What are Linear Electrical Networks?

A network is a collection of interconnected elements. An electrical element is an idealized component that has a well defined relationship among its currents and voltages. An electrical network is an interconnection of idealized electrical elements to form loops. Electrical elements are resistors, inductors, capacitors, transmission lines, independent sources, dependent- and independent- current sources, diodes, transistors, switches etc.  Linear electrical elements include resistors, capacitors, inductors, and ideal active elements, namely, independent, dependent (controlled) voltage and current sources, amplifiers and Op Amps.  Linear electrical network is

Resistors, capacitors and inductors are passive electrical elements. Resistor (R) or conductor (G) satisfies Ohms law and has a proportional V-I relationship. Inductor (L) has V-I differential/integral relationship. Capacitor (C) has V-I integral/differential relationship. An independent current source is capable of developing any voltage across it depending upon the series voltages that are determined by the passive elements connected in series. An independent voltage source is capable of supplying any amount of current as determined by the currents passing through passive elements connected in parallel.

Dependent (current controlled) Voltage Source (also known as transconductor amplifier) is capable of accepting any independent current input with input power equal to zero, as input voltage is equal to zero.  Power gain is infinity while the conductance gain is g_f.  Voltage controlled Voltage Source (voltage amplifier) is capable of accepting any input voltage with input power being zero because of open circuit at the input.  Power gain is infinity while the voltage gain is A_vf.

Dependent (voltage controlled) Current Source (also known as trans-resistor amplifier) is capable of accepting any independent input voltage with input power equal to zero, as input current is equal to zero.  Power gain is infinity while the resistive gain is r_f. Current controlled current source (current amplifier) is capable of accepting any independent current input with input power equal to zero, as input voltage is equal to zero.  Power gain is infinity while the current gain is A_if.

Passive linear electrical networks are interconnected passive linear electrical elements, and they can enhance signal-to-noise ratio, but cannot enhance power levels. Active linear networks are interconnections of linear active and passive electrical elements and can enhance power levels, as well perform signal conditioning and generate signals.

The electrical relationships of linear passive and active network elements will be explored in this Module. The voltage (v)-current (i) relationships of network elements are:

• Resistor: v = R i  where R is the value of resistor and has the units Ohms (W)
• Inductor: v = L where L is the value of inductor and has the units Henries (H)
• Capacitor: v = where C is the value of capacitor and has the units Farads (F)

The structural features of linear electrical networks, namely, nodes, loops, input ports and output ports will be identified.  A node is point at which terminals of more than one electrical network components are connected. The component(s) joining two nodes is called a branch. A group of branches within a network joined so as to form a closed connection (starting at a node and closing on the same node) is called a loop or mesh. A port is two terminals where the current into one is identical to the current out of the other.  (Any two nodes constitute a port where input can be given or output can be taken).  A source (independent/dependent voltage/ current) is connected to an input port.  A passive load is connected to output port.