Transients in Circuits with C, L and R.
Time constants
Aims:
To know:
•Basic features of circuits undergoing a step function or
transient response.
•Role of time constant and how this may influence
measurements.
To be able:
•To analyse some basic circuits.
Lecture 98
Lecture
1
Charging a capacitor through a resistor
What happens when we close
the switch (t=0)?
I
KVL through this
circuit gives
R
VB
C
V
A TRANSIENT
is a one-off change in a
system, like flipping a switch
Differentiating
w.r.t. time:
This is a differential
equation that we can
solve by integration:
Where i0 is the current at time t = 0 (VB/R)
Lecture 98
Lecture
2
1
Charging a capacitor through a resistor
What happens when we close
the switch (t=0)?
KVL through this
circuit gives
I
R
VB = iR + V
VB = iR +
Q
C
VB
C
Differentiating
w.r.t. time:
V
A TRANSIENT
is a one-off change in a
system, like flipping a switch
This is a differential
equation that we can
solve by integration:
R
di i
+ =0
dt C
di
dt
= −∫
i
RC
−t
log e i =
+c
RC
∫
⎛ −t ⎞
i = i0 exp ⎜
⎟
⎝ RC ⎠
Where i0 is the current at time t = 0 (VB/R)
Lecture 98
Lecture
3
Time constant
⎛ −t ⎞
i = i0 exp ⎜
⎟
⎝ RC ⎠
This tells us that the current in the
circuit decays exponentially
The current falls to 1/e of its initial
value at a time τ=RC (seconds
when R is in Ω and C in Farads)
current
I0
I0/e
τ is called the TIME CONSTANT
of the circuit
RC
2RC
time
This is the same expression as for radioactive decay
Lecture 98
Lecture
4
2
Voltage profile …
⎛ −t ⎞
i = i0 exp ⎜
⎟
⎝ RC ⎠
V B/R
VB
Voltage
Current
time
⎛
⎛ −t ⎞ ⎞
V = VB ⎜1 − exp ⎜
⎟⎟
⎝ RC ⎠ ⎠
⎝
Lecture 98
Lecture
5
Voltage profile …
⎛ −t ⎞
i = i0 exp ⎜
⎟
⎝ RC ⎠
We can get the voltage across the capacitor from
V=
i0
⎛ −t ⎞
exp ⎜
⎟dt
C∫
⎝ RC ⎠
i=C
V B/R
VB
Voltage
⎛
⎞
⎛ −t ⎞
V = −VB ⎜ exp ⎜
⎟+ K ⎟
⎝ RC ⎠
⎝
⎠
We know that when t→∞, V = VB,
so K=-1 and
⎛
⎛ −t ⎞ ⎞
V = VB ⎜1 − exp ⎜
⎟⎟
⎝ RC ⎠ ⎠
⎝
dV
dt
Current
time
This is another exponential with a time constant
τ = RC
Lecture 98
Lecture
6
3
Time constant of an LR circuit
In this case KVL gives
I
R
VB
L
or
V
Again we can solve this
equation by integration:
This gives
The time constant of an LR circuit is τ=L/R
(seconds with R in Ω and L in Henry)
Lecture 98
Lecture
7
Time constant of an LR circuit
In this case KVL gives
I
VB = iR + L
R
VB
L
i−
or
V
Again we can solve this
equation by integration:
This gives
di
dt
VB
L di
=−
R
R dt
di
R
= − ∫ dt
VB
L
i−
R
⎛
⎛ t ⎞⎞
i = i0 ⎜1 − exp ⎜ − ⎟ ⎟
⎝ τ ⎠⎠
⎝
∫
where i0 = VB / R and τ =
The time constant of an LR circuit is τ=L/R
(seconds with R in Ω and L in Henry)
Lecture 98
Lecture
L
R
8
4
Voltage profile …
We can get the voltage
across the inductor from
⎛
⎛ − Rt ⎞ ⎞
i = i0 ⎜1 − exp ⎜
⎟⎟
⎝ L ⎠⎠
⎝
V = −L
⎛ − Rt ⎞
V = VB exp ⎜
⎟
⎝ L ⎠
di
dt
I0
VB
Current
Voltage
time
Lecture 98
Lecture
