The voltage and the current relationships in circuit analysis
i = d q d t i= \frac{dq}{dt}
i = d t d q
voltage drop across the inductor
V L = L d i d t = L d 2 q d t 2 V_L =L\frac{di}{dt}=L\frac{d^2q}{dt^2} V L = L d t d i = L d t 2 d 2 q
voltage drop across the resistor
V R = R i = R d q d t V_R=Ri=R\frac{dq}{dt} V R = R i = R d t d q
voltage drop across the capacitor
V C = q C V _C=\frac{q}{C} V C = C q
Kirchhoff's Second Law
L d 2 q d t 2 + R d q d t + q C = E ( t ) L\frac{d^2q}{dt^2}+R\frac{dq}{dt}+\frac{q}{C}=E(t) L d t 2 d 2 q + R d t d q + C q = E ( t )
Given that L = 1 L=1 L = 1 H, R = 1000 R=1000 R = 1000 ohm, C = 1 0 − 6 C=10^{-6} C = 1 0 − 6 F, E ( t ) = 12 E(t)=12 E ( t ) = 12 V
1 d 2 q d t 2 + 1000 d q d t + 1 1 0 − 6 q = 12 d 2 q d t 2 + 1000 d q d t + 1000000 q = 12 1\frac{d^2q}{dt^2}+1000\frac{dq}{dt}+\frac{1}{10^{-6}}q=12\\
\frac{d^2q}{dt^2}+1000\frac{dq}{dt}+1000000q=12\\ 1 d t 2 d 2 q + 1000 d t d q + 1 0 − 6 1 q = 12 d t 2 d 2 q + 1000 d t d q + 1000000 q = 12
The characteristic equation is
r 2 + 1000 r + 1000000 = 0 D = ( 1000 ) 2 − 4 ⋅ 1000000 = − 3000000 r 1 = − 1000 + i 1000 3 2 = − 500 + 500 3 i r 2 = − 1000 − i 1000 3 2 = − 500 − 500 3 i q C = e − 500 t ( c 1 cos ( 500 3 t ) + c 2 sin ( 500 3 t ) ) r^2+1000r+1000000=0\\
D=(1000)^2-4\cdot1000000=-3000000\\
r_1=\frac{-1000+i1000\sqrt3}{2}=-500+500\sqrt3i\\
r_2=\frac{-1000-i1000\sqrt3}{2}=-500-500\sqrt3i\\
q_C=e^{-500t}(c_1\cos(500\sqrt3t)+c_2\sin(500\sqrt3t)) r 2 + 1000 r + 1000000 = 0 D = ( 1000 ) 2 − 4 ⋅ 1000000 = − 3000000 r 1 = 2 − 1000 + i 1000 3 = − 500 + 500 3 i r 2 = 2 − 1000 − i 1000 3 = − 500 − 500 3 i q C = e − 500 t ( c 1 cos ( 500 3 t ) + c 2 sin ( 500 3 t ))
Find the particular solution of the non-homogeneous differential equation using the method of the undetermined coefficients.
Assume that Q ( t ) = A Q(t)=A Q ( t ) = A is a solution of the non-homogeneous differential equation where E ( t ) = 12 E(t)=12 E ( t ) = 12
Differentiate the assumption with respect to t t t
Q ′ = 0 , Q ′ ′ = 0 Q'=0, Q''=0 Q ′ = 0 , Q ′′ = 0
Substitute in the original equation
0 + 0 + 1000000 A = 12 A = 1.2 ⋅ 1 0 − 5 Q ( t ) = 1.2 ⋅ 1 0 − 5 q ( t ) = e − 500 t ( c 1 cos ( 500 3 t ) + c 2 sin ( 500 3 t ) ) + + 1.2 ⋅ 1 0 − 5 0+0+1000000A=12\\
A=1.2\cdot10^{-5}\\
Q(t)=1.2\cdot10^{-5}\\
q(t)= e^{-500t}(c_1\cos(500\sqrt3t)+c_2\sin(500\sqrt3t))+\\
+1.2\cdot10^{-5} 0 + 0 + 1000000 A = 12 A = 1.2 ⋅ 1 0 − 5 Q ( t ) = 1.2 ⋅ 1 0 − 5 q ( t ) = e − 500 t ( c 1 cos ( 500 3 t ) + c 2 sin ( 500 3 t )) + + 1.2 ⋅ 1 0 − 5
