Basic Circuit Analysis

Notes

Afandi Bin Ahmad

Department of Computer Engineering

Faculty of Electrical & Electronic Engineering

Prepared By

Associate Professor Dr Afandi Bin Ahmad

Department of Computer Engineering

Faculty of Electrical & Electronic Engineering

Universiti Tun Hussein Onn Malaysia

afandia@ut hm.edu.my

Graphic & Revision

Muhammad Muzakkir Bin Mohd Nadzri

Muhammad Shafiq Bin Nizar

Mohd Shafie Bin Nemmang

Muhamad Fakhrul Amri Bin Husainy

All Right Reserved 2018

No part of this publication may be reproduced, distributed, or transmitted in

any form or by any means, including photocopying, recording, or other

electronic or mechanical methods, w ithout the prior w ritten permission of the

author and Univ ersiti Tun Hussein Onn Malaysia.

CONTENT 07

08

Circuit Variables & Elements 09

Resistive Circuits 10

Circuit Analysis 11

Network Theorem 12

Capacitor 13

Inductor 15

First Order Circuit

Second Order Circuit

Circuit Variables & Elements

Important Terminology SI Units C harg e

Coulomb (C): The basic unit QUANTITY BASIC UNIT SYMBOL Is an electrical property of the atomic

used to measure electric charge particles of which matter consist,

Length Meter M measured in Coulomb (C).

Joule (J): A joule is the work

done by a constant 1-N force Mass Kilogram Kg q to idt

applied through a 1-m distance. t

Time Second S

Ampere (A): One ampere or amp Electric Current

is the current that flows when 1 Electric current Ampere A

Coulomb of charge passes each Electric current is the time rate of

second (1 A = 1 C/s) Thermodynamic temperature Kelvin K change of charge, measured in

amperes (A).

Volt (V): If a charge of 1 Coulomb Luminous intensity Candela cd

may be moved between two i dq

points in space with expenditure Topic 1 dt

of 1 Joule of work, 1 Volt is said to Circuit Variables &

be a potential difference existing Two common type of currents:

between these points (1 V = 1 J/ Elements

C)

Watt (W): The rate at which work

is done or energy expended. The

watt is defined as 1 Joule per

second (1 J/s).

Direct current Alternating current

Passive and Active Elements Voltage

The voltage vab between two points a and

b in an electric circuit is the energy (or

work) needed to move a unit charge from

a to b; mathematically,

Passive element is an electrical element that

absorbs or stores energy.

Examples of passive element: resistor,

inductor and capacitor.

[email protected] Currentdirection ofa passive elementstarts

Passive Sign Convention

from positive terminal to negative terminal. Whenever reference direction dw

for current direction in an dq

i element is in the direction of vab

reference voltage drop across

+ Passive element _ the element, use a positive

+v Ener gy

– sign in any expression that Energy is the fundamental ability to do

relates voltage to current.

Active element is an electrical element that

supplies energy to other elements in a circuit. Otherwise, use a negative work and produce action.

Examples of active element: voltage source, sign. Power is a measure of how fast energy is

current source, transistor. being used.

Current direction for an active element is In other words, power is the rate at which

going out of positive terminal into the energy is used.

negative terminal. p energy W

time t

i

+ Active element _

+v Power

–

Electrical Sources Power is the time rate of expending or

absorbing energy, measured in watts (W).

An electrical source is a device that is converting

non-electrical energy to electrical energy and vice +power absorbed = - power supplied

versa. Power can be delivered or absorbed as

The sources are either deliver or absorb electrical defined by the polarity of the voltage and the

power in order to maintain the voltage or current. direction of the current.

There are two types of electrical sources; voltage -+ Power delivered or supplied

and current sources. V by voltage source

There are four types of electrical sources used in I Power absorbed by resistor

circuit analysis.