9
Comparison
RC
Driving voltage
RL
Driving voltage
VB
Capacitor voltage
VB
Inductor
voltage
Current
Current
0
time
0
Lecture 98
Lecture
time
10
5
Universal Expressions for Step Functions
Discharging: x = x0 exp(−t / τ )
Charging:
x0 is the starting value
x = x f (1 − exp(−t / τ ) )
xf is the steady state value
τ is the time constant
Capacitors: τ=RC
Inductors: τ=L/R
Time (t=nτ)
τ
2τ
3τ
4τ
5τ
Discharging
exp(-t/τ)
0.367
0.135
0.050
0.018
0.007
Charging
1-exp(-t/τ)
0.633
0.865
0.950
0.982
0.993
Lecture 98
Lecture
11
An application of RC circuits: a flashing light
A NEON LAMP contains low pressure neon
gas in a glass envelope with two isolated
electrodes.
The lamp is a perfect insulator until the
applied voltage reaches a certain value
(typically 60 V) when the gas breaks down,
becomes a good conductor and emits
red/orange light
I
Light
emission
Breakdown
voltage
V
Simplified I-V curve
Lecture 98
Lecture
12
6
An application of RC circuits:
a flashing light
Lamp voltage
Normal charging curve
100 V
60 V
10 kΩ
V
100 V
Current in
lamp
100μF
Neon
lamp
0
T
2T
time
Lecture 98
Lecture
13
Calculating the flashing period
of the circuit
The charging equation is:
V = 100 (1 − exp(−t / RC ) )
RC = 10kΩ × 100μ F = 1 sec,
V = 100(1 − exp(−t ))
The lamp triggers when the voltage
reaches 60V, at time T
60 = 100(1 − exp(−T ))
60
100
T = − log e (0.4)
exp(−T ) = 1 −
T = 916 ms
The lamp flashes at intervals of almost 1 sec
Lecture 98
Lecture
14
7
RC circuit: Response to Pulses
Many test instruments (e.g.
oscilloscope) have an input
equivalent circuit like this.
How does this affect the measurement of
fast voltage changes?
R
V IN
VIN
C
VOUT
VOUT
Short time constant
Output similar to input
Lecture 98
Lecture
15
VIN
VOUT
Long time constant:
output voltage is
distorted
τ
time
Typical values for an oscilloscope probe: R = 10 MΩ, C = 10 pF
Time constant t = RC =
We need to take care in designing probes for measuring fast changes
Lecture 98
Lecture
16
8
VIN
VOUT
Long time constant:
output voltage is
distorted
time
τ
Typical values for an oscilloscope probe: R = 10 MΩ, C = 10 pF
Time constant t = RC = 10 x 106 x 10 x 10-12 = 100 μs.
We need to take care in designing probes for measuring fast changes
Lecture 98
Lecture
C-R Circuit
17
C
VIN
R
Rising voltage step
VOUT
Falling voltage step
VIN
VIN
VOUT = iR
VOUT = iR
time
time
Vout
⎛ −t ⎞
= i0 R exp ⎜
⎟
⎝ RC ⎠
⎛ −t ⎞
Vout = −i0 R exp ⎜
⎟
⎝ RC ⎠
Lecture 98
Lecture
18
9
AC Coupling
VIN
C
VIN
R
VOUT
VOUT
Short time
constant
With a short time constant we
just see the charging and
discharge pulses
VOUT
With a long time constant we see
an alternating voltage similar to
the driving voltage
Long time
constant
Capacitors are used to transmit
alternating voltages with no direct
resistive contact
Lecture 98
Lecture
19
10