Circuit Theory Notes

Kirchhoff Current Law (KCL)

All the current entering a node is equal to the current exiting the node

Where n is the number of nodes. You can write n - 1 independent KCL equations

Kirchhoff Voltage Law (KVL)

The Voltages in any closed loop of a circuit is equal to zero

Where n is the number of nodes and b is the number of branches (components). You can write b - n + 1 independent KVL equations

Power

\[\begin{align*} Power &= Current &&+ Voltage\\ P &= I &&+ V .\end{align*}\]

Passive sign convention

Conservation of Power or Tellgen’s theorem

The sum of the power delivered to all the elements of the circuit is equal to zero at all times

DEFINITION:

\[\sum_{\text{all elements}} p_k = \sum_{\text{all elements}} v_k i_k = 0\]

Basic Circuit Elements

Two-terminal circuit elements

Characteristic

An equation describing how the Voltage and the current are related

e.g.

Independent Voltage source

The characteristic equation of this component constrains the voltage as a function of time(t)

$f(v,i) = v - 3 = 0$
Rearanged gives $v = 3$
In other words this means that no matter what the current flowing into it is the output voltage will always be 3V

Short Cicuit

Characteristic of Short Circuit of a Independent Voltage source $v_s(t) = 0$
No matter the current the independent voltage source is not outputting a voltage (it is acting like a wire)

Independent Current source

The characteristic equation of this component constrains the current as a function of time(t)

$f(v,i,t) = i(t) = 3A$

Power of Voltage & Current Sources

For absorbing $p(t) = v(t)i(t) \ge 0$

Ideal Resistor

A resistor has a value called resistance which is links to the voltage and the current passing through it by Ohm’s Law Resistance is measured in Ohm’s ($\Omega$)

Special Cases

If the voltage is zero or the current is zero the resistor behave like a short circuit
ideal resistor with a resistance of $0\Omega$ it behaves like a short circuit
Ideal resistor wiht a resistance of $\infty\Omega$ it behaves like an open circuit

Ohm’s Law

$v(t) = Ri(t)$ : where R is the resistance of the component

Second lecture (Lecture 4 in reality)

For sources the passive sign convention should be used only when attempting to calculate the power consumed by the source.\

Single-loop circuit without current sources

\[I = \sum_{i} \dfrac{v_i}{R_i}\]

Single node-pair cirucit

\[V = \dfrac{\sum(\text{Current sources *pushing* in the direction of }+V)}{\sum(\text{Conductances})}\]

Conductance $= 1/R_k$