^{Capacitors, Inductors and AC Circuits}

^{Capacitors, Inductors and AC Circuits}

**Wave Form (Signal) **

Signal is information in the form of changes in current or voltage. Changing the form of a signal with respect to a function of time or waveform is a very important part of electronics. The signal waveforms that we often encounter are as shown in Figure 1

Figure 1 Various forms of important signals in electronic systems

Direct or continuous voltage is generated by a DC current generator battery. Flow steps (step) to flow when a switch is turned on that produced

Direct voltage, such as when a radio is turned on. Pulse current when a switch is turned on (ON) then turned off (OFF), is used for information systems on computers. The saw wave rises linearly then resets. Exponential (decreasing) current flows when energy is stored in an electric field in a capacitor and allowed to leak through a resistor. The sine voltage is obtained when a coil is rotated at constant speed in an electric field.

**Capacitor **

Basically a capacitor is two pieces of conductor separated by an insulator (air, vacuum or a certain material). Schematically, a parallel strip capacitor can be described as in Figure 2.

Figure 2 Parallel chip capacitor

Suppose DC voltage is applied to the two chips as shown in Figure 2. Since the two pieces are separated by an insulator, basically no electrons can cross the gap between the two pieces. When the battery is not connected, the two pieces will be neutral (not yet met).

When a battery is connected, the point where the wire at the negative end is connected repels the electrons, while the point where the positive pole is connected attracts electrons. The electrons will be scattered throughout the capacitor. For a moment, electrons flow into the right discs and electrons flow out of the left discs; in this condition the current flows through the capacitor even though in fact no electrons flow through the gaps in the two pieces.

After the outer part of the chip is loaded, it will gradually reject the new charge from the battery. Therefore, the current in the chip will decrease in magnitude with time until the two pieces are at the voltage that the battery has. The rightwill have an excess of electrons measured with a charge ofcharge *-Q *chipand the left chip will have a *+ Q*. The amount of charge *Q *is therefore proportional to *V *or

*Q *∝*V. *

The proportionality constant is expressed as capacitance or *C*

Q = *CV *(1)

where the capacitance unit is expressed in farads (F).

In general, the relationship between charge and voltage for a capacitor can be written as

*q *= *C v *(2)

thus the current *i *flowing is given by

*i *= *dq */ *dt *= *C dv */ *dt *(3)

or

*v *= *q */*C *

$=\frac{1}{C}\int_{0}^{t}i dt +V_{0}$

## Inductor

It is known that a moving electron or an electric current that flows will produce a magnetic field. But on the contrary, to produce an electric current (induced current) it is necessary to change the magnetic field.

A very simple experiment can be carried out as shown in Figure 3. When the switch*(switch)*is closed and current flows in the coil is fixed to the bottom, then there is no induced current flowing in the upper coil. However, when the switch is closed (or opened) so that the resulting magnetic field changes, the voltmeter will show a change in induced voltage. The magnitude of the resulting voltage is proportional to the change in induced current, it can be written as:

*v *= *L di */ *dt *

where the value of proportionality *L is *called self-induction or inductance in henry units (H).

Figure 3 A simple experiment of self-induction in inductors

Figure 4 The occurrence of transient currents in RC circuits

** Transient currents in RC circuits **

Figure 4 explains the process of loading and discharging a capacitor. If at first the switch is in position 1 for a relatively long time, the capacitor will be charged by V volts. In this state we note it as t = 0.

When the switch is moved to position 2, the capacitor’s charge begins to(*dischargedischarge*) so that the voltage on the capacitor begins to decrease. When the voltage on the capacitor starts to decrease, the stored energy will be dissipated to heat through the resistor. Since the voltage across the capacitor is the same as the voltage across the resistor, the current through the circuit will also decrease. This process continues until the entire charge is stripped or the voltage and current are zero so that the circuit is in a stable state (*steady-state*).

To determine the voltage and current equation when the capacitor charge is stripped, Kirchhoff’s hk can use the following about the current.

i_{c}(*t*) + i_{R}(*t*) = 0 (5)

Using the relationship *V-I *on *C *and *R, it is *obtained

$C \frac{^{dV_{c}}}{dt}+ \frac{^{V_{c}}}{R}$ (6)

Divided by C and by defining τ = *RC *, we get

* *$C \frac{^{dV_{c}}}{dt}+ \frac{^{V_{c}}}{\tau }$ (7)

equation 7 applies for t> 0 and have requirements initial condition

V_{C}(0)= V_{1}

*
*The solution of the equation for t> 0 can be shown as

(8)

1

is an exponential equation where

*V*_{C}= is the instantaneous value

*V*_{1 }= amplitude or maximum value

*e *= 2.718 ………………

*t *= time in seconds

τ = time constant in seconds

Figure 5 Capacitor voltage discharge plot

This exponential equation describes how the capacitor condition when its charge is stripped. Graphically this equation can be plotted as shown *v *), the value of the capacitor voltage

in Figure 5. It can be seen that the final condition (V_{c}(∞)*)*

is zero. It can be explained, for the process of charging the capacitor:

(9)

**Differentiator Circuit**

The circuit *RC *in Figure 6-a can function as a differentiator circuit, where the output is a derivative of the input. In the case of the input voltage in the form of a square wave, the output voltage is proportional to the loading and unloading process as a reaction to the*step voltage*. In this case thecircuit *RC *functions as a converter of a square wave into a series of pulses if thetime constant *RC *is less than the period of the input wave.

By making an approximation and using Kirchhoff’s hk about the voltage obtained:

(10)

*if v _{R} is *considered very small compared to v

_{C}because i = C dv

_{C}/ dt

(11)

It can be seen that the output (*output*) is proportional to the derivative of the input (*input*).

Fig 6 RC Circuit as Differensiator and Integrator

**Integrator Circuit**

Thecircuit *RC *can also be used as an integrator circuit as shown in Figure 6-b. In general terms,

*v*_{1 }= *v** _{R }*+

*v*

*≅*

_{C }*v*

*=*

_{R }*iR*(12)

if V_{C }* is very small compared to * V* _{R } *(i.e. if

*RC> T*). Because

the capacitor voltage is proportional to the integral *i *≅ *v*_{1}/ *R *,

(13)

and the output is the integral value of the input.