Circuit Theory/First Order Circuits

First Order Circuits

First order circuits are circuits that contain only one energy storage element (capacitor or inductor), and that can, therefore, be described using only a first order differential equation. The two possible types of first-order circuits are:

1. RC (resistor and capacitor)
2. RL (resistor and inductor)

RL and RC circuits is a term we will be using to describe a circuit that has either a) resistors and inductors (RL), or b) resistors and capacitors (RC).

RL Circuits

An RL parallel circuit

An RL Circuit has at least one resistor (R) and one inductor (L). These can be arranged in parallel, or in series. Inductors are best solved by considering the current flowing through the inductor. Therefore, we will combine the resistive element and the source into a Norton Source Circuit. The Inductor then, will be the external load to the circuit. We remember the equation for the inductor:

$v(t)=L{\frac {di}{dt}}$

If we apply KCL on the node that forms the positive terminal of the voltage source, we can solve to get the following differential equation:

$i_{source}(t)={\frac {L}{R_{n}}}{\frac {di_{inductor}(t)}{dt}}+i_{inductor}(t)$

We will show how to solve differential equations in a later chapter.

RC Circuits

A parallel RC Circuit

An RC circuit is a circuit that has both a resistor (R) and a capacitor (C). Like the RL Circuit, we will combine the resistor and the source on one side of the circuit, and combine them into a thevenin source. Then if we apply KVL around the resulting loop, we get the following equation:

$v_{source}=RC{\frac {dv_{capacitor}(t)}{dt}}+v_{capacitor}(t)$

First Order Solution

Series RL

The differential equation of the series RL circuit

$L{\frac {dI}{dt}}+IR=0$

${\frac {dI}{dt}}=-I{\frac {R}{L}}$

${\frac {1}{I}}dI=-{\frac {R}{L}}dt$

$\int {\frac {1}{I}}dI=-{\frac {R}{L}}\int dt$

$lnI=-{\frac {R}{L}}t+C$

$I=e^{-{\frac {R}{L}}t+C}$

$I=Ae^{-{\frac {R}{L}}t}\quad A=e^{C}$

t	I(t)
1 ${\frac {R}{L}}$	36% A
2 ${\frac {R}{L}}$	14% A
3 ${\frac {R}{L}}$	5% A
4 ${\frac {R}{L}}$	2% A
5 ${\frac {R}{L}}$	0.7% A

Series RC

The differential equation of the series RC circuit

$C{\frac {dV}{dt}}+{\frac {V}{R}}=0$

${\frac {dV}{dt}}=-V{\frac {1}{RC}}$

${\frac {1}{V}}dV=-{\frac {1}{RC}}dt$

$\int {\frac {1}{V}}dV=-{\frac {1}{RC}}\int dt$

$lnV=-{\frac {1}{RC}}t+C$

$V=e^{-{\frac {1}{RC}}t+C}$

$V=Ae^{-{\frac {1}{RC}}t}\quad A=e^{C}$

t	V(t)
1 ${\frac {1}{RC}}$	36% V
2 ${\frac {1}{RC}}$	14% V
3 ${\frac {1}{RC}}$	5% V
4 ${\frac {1}{RC}}$	2% V
5 ${\frac {1}{RC}}$	0.7% V

Time Constant

The series RL and RC has a Time Constant

$T={\frac {L}{R}}$

$T={\frac {RC}{1}}$

In general, from an engineering standpoint, we say that the system is at steady state ( Voltage or Current is almost at Ground Level ) after a time period of five Time Constants.