Electrical oscillation with external excitation
Damped LC tank with external excitation
We consider a resistor, an inductance and a cap in series excited by an external AC signal.
This circuit is mathematically described by
U G U R U L U C
U 0 η t R I L I' 1 C Q
Q'' R L Q' 1 C L Q U 0 L η t
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width=4cm
2013-05-29 16:58:20
tiff
1000432
Solving the homogeneous part of the differential equation
We solve the homogeneous part of the differential equation Eq. 1 first
Q'' R L Q' 1 C L Q 0
using the approach
Q K e i σ t
Q' K i σ e i σ t
Q'' K σ 2 e i σ t
Substitution into the differential equation gives
K σ 2 e i σ t R L K i σ e i σ t 1 C L K e i σ t 0
σ 2 R L i σ 1 C L K e i σ t 0
This equation is true for all t only if
σ 2 R L i σ 1 C L 0
is true. We solve this as follows:
p R L i
q 1 C L
σ 1 p 2 p 2 2 q
σ 2 p 2 p 2 2 q
σ 1 R L i 2 R L i 2 2 1 C L
σ 1 R 2 L i R 2 L 2 1 C L
σ 2 R 2 L i R 2 L 2 1 C L
We define
ω R 2 L 2 1 C L
and thus get the following two solutions:
Q K e i σ 1 t
Q K e i R 2 L i ω t
Q K e R 2 L t e i ω t
Q K e R 2 L t ω t i ω t
Q K e i σ 2 t
Q K e i R 2 L i ω t
Q K e R 2 L t e i ω t
Q K e R 2 L t ω t i ω t
The general solution for the homogeneous differential equation can therefore be written as
Q A e R 2 L t ω t i ω t B e R 2 L t ω t i ω t
Q e R 2 L t A ω t A i ω t e R 2 L t B ω t B i ω t
Q e R 2 L t A ω t A i ω t B ω t B i ω t
Q e R 2 L t A B ω t i A B ω t
or after replacing the factors
Q e R 2 L t K 1 ω t i K 2 ω t
Since the differential equation is linear we can rewrite this as follows
Q e R 2 L t K 1 ω t K 2 ω t
or even shorter like so
Q e R 2 L t K ω t φ
ω ω 0 2 ρ 2
ω 0 2 1 C L
ρ R 2 L
with the two freely choosable constants K und φ . Please note the dependence of the resonance frequency from the resistance.
f R 1 2 π R 2 L 2 1 C L
{
ordinateAuto = 0;
xScale = 1;
xmarkDensity = 7;
yScale = 1;
ymarkDensity = 4;
}
0;70;0;100;0;;Fine
2013-05-29 16:36:11
C=50E-6; L=50E-3
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1000555
C=50E-6;L=50E-3
R
f (Hz)
1
Finding a particular solution for the inhomogeneous differential equation
We find a particular solution for the inhomogeneous differential equation by trying a suitable approach.
Q'' R L Q' 1 C L Q U 0 L η t
A suitable approach for the above equation is
Q t D e i η t δ
Q' t D i η e i η t δ
Q'' t D η 2 e i η t δ
Substituted into the differential equation we get
Q'' t R L Q' t 1 C L Q t U 0 L e i η t
D η 2 e i η t δ R L D i η e i η t δ 1 C L D e i η t δ U 0 L e i η t
D e i η t δ R L i η 1 C L η 2 U 0 L e i η t
D e i δ R L i η 1 C L η 2 U 0 L
D e i δ U 0 L 1 C L η 2 R L i η
D e i δ U 0 1 C L η 2 R L i η L 1 C L η 2 R L i η 1 C L η 2 R L i η
D e i δ U 0 L 1 C L η 2 R L i η 1 C L η 2 2 R L η 2
D e i δ U 0 L 1 C L η 2 2 R L η 2 1 C L η 2 R L η i
The value D is the length of the complex number on the right.
D U 0 L 1 C L η 2 2 R L η 2 1 C L η 2 2 R L η 2
D U 0 L 1 C L η 2 2 R L η 2
The angle δ is determined as follows:
δ D I D R
δ D I D
δ D R D
D R U 0 1 C L η 2 L 1 C L η 2 2 R L η 2
D I U 0 R L η L 1 C L η 2 2 R L η 2
D U 0 L 1 C L η 2 2 R L η 2
δ R η L 1 C L η 2
δ R L η
δ 1 C L η 2
Q t D η t δ
Combining the general and the particular solution
We combine the general solution for the homogeneous differential equation
Q e R 2 L t K ω t φ
ω ω 0 2 ρ 2
ω 0 1 C L
ρ R 2 L
and the particular solution of the inhomogeneous differential equation
Q t D η t δ
D U 0 L 1 C L η 2 2 R L η 2
δ R η L 1 C L η 2
into a total solution by summation:
Q t e R 2 L t K ω t φ D η t δ
Since the e-Function approaches 0 for large values of t only the particular solution remains after an initiation period.
