2. CAPACITANCE AND INDUCTANCE IN D.C. CIRCUITS
Main things to learn
• Transients in CR and LR circuits
• Time constant
C
E
R
L
E
R
In a steady d.c. circuit: C is equivalent to the open circuit
L is equivalent to the short circuit
Our aim is to study the transient processes in CR and LR circuits
What will be happening in such circuits just after the switch is turned on / off ?
TRANSIENT IN A CR CIRCUIT
A
I
E
R
+q
−q
C
B
q - charge at the plates of the capacitor
E - electromotive force in the battery
I - current in the direction of the arrow
UAB - potential difference between A and B
q = 0 , UAB = 0 , I = 0
UAB = E , q = C·E , I = 0
Before the switch is turned on:
Long time after (established):
After the switch is turned on, a transient from q = 0 to q = C·E takes place
How to analyse the transient? - The transient is slow, therefore
at every moment of time t , the 2nd Kirchhoff’s law is valid
UAB(t) - E = - I (t)·R
q(t) = C ·UAB(t)
Relation between charge and current (note sign): I (t ) =
dq (t )
dt
These three equations result in a differential equation
q (t )
dq (t )
We denote
− E = −R
C
dt
y (t ) = q(t ) − CE
dq (t )
1
=−
(q(t ) − CE )
τ = RC
dt
RC
dy
y
=−
y = y 0 exp( − t )
τ
dt
τ
( y 0 - value of y at t = 0 : y 0 = −CE )
∴
q(t ) = CE [1 − exp( − t
U AB = E [1 − exp( − t
I (t ) =
τ = RC
- time constant
E
R
exp( − t
RC
)] = CE [1 − exp( − t )]
RC
RC
τ
)] = E [1 − exp( − t )]
τ
)=
E
R
exp( − t )
τ
2τ
3τ
4τ
t
τ
exp (-t/ τ ) 0.37 0.14 0.05 0.02
ANALYSIS OF OBTAINED RESULTS
UAB
I
E
E
R
t
t
At t = 0 :
q = 0; U AB = 0; I = E
At t = ∞ :
q = CE ; U AB
R
=E; I =0
Though the capacitor provides the open circuit,
the current at t = 0 is such as if it is a short circuit !
This happens because at t = 0, UAB = 0
τ = RC
R = 1 MΩ and C = 1 µF :
τ = 1 sec
TRANSIENT IN A LR CIRCUIT
I
E
L
R
L - self-inductance of the coil
Ei -EMF of the electromagnetic induction
Φ - flux of magnetic field through the coil
Φ = L·I
Reminder: Electromagnetic induction in the coil
dΦ(t )
dI (t )
= −L
Ei = −
dt
dt
I=0
I=E⁄R
Before the switch is turned on:
Long time after (established):
After the switch is turned on, a transient from I = 0 to I = E
⁄ R takes place
At every moment of time t , the 2nd Kirchhoff’s law is valid
E + Ei = I ⋅ R
A differential equation is obtained
dI (t )
= I ⋅R
E −L
dt
dI (t )
R
E
= − (I (t ) − )
dt
L
R
dy
y
=−
τ
dt
We denote
y (t ) = I (t ) −
∴
E
and
R
y = y 0 exp( − t )
( y 0 - value of y at t = 0 :
τ
E
y0 = −
R
)
t ⋅R
E
)] = [1 − exp( − t )]
τ
R
L
R
t ⋅R
E i (t ) = E exp( −
) = E exp( − t )
τ
L
I (t ) =
τ =L
E
R
[1 − exp( −
- time constant for an LR circuit
τ =L
R
ANALYSIS OF OBTAINED RESULTS
E
Ei
I
t
R
t
At t = 0 :
At t = ∞ :
−E
I = 0; E i = −E
I=
E
R
; Ei = 0
Though the coil provides the short circuit (its resistance is almost zero),
the current at t = 0 is such as if it is an open circuit
At t = 0, the external EMF is fully compensated by the EMF of self-inductance
τ =L
R
R = 1 Ω and L = 1 henry :
τ = 1 sec
CONCLUSION
If a circuit includes a capacitance or an inductance,
final values of currents or charges cannot establish instantaneously
There are always transient processes which are characterised
by time constants
For an RC circuit
time constant
τ = RC
For an LR circuit
time constant
τ =L
R
ENERGY STORED IN A SELF-INDUCTANCE (*)
E, r
R
R
L
I
L
After the switch is turned off, the current is driven
by the EMF Ei which is due to the self-inductance L
Ei = −L
∆I
∆t
(I changes from I 0 to 0)
Work done by the self-inductance EMF
Φ ⋅ I0 L ⋅ I0 2
A = ∑ Ei ⋅ I∆t = −L ∑ I∆I = −∑ Φ∆I =
=
2
2
This work was done due to energy W which was accumulated in the coil
L ⋅I2
W =
2
- energy stored in a self-inductance
ENERGY OF MAGNETIC FIELD (*)
L ⋅I2 1
W =
= ⋅ µ0 ⋅ n 2 ⋅V ⋅ I 2
2
2
B = µ0 ⋅ n ⋅ I
W =
1
2µ 0
⋅ B 2 ⋅ V - energy of magnetic field in the volume V
B2
w=
2µ 0
It is similar to w =
- energy density of magnetic field
ε 0E 2
2
- energy density of electric field
This similarity reflects the fact that both electric and magnetic fields
are manifestations of the electromagnetic field