Passive elements

Most of the Circuit elements have at least two leads (electrical terminals). They are characterized by voltage across the terminals and current flowing through the device (see Fig.2.1); this is V-I characterization of device.

Figure 2.1: Simple element

Resistance: Across this element, if we applied a voltage source and observe the current, then we will observer that $v\propto i$, that is, $v=Ri$, where R is the resistance measured in Ohms (Fig.1.1, 2.2).

Figure 2.2: Graph of V vs I for a Resistance

Higher the value of R, larger the voltage required to achieve the same current. Voltage proportional to current - is Ohm's law (It was deduced hueristically by experiments for metals, by George Simon Ohm). For lamp, this law is not true, as with increase in current, temperature of bulb increases, causing the increase in resistance. Hence Ohms law is not strictly true for lamp. But for most of the practical purposes and for this course the Ohm's law holds true for the resistive circuit elements.

The resistive elements (resistances) can be fixed or variable. Commonly use resistor types are carbon film and wirewound. The example of variable resistor is potentiometer.

For a material with length $l$ and cross-sectional area $A$, the resistance will be $R={\rho l \over A}$, where $\rho$ is specific resistivity (property of material). Inverse of resistance is conductance. $G=\frac{1}{R}$, measured in mhos or Seimens.

Figure 2.3: Bulb as a resistive load

In Fig. 2.4, property of interest is resistance between A and B. It is dependent on details of wire, connector, filament, material, and shape. It can be abstracted as simple resistance (Fig.2.4).

Figure 2.4: Bulb abstracted as a simple resistance

Inductance:

It another important basic circuit element. Current flowing in a wire causes generation of magnetic field intensity ( $\mathcal{H}$). $\mathcal{H}$ is independent of material medium surrounding the current carrying wire. The $\mathcal{H}$ leads to magnetic flux density $B=\mu_r \mu_0 \mathcal{H}$. Here $\mu_0$ is absolute permeability of vaccum. $\mu_r$ is relative permeability of material where $B$ is measured. The flux flowing around wire links with the conducting wire. And if the flux linkage changes it lead to generation of EMF (electromotive force) which try to oppose the change in flux. This means it tries to nullify the change in current.

The current $i$ causes production of magnetic flux. $E=\frac{d\phi }{dt}$ is the EMF of a single turn. Here, $B A=\phi$. Thus, the total EMF $v=N\frac{d\phi}{dt}$, where $N$ is the number of turns.

Defining the inductance - $v=L{di \over dt}$, $L$ is inductance and measured in Henry. See the output current of sinusoidal $v$ applied across an inductor in Fig.2.5

Figure 2.5: V and I as function of t for an Inductor

Figure 2.6: Physical Implementation of an inductor

In lumped model, inductance is considered only due to element. The inductance due to wires connecting it to other elements is neglected (Fig.2.6)

Analysis of circuit: To find voltage or currents in an element of interest. One can also find voltage and current in all the elements of circuit.

Lumped simplified model of resitance, inductance and capacitance.

Capacitance: $i=c{dv \over dt}$, $c$ capacitance. The magnitude of charge on either plate is given by $q=cv$.

Figure 2.7: Symbols for resitance, inductance and capacitance

Lumped model: Shape, material, wire, connectors - effect of each is assumed to be due to single entity shown by the symbols in the diagram. In actual resistance, inductance and capacitance are distributed all across the circuits. For most practical purpose, lumped model- satisfactory.

Series and Parallel connections

Figure 2.8: A series connection of resistors

Kirchoff's volatage law: In a circuit, if your start from a point A and tranverses the circuit in any fashion and reaches back to point A, the total sum of potential changes should be zero. This has to be true since, same point cannot have two different potentials.

Kirchoff's current law: At any point in the network, total amount of current entering and leaving the point has to be equal to rate of accumulation of charge at the point. Since, in the circuits ordinarily the points where one circuit element is connected to other circuit element (these points are called nodes) do not store charge sum of incomming current has to be equal to sum of outgoing currents.

