c21b2cd2-069d-48ac-abfe-3e915ee8feda 10 4 5 decimal

&[DATE] &[TIME] - &[FILENAME]

mΩ 103 -^Ω * :

μs 106 -^s * :

Enter:

Component values (ohms, henrys, farads) R 6.4 mΩ * : L 0.394 μH * : C 480 μF * : 13 mat

initial value of voltage on capacitor (V) v.Co 8 - kV * :

initial value of current in inductor (A) i_.Lo 0 A * :

value of step voltage (V) V.f 0 V * :

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Damping Term R α R 2 L * / :

Define:

R α 8121.8274 Hz *

Undamped Natural Frequency ω.o 1 L C * sqrt / :

ω.o 72716.2159 Hz *

R α ω.o 21 mat 8121.8274 1 s / * 72716.2159 1 s / * 21 mat

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R s.1 R 2 L * / - R 2 L * / (2^1 L C * sqrt / (2^- sqrt + :

R s.1 #

R s.2 R α - R α 2^ω.o 2^- sqrt - :

R s.2 #

R A.2 1 R s.2 R s.1 - 0.00001 R s.1 * + / i_.Lo C / R s.1 v.Co V.f - (* - (* :

R A.2 #

R A.1 v.Co R A.2 - V.f - :

R A.1 #

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R ω.d ω.o 2^R α 2^- sqrt :

R ω.d 72261.2204 Hz *

B.1 v.Co V.f - :

R B.2 1 R ω.d 0.000001 R α * + / i_.Lo C / R α B.1 * + (* :

B.1 8000 V * -

R B.2 899.1629 V * -

R D.1 i_.Lo C / R α D.2 * + :

D.2 v.Co V.f - :

R D.1 V 6.4975 107^* Hz * -

D.2 8000 V * -

For overdamped response (i.e.,wo < a )

t R v.C1 V.f R A.1 R s.1 t * exp * R A.2 R s.2 t * exp * + (+ :

R α Hz 8121.8274

For critically damped response (i.e.,wo = a):

t R v.C2 V.f R D.1 t * R α - t * exp * + D.2 R α - t * exp * + :

ω.o Hz 72716.2159

For underdamped response (i.e.,wo > a )

t R v.C3 V.f R α - t * exp B.1 R ω.d t * cos * R B.2 R ω.d t * sin * + (* + :

t R v.C R α ω.o > t R v.C1 R α ω.o < t R v.C3 t R v.C2 if if :

Then the voltage across the capacitoris

Check the limiting values:

0 s * R v.C 8000 V * -

vC(∞) =

100000 s * R v.C 0

t.m 1 R s.1 abs / 1 R s.2 abs / + 1 R α / + (:

For plotting purposes, let time scale be 0 to

R 0.0064 Ω *

and use k as a parmeter for R:

k 2 :

x v.C t R v.C :

100 v.C #

t 0.01 t.m * (t.m 0.02 t.m * range :

UNIT kg m 2^* s 3^A * / :

X t μs / :

Y1 t R v.C vectorize UNIT / :

Y2 t R k / v.C vectorize UNIT / :

Y3 t k R * v.C vectorize UNIT / :

plot X Y1 augment X Y2 augment X Y3 augment 31 sys :

plot

MathCad graph

and the current in the circuit is t R i_C t R v.C t diff * :

undamped impedance Z.0 L C / sqrt :

Z.0 mΩ 28.6502

damping ratio (if < 1 then oscillatory) δ R 2 Z.0 * / :

δ 0.1117

undamped peak current I.0 v.Co abs Z.0 / :

I.0 kA 279.2303

undamped quarter-cycle rise time τ.0 π 2 / L C * sqrt * :

τ.0 μs 21.6017

damped time to peak τ.1 2 τ.0 * δ cos 1 -^* π 1 δ 2^- sqrt * / :

τ.1 μs 13.9255

damped current peak I.1 I.0 π τ.1 * δ * 2 τ.0 * / - exp * :

I.1 2.4937 105^* A *

The energy stored in the inductor and capacitor and the total energy dissipated in the resistor is

t R w.C C 2 / t R v.C 2^* :

t R w.L L 2 / t R i_2^* :

Initial stored energy:

w.o 0 s * R w.C 0.00001 t.m * R w.L + :

0 s * R w.C 15360 J *

NOTE: this calculation can take a while!

0.00001 t.m * R w.L 0.0002 J *

w.o 15360.0002 J *

Joules