Page 268 - Mechatronics with Experiments

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V(t) R V(t) L V(t) C

ω
1
ω
ω
Z( j ) = R Z( j ) = j .L Z( j ) =ω j Cω
FIGURE 5.6: Concept of impedance: generalized resistance as impedance.

filter between input and output voltages), the transfer function between them is
t
1
V (t) = i( )d (5.49)
C
C ∫ 0
1
V (jw) = ⋅ i(jw) (5.50)
C
C ⋅ jw
i(jw) = C ⋅ jw ⋅ V (jw) (5.51)
C
and
V (jw)
C 1
= (5.52)
V(jw) 1 + RC jw
which defines a low pass filter between the input voltage and output voltage.
It is rather easy to show that the impedance of a resistor, capacitor, and inductor are
as follows (Figure 5.6),

Z (jw) = R (5.53)
R
1
Z (jw) = ; also called capacitive reactance (5.54)
C
Cjw
Z (jw) = jw L; also called inductive reactance (5.55)
L
Impedance between two points in a circuit is the Fourier transform of the voltage (output)
and current (input) ratio. It is also called the transfer function between voltage (output)
and current (input). Impedance is a complex quantity. It has magnitude and phase, and is a
function of frequency. When we refer to large or small impedance, we generally refer to its
magnitude.

Example Consider the RL and RC circuits shown in Figure 5.5c, d, respectively. A
switch in each circuit is connected to the supply voltage (A side) and B side at specified
time instants. Assume the following circuit parameters; L = 1000 mH = 1000 × 10 −3 H =
1H, C = 0.01 μF = 0.01 × 10 −6 F, R = 10 kΩ. Let the supply voltage be V (t) = 24 VDC.
s
Assume in each circuit, initial conditions on the current is zero and initial charge in the
capacitor is zero. Starting time is t = 0.0 s. At time t = 100 μs the switch is connected
o
1
to the supply, and at time t = 500 μs, the switch is disconnected from the supply and
2
connected to the B side of the circuit. Plot the voltage across each component and current
as function of time for the time period of t = 0.0s to t = 1000 μs.
f
0
For the RL circuit, when it is connected to the supply,
di(t)
V (t) = L ⋅ + R ⋅ i(t); t ≤ t ≤ t 2 (5.56)
s
1
dt