1.Independent Sources +-

2.Ideal Voltage Sources V

Ideal Current Sources Dependent (Controlled)

Sources

1.Voltage Sources

2.Current Sources

7 [email protected]

Resistive Circuits

Resistor Ohm Law Nodes, Branches, Loops

Resistance: physical property or Ohm's law states that the voltage, v A branch:

ability to resist current, behavior of across a resistor is directly proportional A single element such as a voltage source or a

resisting the flow of electric to the current, i flowing through the resistor

charge. resistor. Represents any two-terminal element

Resistance factor:- vαi A branches:

The point of connection between two or more

length, l R is the material property can change branches.

if the internal or external condition of Usually indicated by a dot in a circuit.

the element are altered. A Loops:

To apply v=iR, the current and voltage Any closed path in circuit

must conform with the passive sign

cross - sectional area, A material with resistivity, convention.

This implies that the current flows from

Rρ l a higher potential to a lower potential Current Divider

A (v = iR).

If current flows from a lower to high I

potential, v = -iR

Possible value of R:- + I1 + I2 I1 R2 I

If R = 0, short circuit V1 R1 V2 R2 (R1 R 2 )

v = iR = 0, voltage is zero, current VS

- - I2 R1 I

could be anything Topic 2 (R1 R 2 )

In practical, short circuit perfect Resistive Circuits

conductor resistance approaching

zero

If R = infinity, open circuit

Current is zero, voltage could be

Voltage Divider

anything

Open circuit perfect conductor Equivalent Resistance R1 R2 R3 RN Rseries

+V-

resistance approaching infinity R1 R2 R3 R6

R7 (b)

[email protected]

+ V1 - + V2 - + V3 - + VN - Vs

Vs

Requivalent R4 R5

(a)

Kirchoff's Current Law (KCL) VX RX VS

states that the algebraic sum RT

of current s entering a node (or Req3 Req2 Req1

a closed loop boundary) is

zero. R eq1 (R 6 R 7 )Ω Source Transformation

R eq2 (R eq1//R 5 )Ω

The sum of the currents R eq3 (R eq2//R 4 )Ω x VS IS R b

entering a node is equal to the R equivalent (R eq3 R1 R 2 R 3 )Ω Ra

sum of the currents leaving the VS Is Rb IS VS

node. y Ra

I6 I1

I5 I2 Ra Rb

I4 I3

Ientering Ileaving

Wye-Delta Transformation

I1 - I2 - I3 + I4 - I5 + 16=0 @ I1 + I4 + 16 = I2 + I3 +I5 a RaRc

Rb

R1 Ra Rc

current entering R1 Ra R2 RaRb

Rc Rb Rc

Kirchoff's Voltage Law (KVL) current leaving R2 Ra Rb

states that the algebraic sum of all

voltages around a closed path (or b R3 Rb Rc RbRc

loop) is zero. Ra

R3

c

R1

+ V1 - R2 +

V2 Delta-Wye Transformation

E1 -

- V3 +

E2

a

R3 Ra R1R 2

R2 R3

R1 Ra R2 R1

Rc Rb

Rb R1 R 2R 3 R3

R2

E1 – V1 – V2 – E2 – V3 = 0 @ E1 – E2 = V1 + V2 + V3

b R3 Rc R1R 3

R2 R3

R1

c

afandia@ut hm.edu.my 8

Circuit Analysis

Example

Nodal Analysis Step 1: KCL at node a b IB2

R3

Nodal analysis - based on the systematic application of KCL. a IB1 I1 I2

Analyze any linear circuit by: R2 IB1 I2 IB1 I1 I2 0

I1 0 I1 I2 IB1

1. Obtaining a set of simultaneous equations IB1 R1

2. Solved to obtain the required values (voltage or current) c

3. Solve the simultaneous equation either using Cramer's

Rule or any other Va Vc Va Vb IB1 0

Nodal analysis provides a general procedure for analyzing circuit R1 R2

using node voltages as a circuit variables.

