6
Dynamic Characteristics II
6.1 Introduction
The charge control equations are given for the forward active mode of
the transistor as:
iC 
QF dQBC

F
dt
iB 
QF dQF dQBC dQBE



BF
dt
dt
dt
iE 
QF QF dQF dQBE



F
BF
dt
dt
where all charges and currents are intrinsically taken as instantaneous
functions of time.
Delay Time, td
The above equations do not apply to calculation of the delay time, as
the transistor is not in the forward active mode during this time. The
delay time is, however, very small in relation to the other switching
times and is hence ignored.
Fall-Time, tf : Transistor Turn-On
The fall-time, tf, for the output voltage is obtained by solving the
charge control equations for this condition. To solve the above charge
control equations in this regard, the currents need to be related in
terms of circuit parameters. Since the emitter terminal is grounded,
this cannot be done for the emitter current and hence its charge
control equation is of little benefit. During transistor turn-on in the
forward active mode, the base of the transistor can be taken as sitting
at a potential of VBE ON so that the base current can be assumed
constant during this period as shown in Fig. 6.1.
Hence,
and
iB 
VCC  VBEON
RB
dQBE
dVBE
 CBE
0
dt
dt
1
VCC
iB
VBE ON
RB
CBE
Fig. 6.1
Constant Voltage on Base during Transistor Turn-On
In this case, the charge control equations simplify to:
iC 
QF dQBC

F
dt
iB 
V  VBE ON
QF dQF dQBC


 CC
BF
dt
dt
RB
The equation for iB is expressed in terms of the circuit parameters and
is hence clearly the one to be solved. However, since a solution for ic is
required, substitutions for the QF and dQF/dt terms in terms of ic must
be found. First consider Fig. 6.2
VCC
iC
RC
CBC
+
VO (t)
RB
VBE ON
Fig. 6.2
The Effect of the Collector-Base Junction Capacitance
2
From the circuit:
dQBC
dVBC
d
VBE ON  VO (t)
 CBC
 CBC
dt
dt
dt

But
dQBC
dVO (t)
 CBC
dt
dt
VO (t)  VCC  iCR C

dVO
di
 R C C
dt
dt
dQBC
di
 CBCR C C
dt
dt
Hence
...(1)
Secondly, consider:
iC 
QF dQBC

F
dt
QF  FiC  F
dQBC
dt
so that:
QF  FiC  F CBC R C
Thirdly, take
diC
dt
...2
diC
d2iC
dQF
 F
 F CBC R C
dt
dt
dt 2
During turn-on, the collector current begins to rise towards  F IB but is
limited on reaching saturation to I C MAX   F IB . The rise in collector
current can be seen to have an almost constant rate during this time
as shown in Fig. 6.3 below.
3
diC
 constant
Then, if
dt

d2iC

dt 2
0 and neglecting the second
order derivative gives:
di
dQF
 F C
dt
dt
...3
iC
 FI B
IC
MAX
t
tT URN - ON
Fig. 6.3 Rise in Collector Current During Transistor Turn-On
Substituting equations (1), (2) and (3) to replace the charge terms in
the equation for base current gives:
iB 
Since
V  VBEON
F

di
di
di
iC  F CBCR C C  F C  CBCR C C  CC
BF
BF
dt
dt
dt
RB
F

1
 F 
BF F F F
then this equation can be rearranged as:
1
 di
(VCC  VBE ON)
1
iC   CBCR C  F  CBCR C  C 
F
RB
 F
 dt
4
multiplying by βF gives:
iC  CBCR C  F F  FCBCR C 
(VCC  VBE ON)
diC
 F
dt
RB
If F  1 then (F  1)CBCR C  F CBCR C so that :
iC  F (F  CBCR C )
V - VBE ON
diC
 F CC
dt
RB
This is a 1st order linear differential equation in ic that can be solved by
taking the Laplace Transform. Note that the Laplace Transforms of a
constant, K and a derivative of a function are given as:
L {K} = K/s
L {dy/dt} = s L {y} – y(t=0) = sY(s) – y(t=0)
So that
IC (s)  F (F  CBCR C )[sIC (s) - iC (t  0)]  F
VCC - VBE ON
sR B
IC (s)  F (F  CBCR C )[sIC (s) - iC (t  0)]  F
VCC - VBE ON
sR B
iC(t = 0) is the initial value of collector current during turn-on which
must be taken as zero. Then:
IC (s)1  F (F  CBCR C )s 
IC (s) 
F (VCC  VBE ON)
sR B
F (VCC  VBEON)
sR B 1  F (F  CBCR C )s
5
1
 (V  VBE ON)
F (F  CBCR C )
IC (s)  F CC
RB


1
s s 


(


C
R
)
F
F
BC
C


The solution can be obtained easily by taking the Inverse Laplace
Transform where the Inverse Laplace transform of the term
L -1{a/s(s+a)} = 1 - e-at
which gives
-t

F (VCC  VBE ON) 
F (F  CBCR C )
iC (t) 
1  e

RB


This equation shows that in the linear forward active region, during
transistor turn-on, the collector current rises exponentially from zero
towards a value of
F
VCC  VBE ON
RB
  F IB
The time constant of the expression can be seen to depend on the
transistor properties βF, τF and CBC as well as the value of the external
resistor RC.
6