4—Differential Equations 82These are respectively
V 0 eiωt=IR=I 0 eiωtR
V 0 eiωt=q/C =⇒ iωV 0 eiωt=q/C ̇ =I/C=I 0 eiωt/C
V 0 eiωt=LI ̇=iωLI=iωLI 0 eiωt
In each case the exponential factor is in common, and you can cancel it. These equations are then
V =IR V=I/iωC V=iωLI
All three of these have the same form: V = (something times)I, and in each case the size of the
current is proportional to the applied voltage. The factors ofiimplies that in the second and third
cases the current is± 90 ◦out of phase with the voltage cycle.
The coefficients in these equations generalize the concept of resistance, and they are called
“impedance,” respectively resistive impedance, capacitive impedance, and inductive impedance.
V =ZRI=RI V=ZCI=
1
iωC
I V =ZLI=iωLI (4.39)
Impedance appears in the same place as does resistance in the direct current situation, and this implies
that it can be manipulated in the same way. The left figure shows two impedances in series.
Z 1 Z 2
Z 1
Z 2
I I
I 1
I 2
The total voltage from left to right in the left picture is
V =Z 1 I+Z 2 I= (Z 1 +Z 2 )I=ZtotalI (4.40)
It doesn’t matter if what’s inside the box is a resistor or some more complicated impedance, it matters
only that each box obeysV =ZIand that the total voltage from left to right is the sum of the two
voltages. Impedances in series add. You don’t need the common factoreiωt.
For the second picture, for which the components are in parallel, the voltage is the same on each
impedance and charge is conserved, so the current entering the circuit obeys
I=I 1 +I 2 , then
V
Ztotal
=
V
Z 1
+
V
Z 2
or1
Ztotal
=
1
Z 1
+
1
Z 2
(4.41)
Impedances in parallel add as reciprocals, so both of these formulas generalize the common equations
for resistors in series and parallel. They also include as a special case the formula you may have seen
before for adding capacitors in series and parallel.
In the example Eq. (4.37), if you replace the constant voltage by an oscillating voltage, you have
two impedances in series.