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Unformatted text preview: 4 MA 36600 LECTURE NOTES: MONDAY, MARCH 9 20 Figure 4. Graph of Solution u(t) = uc (t) + U (t) under Resonance 16 12 8 4 0 2.5 5 7.5 10 12.5 15 17.5 20 -4 -8 -12 -16 -20 ii. Ohm’s Law asserts that the voltage across a resistor is proportional to the current:
Vresistor ∝ I =⇒ Vresistor = I R for some positive constant R. We call R the resistance ; it is measured in ohms.
iii. The voltage across a capacitor is proportional to the charge:
Vcapacitor ∝ Q
Vcapacitor = Q
for some positive constant C . We call C the capacitance ; it is measured in farads.
iv. The voltage in the battery is a function of time:
Vbattery = E (t).
We call such an electrical circuit an RLC Circuit. We have chosen our units so that
1 volt = 1 henry × 1
= 1 ampere × 1 ohm = 1 farad−1 × 1 coulomb.
We think of voltage V as being similar to force F . Just as Newton’s Second Law of Motion states that
F = m a is the sum of the forces on an object, Kerchhoﬀ ’s Second Law of Circuits states that the impressed
voltage is the sum of the voltage drops across each of the items in a closed circuit. That is,
Vinductor + Vresistor + Vcapacitor = Vbattery
+ I R + Q = E (t).
We can express everything in terms of the charge Q = Q(t):
L Q + R Q + Q = E (t).
This is a constant coeﬃcient diﬀerential equation. We may identify:
i. Inductance L with mass m
ii. Resistance R with the damping coeﬃcient γ
iii. Capacitance C with the spring constant k
iv. Electromotive Force E (t) with the forcing term F (t)
Say that we charge the capacitor to some initial amount and apply an initial current:
I= Q(t0 ) = Q0 , Q (t0 ) = I0 . ...
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This note was uploaded on 11/30/2011 for the course MATH 366 taught by Professor Edraygoins during the Spring '09 term at Purdue University-West Lafayette.
- Spring '09