Week 11

Basic Facts

A time varying voltage V(t) = V₀ cos(ωt+φ) has amplitude V₀, phase φ and
frequency f where ω = 2πf.

When the voltage is applied to a resistor R a current I(t) flows according to Ohm's law
V(t) = R I(t).

When a time varying voltage V(t) is applied to a capacitor C a current I(t) flows according
to I(t) = 1 dV
             C dt

When a time varying voltage V(t) is applied to an inductor L a current I(t) flows according
to V(t) = L dI
                  dt

In any circuit made up only of resistors, capacitors, and inductors there is always a Linear
relationship between the current and voltage. Such a circuit is called a Linear Circuit. If we
apply a sinusoidal voltage with frequency f to a Linear circuit then the current that flows will
be simusoidal with frequency f but possibly with a different phase.

When studying a linear circuit (or other linear system) it is convenient to any sinusoidally varying
quantity by a complex exponential. If we represent the driving the voltage
V₀ cos(ωt+φ)
by the complex exponential
V₀ e^j(ωt+φ) where j² = -1
and we perform all calculations using the complex voltage and current, then the correct results
will obtained by taking the Real part of each complex quantity.

In complex notation the resistor, capacitor, and inductor rules become
V = I R for a resistor, V = I  1  for a capacitor and V = j ω L I   for an inductor.
                                           jωC

We introduce the complex quantity Z, the impedance, as an extension of the resistance and say
that a resistor has Z = R, a capacitor has Z =  1 , and an inductor Z = j w L.
                                                                    jωC

Impedance obeys the same rules as resistance. For Z1 in series with Z2 we have Z = Z1 + Z2
and for Z1 in parallel with Z2 we have
1 = 1 + 1 .
Z   Z1   Z2