The initial charge of the capacitor is zero
q ( 0 ) = e − 500 ⋅ 0 ( c 1 cos ( 500 3 ⋅ 0 ) + + c 2 sin ( 500 3 ⋅ 0 ) ) + 1.2 ⋅ 1 0 − 5 = 0 c 1 = − 1.2 ⋅ 1 0 − 5 q(0)= e^{-500\cdot 0}(c_1\cos(500\sqrt3\cdot 0)+\\
+c_2\sin(500\sqrt3\cdot 0))+1.2\cdot10^{-5}=0\\
c_1=-1.2\cdot10^{-5} q ( 0 ) = e − 500 ⋅ 0 ( c 1 cos ( 500 3 ⋅ 0 ) + + c 2 sin ( 500 3 ⋅ 0 )) + 1.2 ⋅ 1 0 − 5 = 0 c 1 = − 1.2 ⋅ 1 0 − 5
The circuit is closed at t = 0 : i ( 0 ) = q ′ ( 0 ) = 0 t=0: i(0)=q'(0)=0\\ t = 0 : i ( 0 ) = q ′ ( 0 ) = 0
q ′ ( t ) = − 500 e − 500 t ( − 1.2 ⋅ 1 0 − 5 ) ( cos ( 500 3 t ) − − 500 e − 500 t ( c 2 sin ( 500 3 t ) ) + + 1.2 ⋅ 1 0 − 5 ( − 500 3 ) e − 500 t ( sin ( 500 3 t ) + + c 2 ⋅ 500 3 e − 500 t cos ( 500 3 t ) ) q ′ ( 0 ) = − 500 ( − 1.2 ⋅ 1 0 − 5 ) + c 2 ⋅ 500 3 = 0 c 2 = − 0.4 3 ⋅ 1 0 − 5 q ( t ) = e − 500 t ( − 1.2 ⋅ 1 0 − 5 cos ( 500 3 t ) − − 0.4 3 ⋅ 1 0 − 5 sin ( 500 3 t ) ) + 1.2 ⋅ 1 0 − 5 q'(t)= -500e^{-500t}(-1.2\cdot10^{-5})(\cos(500\sqrt3t)-\\
-500e^{-500t}(c_2\sin(500\sqrt3t))+\\
+ 1.2\cdot10^{-5}(-500\sqrt3)e^{-500t}(\sin(500\sqrt3t)+\\
+c_2\cdot500\sqrt3e^{-500t}\cos(500\sqrt3t))\\
q'(0)=-500(-1.2\cdot10^{-5})+c_2\cdot500\sqrt3=0\\
c_2=-0.4\sqrt3\cdot10^{-5}\\
q(t)=e^{-500t}(-1.2\cdot10^{-5}\cos(500\sqrt3t)-\\
-0.4\sqrt3\cdot10^{-5}\sin(500\sqrt3t))+1.2\cdot10^{-5} q ′ ( t ) = − 500 e − 500 t ( − 1.2 ⋅ 1 0 − 5 ) ( cos ( 500 3 t ) − − 500 e − 500 t ( c 2 sin ( 500 3 t )) + + 1.2 ⋅ 1 0 − 5 ( − 500 3 ) e − 500 t ( sin ( 500 3 t ) + + c 2 ⋅ 500 3 e − 500 t cos ( 500 3 t )) q ′ ( 0 ) = − 500 ( − 1.2 ⋅ 1 0 − 5 ) + c 2 ⋅ 500 3 = 0 c 2 = − 0.4 3 ⋅ 1 0 − 5 q ( t ) = e − 500 t ( − 1.2 ⋅ 1 0 − 5 cos ( 500 3 t ) − − 0.4 3 ⋅ 1 0 − 5 sin ( 500 3 t )) + 1.2 ⋅ 1 0 − 5
Coulomb:
q ( 1 ) = e − 500 ⋅ ( 1 ) ( − 1.2 ⋅ 1 0 − 5 cos ( 500 3 ⋅ ( 1 ) ) − − 0.4 3 ⋅ 1 0 − 5 sin ( 500 3 ⋅ ( 1 ) ) ) + 1.2 ⋅ 1 0 − 5 ≈ ≈ 1.2 ⋅ 1 0 − 5 ( C ) q(1)=e^{-500\cdot(1)}(-1.2\cdot10^{-5}\cos(500\sqrt3\cdot(1))-\\
-0.4\sqrt3\cdot10^{-5}\sin(500\sqrt3\cdot(1)))+1.2\cdot10^{-5}\approx\\
\approx1.2\cdot10^{-5}(C) q ( 1 ) = e − 500 ⋅ ( 1 ) ( − 1.2 ⋅ 1 0 − 5 cos ( 500 3 ⋅ ( 1 )) − − 0.4 3 ⋅ 1 0 − 5 sin ( 500 3 ⋅ ( 1 ))) + 1.2 ⋅ 1 0 − 5 ≈ ≈ 1.2 ⋅ 1 0 − 5 ( C )
Find the steady state charge.
t → ∞ : q p ( t ) → 1.2 ⋅ 1 0 − 5 C t\to\infty:q_{p}(t)\to1.2\cdot10^{-5}C t → ∞ : q p ( t ) → 1.2 ⋅ 1 0 − 5 C
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