Phy 6
Uploaded by
Ramo ApuPhy 6
Uploaded by
Ramo ApuVoltage, series circuits: [V]
V
q
C
C
= V IR
R
=
V
X
V
R
I
X R
= =
V V V
R X
2 2 2
= +
VC =voltage across capacitor [V]
q =charge on capacitor [C]
fR =Resonant Frequency [Hz]
L =inductance [H]
C =capacitance in farads [F]
R =resistance [ ]
I =current [A]
V =supply voltage [V]
VX =voltage across reactance [V]
VR =voltage across resistor [V]
Phase Angle of a series RL or RC circuit: [degrees]
tan = =
X
R
V
V
X
R
cos = =
V
V
R
Z
R
( would be negative
in a capacitive circuit)
=Phase Angle [degrees]
X =reactance [ ]
R =resistance [ ]
V =supply voltage [V]
VX =voltage across reactance [V]
VR =voltage across resistor [V]
Z =impedance [ ]
Impedance of a series RL or RC circuit: [ ]
Z R X
2 2 2
= +
E I Z =
Z
V
X
V
R
V
C
C R
= =
Z R X = j
=Phase Angle [degrees]
X =reactance [ ]
R =resistance [ ]
V =supply voltage [V]
VX =voltage across reactance [V]
VX =voltage across resistor [V]
Z =impedance [ ]
Series RCL Circuits:
The Resultant Phasor X X X
L C
= is
in the direction of the larger reactance
and determines whether the circuit is
inductive or capacitive. If XL is larger
than XC, then the circuit is inductive
and X is a vector in the upward
direction.
In series circuits, the amperage is the
reference (horizontal) vector. This is
observed on the oscilloscope by
looking at the voltage across the
resistor. The two vector diagrams at
right illustrate the phase relationship
between voltage, resistance, reactance,
and amperage.
X
C
X
L
I
R
V
L
C
V
I
R
V
Series RCL
Impedance
Z R X X
L C
2 2 2
= + ( )
Z
R
=
cos
Impedance may be found by adding the components using
vector algebra. By converting the result to polar notation,
the phase angle is also found.
For multielement circuits, total each resistance and reactance
before using the above formula.
Damped Oscillations in an RCL Series Circuit:
q Qe t
Rt L
= ' +
/
cos( )
2
where
' =
2 2
2 ( / ) R L
= 1/ LC
When R is small and e' ~ e:
U
Q
C
e
Rt L
=
2
2
/
q =charge on capacitor [C]
Q =maximum charge [C]
e =natural log
R =resistance [ ]
L =inductance [H]
=angular frequency of the
undamped oscillations
[rad/s]
=angular frequency of the
damped oscillations
[rad/s]
U =Potential Energy of the
capacitor [J]
C =capacitance in farads [F]
Parallel RCL Circuits:
I I I I
T R C L
= +
2 2
( )
tan =
I I
I
C L
R
V
I
L
I
R
C
I
To find total current and phase angle in multielement circuits,
find I for each path and add vectorally. Note that when
converting between current and resistance, a division will
take place requiring the use of polar notation and resulting
in a change of sign for the angle since it will be divided into
(subtracted from) an angle of zero.
Equivalent Series Circuit: Given the Z in polar notation of a
parallel circuit, the resistance and reactance of the
equivalent series circuit is as follows:
R Z
T
= cos X Z
T
= sin
AC CIRCUITS
Instantaneous Voltage of a Sine Wave:
V V ft =
max
sin2t
V =voltage [V]
f =frequency [Hz]
t =time [s]
Maximum and rms Values:
I
I
m
=
2
V
V
m
=
2
I =current [A]
V =voltage [V]
RLC Circuits:
V V V V
R L C
= +
2 2
( ) Z R X X
L C
= +
2 2
( )
tan =
X X
R
L C
P IV
avg
= cos
PF = cos
Conductance (G): The
reciprocal of resistance in
siemens (S).
Susceptance (B, BL, BC): The
reciprocal of reactance in
siemens (S).
Admittance (Y): The reciprocal
of impedance in siemens (S).
Y B
S
u
s
c
e
p
t
a
n
c
e
Conductance
A
d
m
i
t
t
a
n
c
e
G