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National Taiwan University
Department of Electrical Engineering
電路學 (Electric Circuit)
Chapter 5
Energy Storage and Dynamic Circuits
Ch5 (^2) EC ; Hsin-Shu Chen
Chapter 5 Outline
Energy Storage and Dynamic Circuit (Time Varying Characteristics)
Capacitor Branch (Device) Equation in Differential Form Power and Stored Energy Memory and Continuity Parallel and Series
Inductor(Dual)
Dynamic Circuit (Differential Equation) Natural Response (No Input, Homogeneous Solution) Forced Response(No Initial Condition, Particular Solution) Complete Response = Natural Response+ Forced Response Transient Response, Steady-State Response
Ch5 (^3) EC ; Hsin-Shu Chen
5.1, 5.2 Self Study
5.1,5.2 self study with quick review
R → Branch (or Device) equation R = v / i Memoryless → v = i R : independent of time Time-invariant currents or voltages C and L Memory Time-varying currents or voltages The fourth element: Memristor M(q)
Ch5 (^4) EC ; Hsin-Shu Chen
5.1 Capacitor
Definition of capacitance q: charge; C: capacitance in farad (F) Branch equation
vC: continuity of voltage when i is finite DC open circuit (i or iC=0) Instantaneous power
Instantaneous stored energy
Electrical memory
C q
v
; (or C)
dq dCv dv
i C v v
dt dt dt
is the preferred variable for capacitor
dt
P vivCdv
w Cv
(^) (^) t t
t i d C i d vt C
v (t )^1 () ( 0 )^10 ( )
Ch5 (^7) EC ; Hsin-Shu Chen
5.2 Memristor
The fourth two-terminal passive element: “Memory Resistor” Memristor First theoretically proposed by L. O. Chua(蔡少棠) in 1971 After almost 40 years, in 2008, Strukov et al. experimentally made a nano-scale device to prove its existence. Definition of memristor : flux linkage; q: charge M is a function of q, Memristor is a nonlinear element with hysteretic current-voltage behavior Memristance effect increases as the inverse square of device size D Work under AC conditions. Example device:
Ref: D. B. Strukov, et al, “The missing memristor found,” Nature 453, 80–83 (2008)
M q( ) d
dq
^
Ch5 (^8) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (1)
A circuit is dynamic when currents or voltages are time-varying. Dynamic circuits are described by differential equations. Order of the circuit is determined by order of the differential equation or the number of energy-storage elements.
The differential equations are derived based on Kirchhoff’s laws (KCL, KVL) Device (branch) equations ( ).
dt
v Ldi
dt
v iR,iCdv,
Ch5 (^9) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (2)
First-order Dynamic Circuits in-series KVL: vL+vR=v
inhomogeneous linear first-order differential equation Generic form
a 1 , a 0 : constants related to element values f(t) : forcing function one resistance and one energy-storage element first-order use i=iL
di
L Ri v
dt
dy
a a y f t
dt
Ch5 (^10) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (3) – Ex: First-Order Circuits
in parallel in series
KCL: KVL: R C C
C C
dv
C v i v v v Ri v
dt R
dv
RC v
dt
v=vC i → vC
Generally, use vC or iL
Ch5 (^13) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (6) – Example 5.8: Second-Order CCCS
Amplifier Circuits
2 2
KV L: in KCL: out 0
out out in
di dv
v L Ri i C
dt dt
d v dv
LC RC v
dt dt
Find vout vs. vin two independent energy-storage elements second order In series In parallel
Either i or vout can be chosen. Here vout is used. Combine the two differential equations by replacing i
Ch5 (^14) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (7) - Complete Response
Generic form of dynamic circuits
First-order
Second-order
y can be v(t) or i(t)
Complete response is the sum of the natural response and the force response, i.e. y(t) = yN(t) + yF(t). Natural response yN(t): no input (homogeneous equation) and homogeneous solution Force response yF(t): no initial condition (inhomogeneous equation) and particular solution
a dy a y f t dt
2 (^2) dt ay ft ady dt a dy
Ch5 (^15) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (8) - Natural Response
Natural response yN(t) is the solution of the circuit equation with the forcing set to zero, also called the complementary solution or homogeneous solution. The differential equation becomes homogeneous with the forcing function set to zero. A homogeneous differential equation can be solved using the characteristic equation along with the initial condition. Order of the circuit corresponds to the number of initial conditions, ( or the number of energy-storage elements )
Ch5 (^16) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (9) - Natural Response
First-order circuits ─ Homogeneous first-order differential equation with initial condition yN (0+) Characteristic equation
A can be determined by initial condition yN (0+)
the natural response of a first-order circuit with a 0 /a 1 >0 starts at the initial value Y 0 and decays exponentially toward zero as t
(^01)
(^01)
1 0
1 0 0 1
0 0
0 one root
( )
aa
aa
N N
st t N
t N N
a dy a y dt
a s a s a a y t Ae Ae
y A Y y t Y e t
Ch5 (^19) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (12) – Example 5.9: Capacitor Discharge
v V
R M
C F
600
600
0 with (0 ) 1000
: time constant; unit in second
(0 ) (0 ) 1000 (continuity)
t
t
N N
N N N
N N N N
KCL v^ Cdv
R dt
RC dv v v V
dt
RCs s
RC
RC
v t Ae
A v v
v t e V t
Find vN(t), t > 0 page 10
Ch5 (^20) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (13) - Forced Response
Forced response yF(t) is the solution of the inhomogeneous differential equation (i.e. the forcing function is not zero), independent of any initial conditions, also called particular solution.