Important key idea resistance is a passive element, by the Va 0 Va Vb IB1 ...............[1]

passive sign convention, current must always flow from a higher R1 R2

potential to a lower potential

i vhigher vlower

R

Supernode Step 2: KCL at node b Step 3: Simplify equation [1] and [2]

Formed by enclosing a (dependent or Vb Vc Vb Va IB2 0 Step 4: Solve the equation

independent) voltage source connected R3 R2

between two nonreference nodes and any

elements connected in parallel with it. Vb 0 Vb Va IB2 ...............[2] Va Va Vb IB1 ...............[3]

4 R3 R2 R1 R2 R2

Supernode Vb Vb Va IB2 ...............[4]

R3 R2 R2

5V 1 1 1 Va IB1

12 2 3 R1 R2 R2

2ix

[email protected] 1

Topic 3 R3 Vb IB2

4 Circuits Analysis 1 1

R2 R2

How to deal with Supernode? Mesh Analysis

Supernode equation combination of

Mesh analysis provides another general procedure

KCL equation for the respective for analyzing circuits.

nodes.

Support equation equation for the Recall that a loop is a closed path with no node

voltage drop in between the combined passed more than once.

nodes compared to the voltage

source. A mesh is a loop that does not contain any other

loop within it.

Example

Mesh analysis apply KVL to find unknown currents

Write the supermesh and support Example R5

equation for the following circuit.

R1 R3

R1 R3

Mesh Analysis with Current Sources VB1 R2 R4 VB2

VB1 I1 R2 I2 VB2 Case 1: Loop 1,

When a current source exists only in one mesh I1R1 (I1 - I2 )R 2 - VB1 0

Supermesh equation, I1R1 (I1 - I2 )R2 VB1 ................ [1]

I1R1 I2R3 - VB2 - VB1 0 ............... [1] 10Ω 4Ω

Support equation,

I1 - I2 Ib (I1 I2 IB) ............... [2] 24V I1 12Ω I2 3A

9 [email protected] Loop 2, ................. [2]

(I2 - I1)R 2 I2R 3 (I2I3 )R 4 0

I2R 2 - I1R 2 I2R 3 I2R 4 I3R 4 0

I2=-3A Loop 3, ................. [3]

(I3 - I2 )R 4 I3R 5 -VB2

Case 2: I3R 4 - I2R 4 I3R 5 -VB2

When a current source exists in between of two meshes

create supermesh.

Set the following equation:

1. Supermesh equation

2. Support equation

Network Theorem

Superposition Theorem Thevenin's Theorem Norton's Theorem

Definition: Definition: Definiton:

Superposition theorem states that the voltage Thevenin's Theorem states that a linear two- Norton's Theorem states that a linear two-

across (or current through) an element in a terminal circuit can be replaced by an terminal circuit can be replaced by equivalent

circuit is the algebraic sum of the voltages equivalent circuit consisting of: circuit consisting of:

across (or current through) that elements A voltage source VTh in SERIES with A current source IN in PARALLEL with

due to each independent source acting A resistor RTh A resistor RN

alone.

Where; Where;

How to apply? VTh is the open-circuit voltage at the terminals. IN is the SC current through the terminals.

1. Consider one independent source at a RTh is the input or equivalent resistance at the RN is the input or equivalent resistance at the

time, while other independent sources are terminals when the independent sources are terminals when the independent sources are

turned-off. [short-circuit for voltage source and turned off. turned-off.

open-circuit the current source]

2. Dependent source are left intact because

they are controlled by the other circuit

variables.

Topic 4

Network Theorems

Thevenin's and Norton's Theorems with Maximum Power Transfer Theorem

Dependent Sources

Thevenin equivalent - useful in finding the maximum

power a linear circuit can deliver to a load.

Consider the following circuit for maximum power

transfer.

R Th a

Notes:

1. To get VTh and IN, we can analyze the circuit

using any method same with independent

sources.

a[email protected]. But, for RTh and RN, we can not go directly

(because the behavior of dependent sources). To VTh i p i2R VTh 2 R L

deal with this situation, let's go through the RL R Th R L

following ideas.

How to deal with dependent source in Thevenin's b

and Norton's Theorems?