Thus, for series model, we get relations:

$$v_1+v_2=v$$

$$iR_1+iR_2=v$$

$$i(R_1+R_2)=v$$

Thus equivalent resistance of series connection is $R=R_1+R_2$.

For parallel connection, we get the relations:

$$v_1=v_2=v$$

$$i_1R_1=i_2R_2=v$$

$$i=i_1+i_2=\frac{v}{R_1}+\frac{v}{R_2}$$

$$i\frac{R_1R_2}{R_1+R_2}=v$$

Thus, equivalent resistance here is: $R=\frac{R_1R_2}{R_1+R_2}$. Similarly, we calculate equivalent inductance and capacitance for series and parallel cases.

Inductances in series:

$$v=v_1+v_2=L_1\frac{di}{dt}+L_2\frac{di}{dt}$$

$$v=(L_1+L_2)\frac{di}{dt}$$

$$Thus:\;\; L=L_1+L_2$$

Inductances in parallel:

$$v_1=v_2=v$$

$$v_1=L_1\frac{di_1}{dt}$$

$$v_2=L_2\frac{di_2}{dt}$$

$$\frac{v_1}{L_1}+\frac{v_2}{L_2}=v(\frac{1}{L_1}+\frac{1}{L_2})=\frac{d(i_1+i_2)}{dt}=\frac{di}{dt}$$

$$Thus, \;\; L=\frac{L_1L_2}{L_1+L_2}$$

Figure 2.9: Series connection of capacitances

Capacitances in series:

$$i=C_1\frac{dv_1}{dt}=C_2\frac{dv_2}{dt}$$

$$i(\frac{1}{C_1}+\frac{1}{C_2})=\frac{d(v_1+v_2)}{dt}$$

$$i=\frac{C_1C_2}{C_1+C_2}\frac{dv}{dt}$$

$$Thus,\;\; C=\frac{C_1C_2}{C_1+C_2}$$

Capacitances in parallel

$$i_1=C_1\frac{dv}{dt}$$

$$i_1=C_1\frac{dv}{dt}$$

$$i=i_1+i_2=(C_1+C_2)\frac{dv}{dt}$$

$$Thus, \;\; C=C_1+C_2$$

Inductance of a Solenoid (Coil)

Figure 2.10: A Solenoid diagram showing magnetic circuit path lengths

Define $l$=magnetic circuit path length
A=magnetic circuit crossectional area.
Inductance: (Assuming that ${\cal H}$ is same in the closed path of length $l$.)

$$\oint\vec{\cal H}.d\vec{l}=Ni$$

$${\cal H}.l=Ni$$

$${\cal H}=\frac{Ni}{L}$$

$$B=\mu {\cal H}$$

$$B=\frac{\mu Ni}{l}$$

$$Thus, \;\;\phi=\frac{\mu NiA}{l}$$

$$Now,\;\; v=L\frac{di}{dt}=N\frac{d\phi}{dt}$$

$$Thus, \;\; v= \frac{\mu N^2A}{l}\frac{di}{dt}$$

$$Which\;gives\;\;L=\frac{\mu N^2A}{l}$$

Here ${\cal H}.l=N i$ is magnetomotive force (equivalent of eletromotive force - EMF in magnetic domain), and flux $\phi$ is equivalent of current in magnetic domain. The Magnetic reluctance= $\frac{l}{\mu A}$ is then equivalent of resistance in magentic domain.

Linearity: when elemental change in cause $\Delta C$ , always leads to same elemental change in effect i.e., $\Delta E$ , then the system is said to be linear.

In general, for a system let input $x(t)$ lead to output $y(t)$, and a small perturbation in $\delta x(t)$ causes a small perturbation $\delta y(t)$ in output. If perturbation $\delta x(t)$ alwasy leads to same perturbation of $\delta y(t)$ in the output irrespective of any $x(t)$, then the system is linear.

The implication of the above is that if input $x_1(t)$ causes output $y_1(t)$, and $x_2(t)$ causes $y_2(t)$, then $A x_1(t)+B x_2(t)$ will cause an output of $A y_1(t) + B y_2(t)$. This is principal of superposition.