fundamentals of engineering supplied-reference handbook - Ventech!

RC AND RL TRANSIENTS
v
R
+
V 1
−
V
V
I1
i(t)
i(t)
t = 0
t ≥ 0; vC(t) = vC(0)e –t/RC + V(1 – e –t/RC )
i(t) = {[V – vC(0)]/R}e –t/RC
vR(t) = i(t) R = [V – vC (0)]e –t/RC
t = 0
−Rt
L V −Rt
L
t ≥ 0; i()
t = i(
0 ) e + ( 1−
e )
R
vR(t) = i(t) R = i(0) Re –Rt/L + V (1 – e –Rt/L )
vL(t) = L (di/dt) = – i(0) Re –Rt/L + Ve –Rt/L
where v(0) and i(0) denote the initial conditions and the
parameters RC and L/R are termed the respective circuit
time constants.
Two-Port
Network
R
v R
R
C
L
v L
v C
I2
+
V2
−
170
ELECTRICAL AND COMPUTER ENGINEERING (continued)
RESONANCE
The radian resonant frequency for both parallel and series
resonance situations is
1
ω o = = 2πf
LC
Series Resonance
1
ωoL
=
ω C
o
Z = R at resonance.
ωoL
1
Q = =
R ω CR
BW = ωo/Q (rad/s)
Parallel Resonance
1
ω oL
=
ω C
TWO-PORT PARAMETERS
A two-port network consists of two input and two output terminals as shown below.
o
o
o
and
Z = R at resonance.
R
Q = ωoRC
=
ω L
BW = ωo/Q (rad/s)
o
( rad s)
A two-port network may be represented by an equivalent circuit using a set of two-port parameters. Three commonly used sets
of parameters are impedance, admittance, and hybrid parameters. The following table describes the equations used for each of
these sets of parameters.
Parameter Type Equations Definitions
Impedance (z) V1
= z11I
1 + z12I
2
V = z I + z I
z
V
=
I
z
V
=
I
z
V
=
I
z
V
=
I
Admittance (y)
Hybrid (h)
I
I
2
1
2
2
21
21
1
1
22
= y11V1
+ y12V2
= y V + y V
21 1
22
V1
= h11I1
+ h12V2
I = h I + h V
22
2
2
2
1 1 2 2
11 I2= 0 12 I1= 0 21 I2= 0 22 I1=
0
1 2 1 2
I I I I
y y y y
1 1 2 2
11 = V2= 0 12 = V1= 0 21 = V2= 0 22 = V1=
0
V1 V2 V1 V2
I V I I
h h h h
1 1 2 2
11 = V2= 0 12 = I1= 0 21 = V2= 0 22 =
I1=
0
V1 V2 I1 V2