t0 L
+ v(t)  t0 p(τ ) dτ 1
w(t) = 2 Li2 (t) 1
1
= 2 Li2 (t) − 2 Li2 (t0 ) w(t) − w(t0 ) = t Inductors: Deﬁning Equations
i(t) v(τ ) dτ + i(t0 ) di(t)
v(t) =L
dt
i(t) = p(t) =v(t) i(t) ECE 221 ECE 221 Capacitors and Inductors L Capacitors and Inductors  Ver. 1.46 Ver. 1.46 28 26 We can not extract energy from the inductor when i = 0, so we use
this as our reference for 0 energy. 25 Portland State University i(t) + v(t) Inductors: Key Concepts
di(t)
v(t) = L
dt
• If the current i is constant, v = 0
• Thus, if a DC current is applied, v = 0 27 Portland State University • How much energy does an inductor dissipate? • What other circuit element has this property?
• How quickly can the voltage v change? Ver. 1.46 • Physical (i.e. practical, nonideal) inductors
– Range is typically from a few µH to tens of henrys
– Resistance is usually not negligible (0.1–100 Ω) Capacitors and Inductors • How quickly can the current i change? ECE 221 • Capacitors and inductors are duals Portland State University i(t) 1 i(t) (A) 1 2 t (s) ECE 221 L2
v2  +
vs
 is Capacitors and Inductors L3 Inductors: Series Equivalent
v1 v3  is Left circuit: = 2 t (s) Ver. 1.46 Leq Ver. 1.46 = Leq Right circuit,
vs (t)
Leq Capacitors and Inductors vs (t) = v1 (t) + v2 (t) + v3 (t)
dis (t)
dis (t)
dis (t)
= L1
+ L2
+ L3
dt
dt
dt
dis (t)
dt ECE 221 = (L1 + L2 + L3 ) Portland State University dis (t)
dt We need to ﬁnd the value of Leq that makes the relationship between
vs and is the same for both circuits. + 1 v(t) (mV) Example 6: Inductor Deﬁning Equations 10 mH
v(t)  + Portland State University vs +  + L1 Find an expression for v(t) and plot v(t) versus t. + i(t) i(t) v(t) (mV)
10
1 2 t (s) i(t) (A) 1 Example 7: Inductor Deﬁning Equations 10 mH
v(t)  L1
+ v1  +
v2
 29 Portland State University v(t) ECE 221 i(t) Capacitors and Inductors Capacitors and Inductors Leq Example 8: Inductors in Series v(t) ECE 221 L2 Find v1 (t) and v2 (t). 31 Portland State University 2 t (s) Ver. 1.46 Ver. 1.46 32 30 Find an expression for i(t) and plot i(t) versus t. The initial current in
the inductor at t = 0 is 0.5 A. + Capacitors and Inductors Example 8: Workspace ECE 221 ECE 221 L2
i2 i(t) Leq Capacitors and Inductors +
v(t)
 Example 9: Inductors in Parallel L1
i1 Portland State University i(t) +
v(t)
 Find i1 (t) and i2 (t). Portland State University Ver. 1.46 Ver. 1.46 vs Left: is (t) is =
= Right: is (t) = = = Leq 33 Portland State University 35 Portland State University i1 i2
L2 i3
L3 1
L2
t
−∞ −∞ vs is vs (τ ) dτ + vs (τ ) dτ 1
L3 Capacitors and Inductors vs (τ ) dτ t Inductors: Parallel Equivalent L1 t
−∞ vs (τ ) dτ + i1 (t) + i2 (t) + i3 (t)
1
L1 −∞ t 1
1
1
+
+
L1
L2
L3
1
Leq ECE 221 Capacitors and Inductors Example 9: Workspace ECE 221 Leq t 34 vs (τ ) dτ Ver. 1.46 36 −∞ Ver. 1.46 Inductors: Series & Parallel Summary
• The equivalent inductance of N inductors connected in series is
the sum of the individual inductances
Leq = L1 + L2 + · · · + LN
• The equivalent inductance of N inductors connected in parallel is
the reciprocal of the sum of reciprocals of the individual
inductances
1
1
1
1
=
+
+ ···+
Leq
L1
L2
LN Leq = L1 L2
L1 + L2 Capacitors and Inductors Capacitors and Inductors Ver. 1.46 Ver. 1.46 • Two inductors in parallel have an equivalent inductance given by ECE 221 Example 10: Workspace ECE 221 • Similar to resistors Portland State University Portland State University Leq 20 mH 60 mH 10 mH 50 mH 40 mH Capacitors and Inductors 70 mH Example 10: Inductor Network Equivalent
50 mH 30 mH C ECE 221 RL Capacitors and Inductors vo Example 11: Capacitive Integrator ECE 221 Find the equivalent inductance Leq . R 37 Portland State University vi Solve for vo (t). 39 Portland State University Ver. 1.46 Ver. 1.46 40 38 vi L Solve for vo (t). R Capacitors and Inductors vo Example 12: Inductive Integrator
R RL ECE 221 L ECE 221 RL Capacitors and Inductors vo Example 14: Inductive Diﬀerentiator Portland State University v
i Solve for vo (t). Portland State University Ver. 1.46 Ver. 1.46 vi C R ECE 221 RL Capacitors and Inductors vo Example 13: Capacitive Diﬀerentiator Solve for vo (t). 41 Portland State University Applications: Comments Ver. 1.46 Ver. 1.46 • Recall: the op amp is useful for implementing a wide range of
equations
• Integrators tend to saturate • Often have a feedback resistor to eliminate this problem Capacitors and Inductors • Will be a key building block in ECE 222
• Diﬀerentiators amplify noise ECE 221 • Rarely used in practice 43 Portland State University 42 44 Summary
• Capacitors and inductors store energy Capacitors and Inductors Ver. 1.46 • They have many dual properties
– Capacitors store charge, inductors store energy in a magnetic
ﬁeld
– Networks of both devices in parallel and series combinations
have a single device equivalent
– Both have a deﬁning equation in the form of a ﬁrstorder
diﬀerential equation
– Capacitors prevent instantaneous voltage changes, inductors
prevent instantaneous current changes ECE 221 • This makes time an important factor in circuit analysis Portland State University 45 ...
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 Fall '08
 MEISSNER
 Energy, Inductor, Portland State University

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