Method of undetermined coefficients TABLE 5.3 Selected Trial Solutions for Forced Response
DC Chapter
a is the same!
AC Chapter6, ω is the same!
0 0 1 1 0 2 2 3 4
( ) ( ) (a constant) (a constant)
cos sin
F
at at
f t y t k K k t K t K k e K e k t k t
K 3 cos t K 4 sint
Ch5 (^21) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (14) – Example 5.10: Sinusoidal Forced
Response
( ) 25sin 30 1 40
0.1 F (^4) F 25sin 30 0 cos 30
R
L H
v t t L L R
KVL L di Ri v dt di (^) i t t dt
voltage source; has no initial condition time constant
inhomogeneous differential equation
Find iF(t), t > 0
page
Ch5 (^22) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (15) – Example 5.10 Sinusoidal Forced
Response
3 4 3 4 3 4
3 4 3 4 3 4 3 4 3
( ) cos30 sin 30 Find and 30 sin 30 30 cos 30
(4 3 ) cos 30 ( 3 4 )sin 30 25sin 30 4 3 0 and 3 4 25 3 and
F F
F F
i t K t K t K K di (^) K t K t dt di (^) i K K t K K t t dt K K K K K K
(1) From Table 5.
substitute and
iF ( )t 3cos30 t 4sin 30 t or i (^) F( )t A cos(30 t ) 5cos(30 t 53.1 )
Note: sum of the two components is another sinusoidal waveform, frequency is the same!
Ch5 (^25) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (18) - Complete Response
Complete response is the sum of the natural response and the force response, i.e. y(t) = yN(t) + yF(t).
The constants in yN(t) are evaluated from the initial conditions (I.C.) on the complete response y(t). (Not just natural response itself)
Ch5 (^26) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (19) – Example 5.12 Complete
Response Calculation (RL)
R=4Ω, L=0.1H, Find complete response 0.1(di/dt) + 4i = v, i(t) = 0 for t < 0, i(0-) = I.C.: i(0+) =i(0-) = 0 (this I.C. is not for iN(t) only) Forcing function: v(t) = 400sin280t, for t > 0 AC input homogeneous solution : iN(t) = Ae–^40 t particular solution : iF(t) = – 14cos280t + 2sin280t complete response : i(t) = iN(t) + iF(t) = Ae –^40 t^ – 14cos280t + 2sin 280t i(0+) = 0 = A – 14 A = 14 Obviously, it is not A = 0
Ch5 (^27) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (20) – Example 5.12 Complete
Response Calculation (RL)
Transient state → iN(t) and iF(t) Steady state → iF(t) Frequencies of input and output are the same! AC output i(t) = 14e –^40 t^ – 14cos280t + 2sin 280t
5x(1/40)
Ch5 (^28) EC ; Hsin-Shu Chen
5.3 Dynamic Circuits (21)
For a stable circuit, y(t) = yF(t) as t since yN(t) 0 as t . The circuit is in a steady state if yN(t) is negligible compared with yF(t). Before arriving the steady state, the circuit is in the transient state, which involves both yN(t) and yF(t).
