Overview
Passive and Active Elements
On electronic circuits, they contain many components such as transistors or operational amplifiers. However, these components can be classified in to either an active element or a passive element. Passive elements are elements which cannot supply energy by themselves, whereas a battery is an active element which generate energy. Resistors, capacitors and inductors are passive elements which we are going to discuss in this chapter.
Before we begin our discussion, let’s see <Table 1> and briefly think about how Resistors, capacitors and inductors work.
| Relation | Resistor | Capacitor | Inductor |
|---|---|---|---|
| Schematic |  |
 |
 |
| Voltage | \(v_R(t)=i_R(t)R\) | \(\frac{1}{C}\int_{t_0}^{t}idt+v(t_0)\) | \(v_L(t)= \frac{di_l(L)}{dt}\) |
| Current | \(i_R(t)=\frac{v_R(t)}{R}\) | \(i=c\frac{dv_c(t)}{dt}\) | \(\frac{1}{L}\int_{t_0}^{t}vdt+i(t_0)\) |
| Power or Energy | \(p=i^2R=\frac{v^2}{R}\) | \(w=\frac{1}{2}cv^2\) | \(w=\frac{1}{2}li^2\) |
| Series | \(R_{eq}=R_1+R_2+... R_n\) | \(C_{eq}=\frac{C_1C_2}{C_1+C_2}\) | \(L_{eq}=L_1+L_2+... L_n\) |
| Parallel | \(R_{eq}=\frac{R_1R_2}{R_1+R_2}\) | \(C_{eq}=C_1+C_2+... C_n\) | \(L_{eq}=\frac{L_1L_2}{L_1+L_2}\) |
| At DC Voltage | Same | Open Circuit | Short Circuit |
Capacitors
Basic Background
Capacitor is composed of two conducting plates separated by a dielectric. When voltage is applied between the plates, +q charges accumulate in one plate while -q charge on the other. Capacitance, which is ability to store an electrical charge, depends on the geometry of the conductors and permittivity of the insulating medium in between the plates. Capacitance of ideal parallel plate capacitor is defined by following equation. \[C=\frac{\epsilon A}{d}\]
Where plate area \(A\) separated by distance \(d\) with a permittivity of \(\epsilon\).
Terminal Characteristics
The relation between charge and capacitance is \[q=Cv\] By looking at this equation, we can know that the plate charge increases as the voltage increases. when we take a derivative for both sides, we can get \[i=\frac{Cdv}{dt}\] where \(i=\frac{dq}{dt}\)
when \(v\) is a constant voltage(which is a DC voltage), then \(i=0\). This means when DC voltage is applied to capacitor, it looks as if it is an open circuit. Meanwhile, abrupt change in it voltage \(v\) is not allowable.
If we integrate equation over time, \[\int_{-\infty}^{t}idt=\int_{-\infty}^{t}C\frac{dv}{dt}dt\] \[v=\frac{1}{C}\int_{-infty}^{t}idt\] \[v=\frac{1}{C}\int_{0}^{t}idt + v(0)\]
\(v(0)\) repr
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