Q t D η t δ
We devide by C to get the voltage in the cap, take first derivatives to get the current and take the second derivative to get the voltage over the coil.
Q t D η t δ
Q' t D η η t δ
Q'' t D η 2 η t δ
U C t Q t C
U L t L Q'' t
I t Q' t
δ R η L 1 C L η 2
We determine the peak voltage in the cap and the peak current with respect to f (resonance frequency):
Q t D η t δ
U s f D C
U s f U 0 L C 1 C L η 2 2 R L η 2
U s f U 0 L C 1 C L 2 π f 2 2 2 π f R L 2
I t D η η t δ
I s f D η
I s f U 0 L 1 C L η 2 2 R L η 2 η
I s f 2 π f U 0 L 1 C L 2 π f 2 2 2 π f R L 2
If we excite the series tank with its resonance frequency we get high voltages over the components and a very high current only limited by the resistance of the wire.
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0;150;0;50;0;;Fine
2013-05-29 23:08:49
C=50E-6; L=100E-3; R=1; U_0=1; Factor I_s(f): 10.00
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PiBdID4+CnN0YXJ0eHJlZgo0MjAzMQolJUVPRgo=
1000555
C=50E-6;L=100E-3;R=1E0;U_0=1E0
Hz
1
{
ordinateAuto = 1;
xScale = 1;
xmarkDensity = 5;
yScale = 1;
ymarkDensity = 4;
}
0;50;-60;60;-3;;Fine
2013-05-29 19:38:52
C=50E-6; L=100E-3; R=1; U_0=1; η=400; Factor I(t): 10.00
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PiBdID4+CnN0YXJ0eHJlZgo1MzU5OQolJUVPRgo=
1000555
C=50E-6;L=100E-3;R=1E0;U_0=1E0;η=400E0
1
Energy Consideration
The power going into the circuit is given by
P 1 T t 0 T U G t I t
P 1 T t 0 T U 0 η t D η η t δ
P U 0 D η T t 0 T η t η t δ
P U 0 D η T t 0 T 2 η t δ 2 η t δ 4 η
P U 0 D η T 2 η T δ 2 η T δ 4 η δ 4 η
P U 0 D 4 T δ 2 η T δ 2 η T δ
η 2 π T
T 2 π η
P U 0 D 4 2 π η δ 2 η 2 π η δ 2 η 2 π η δ
This gets us the following expression for the input power.
P in U 0 D η 8 π δ 4 π δ 4 π δ
δ R η L 1 C L η 2
D U 0 L 1 C L η 2 2 R L η 2
The circulating power in the circuit is defined as
P circ t t W t
with
W t 1 2 C U C t 2 1 2 L I t 2
U C t D C η t δ
I t D η η t δ
P circ t t W t
P circ t t 1 2 C U C t 2 1 2 L I t 2
P circ t t 1 2 C D C η t δ 2 1 2 L D η η t δ 2
P circ t 1 2 C t D C η t δ 2 1 2 L t D η η t δ 2
P circ t C D C η t t 0 η t δ D C η t δ L D η η t t 0 η t δ D η η t δ
P circ t C D C η η t δ D C η t δ L D η 2 η t δ D η η t δ
P circ t D η η t δ D C η t δ L D η 2 η t δ D η η t δ
P circ t D η D C η t δ η t δ L D η 2 D η η t δ η t δ
P circ t D η D C L D η 2 D η η t δ η t δ
P circ t D 2 η 1 C L η 2 η η t δ η t δ
P circ t D 2 η 1 C L η 2 η t δ η t δ
D U 0 L 1 C L η 2 2 R L η 2
δ R η L 1 C L η 2
We have scaled P in with a factor of 100 in the figure above. The Input power is very small compared to the circulating power for R 0.25 Ohm .
{
ordinateAuto = 0;
xScale = 1;
xmarkDensity = 5;
yScale = 1;
ymarkDensity = 4;
}
0;25;-2;2;-3;;Fine
2013-05-30 00:13:26
C=50E-6; L=50E-3; R=0.25; U₀=12; η=314; Factor P_in: 100.00
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1