Transcript Chapter 6-Extra

CHAPTER 5
The Energy Storage
Elements
The capacitor is a device that stores charge
The charge and the applied voltage are related
by the capacitance of the capacitor as
Q = CV
For a time varying source v(t) we have
q(t) =C v(t)
The unit of capacitance is
coulomb/volt or Farad
For plates if size A (m2) separated by distance
d(m) with air in between the capacitance is:
C = 8.84×10¹²
ֿ A/d
Figure 5.1 A parallel-plate capacitor.
F
q  t   C v t 
dq t 
i t  
dt
Passive sign convention
dv t 
i t   C
dt
1
v t  
C
 i  d 
t

1
v t  
C
1
v t  
C

t

i   d 
1
 i  d   C
 i  d 
1
v t 0  
C
i   d 
t0
1
v t  
C

t0

t
t0
 i  d  v t 
t
t0
0
Capacitor Energy
The instantaneous power delivered to the capacitor is:
p t   v t  i t 
 dv t  
dv t 
 v t  C
  Cv t 
dt 
dt

The energy stored in a capacitor at a particular time can be found by:
w t    p   d 
t
Assuming capacitor voltage is zero at t  

dv  
 C  v  
d

d
t
1
1
2
2
 Cv t   Cv   
2
2
1 2
w t   Cv t 
2
Example 5.1
A voltage source is applied to a 5-F capacitor as shown.
Sketch the capacitor current and the stored energy as a
function of time.
dv t 
i t   C
dt
dv t 
5
dt
1
2
w t   C v t 
2
1
2
  5v s t 
2
Example 5.2
A current source is applied to a 5-F capacitor.
Sketch the capacitor voltage as a function of time.
t
1
v t    i s  d
5 
The capacitor voltage is related to
area Under current source.
For example area at t = 1 s is
Area of Triangle is (1)(10)=10
V(1s)=10/5= 2V.
For example area at t=3 s is
10+10+5 = 25 hence
V(3s)=25/5= 5V.
5.1.1 Capacitors in Parallel
The terminal voltage is equal to the voltage of each capacitors
v(t) = v1(t) = v2(t)=….=vn(t)
The current entering the combination is the sum of currents of each capacitor
i(t) = i1(t)+i2(t)+…+in(t)
dv t 
From relation i t   C
d t 
dv t 
dv t 
dv t 
i t   C 1
C 2
 C n
dt
dt
dt
dv t 
dv t 
 C 1  C 2   C n 
C
dt
d t 
C eq  C 1  C 2 
C n
5.1.1 Capacitors in Series
v(t) = v1(t) + v2(t) + …. + vn(t)
i(t) = i1(t) = i2(t) = … = in(t)
t
t
1
1
v t    i 1  d   v 1 t 0  
i 2  d   v 2 t 0  

C1 t0
C 2 t0
 1
1
 

 C1 C 2
t
1 

  i  d   v 1 t 0   v 2 t 0  
C n  t0
t
1

i n  d   v n t 0 

C n t0
 v n t 0 
1
1
1



C eq C 1 C 2

1
Cn
The Inductor
A device that stores energy in its magnetic field
 t   Li t 
The current produces a magnetic field round the wire
The relation between the voltage and current of inductor is
di t 
v t   L
dt
This relation can be inverted to yield
1
i t  
L
t
t
1
v  d   L tv  d   i t 0 
0
The energy stored in an inductor
The instantaneous power of an inductor
di t 
p t   v t  i t   Li t 
dt
Thus the energy stored in an inductor is
t
1 2
1 2
w t    p  d   Li t   Li   
2
2

1 2
w t   Li t 
2
Example 5.3
A current source is applied to a 5-H inductor as shown. Sketch the
voltage across the inductor versus time.
di t 
v t   L
dt
1 2
w t   Li t 
2
Example 5.4
A voltage source is applied to a 5-H inductor as shown.
Sketch the inductor current versus time.
1
i t  
L
t
 v  d 

1
i t  
L
t
 v  d 

Inductors in Series
v t   v 1 t   v 2 t    v n t 
di t 
di t 
di t 
v t   L1
 L2
  Ln
dt
  L1  L 2 
dt
di t 
 Ln 
dt
dt
Leq  L1  L 2 
 Ln
Inductors in Parallel
i t   i 1 t   i 2 t  
 i n t 
t
1
i t   v 1  d   i 1 t 0  
L1 t 0
v t   v 1 t   v 2 t  
1
i t    
 L1
t
1

v n  d   i n t 0 

Ln t0
 v n t 
t
1 
  v  d   i 1 t 0  
Ln  t0
 i n t 0 
1 1
1
  
Lequ  L1 L 2
1 


Ln 
v(t)=2t
V(t)= -2t+4
0<t<1
1<t<2
iL=t2/2
iL= -t2/2+2t
0<t<1
1<t<2