Method 1 Maximum power is transferred to the load when the load

Find the VTh and IN. Then: resistance equals the Thevenin resistance as seen from

the load (RL=RTh).

RN R Th VTh

IN To prove the maximum power transfer differentiate p with

respect to RL and set the results =0

Method 2 The power is maximum when,

1.Introduce one independent voltage/current

source with any value at respective terminal. For dP 0

any dependent sources in the circuit must be dR L

turned-off first.

2.Consider the following circuit, for introducing dP VT h 2 (R Th R L )2 2R L (R Th R L )

any dependent sources. dR L (R Th R L )2

VT h 2 (R Th R L 2R L ) 0

(R Th R L )3

a Introduce VT ,

Linear circuit VT R Th RN VT ()

I

This implies that

b 0 (R Th R L 2R L ) (R Th R L ),[R Th R L ]

Showing that, the maximum power takes place when R L RTh.

a Introduce IT , With R L R Th ,

Linear circuit Vab IT R Th RN Vab () Pmax VT h 2 VT h 2 (watt)

IT 4R L 4R L

b

10afandia@ut hm. edu. my

Capacitor

Definition

Passive elements designed to store energy in its electric field . Working Principles

In general a capacitor is constructed by two plates separated The amount of charge stored = q (directly

by an insulating (dielectric) material

proportional to the applied voltage)

So, C = constant of proportionality called

capacitance of the capacitor

Unit = Farad (F)

Capacitance is the ratio of the charge on one

Properties of a capacitor

plate of a capacitor to the voltage difference

When the voltage across a between the two plates. Measured in Farad (F)

capacitor is not changing with

time (eg. DC voltage), the 1 Farad = 1 Coulomb/Volt

Capacitance:

Ratio of the charge, q per plate to the applied

current through the capacitor voltage, V

is zero. It does not depend on q or V, but depends on

A capacitor is open circuit to

the physical dimensions of the capacitor

DC

However, if a battery (DC

voltage) is connected across a

capacitor, the capacitor Topic 5

charges.

The voltage on the capacitor

must be continuous. Energy Storage Elements

The voltage on a capacitor (Capacitor)

cannot change abruptly.

Current-voltage relationship for a capacitor

Basically, i C dv V(t) 1 t

q CV dt C

dq dCV idt V(t o )

dv i

[email protected] dt dt C to

wh er e V(t o ) q(t o ) the voltage across

C

dq C dV , since i dq dv 1 idt the capacitor at time t o Parallel and Series Capacitor

C * shown that capacitor voltage

dt dt dt

i C dv dv 1 idt depends on the past current

dt C

V(t) 1 t q(t o)

idt i1 i2 i3

1 C to C

v C idt I C1 C2 C3 iN

CN V

Instantaneous power and energy stored

P Vi VC dv CV dv E dw CV dv

dt dt dt dt

i i1 i2 i3 ... iN

dw CV dv i C dv

dt dt dt

dw CV dv dt i C1ddvt C2 ddvt C3 dv ... CNddvt

dt dt

w t dv

-CV dt

where,

t

w C -Vdv Ce C1 C2 C3 ... CN

q

w C V2 t ,we note that v(- ) 0 because the capacitor

2 t- i C1 C2 C3 CN V V1 V2 V3 ... VN

V V1 V2 V3 VN

was uncharged at t -, thus 1 t

Ck

w 1 CV 2(t) ButVk i(t) Vk (to ).

2

to

Therefore,

w(t) 1 CV 2(t), q CV V1 t 1 t

2 C1 V1(to ) C2 V2 (to )

i(t) i(t)

w qV , V q

2C to to

W q2 1 t 1 t

2C C3 V3 (to ) ... CN VN (to )

i(t) i(t)

to to

where,

1 1 1 1 ... 1

Ceq C1 C2 C3 CN

11 [email protected]

Inductor

Definition

Passive element designed to store energy in its magnetic field. Working Principles

To enhance the inductive effect: a practical inductor is usually If current is allowed to pass through an inductor,

formed into a cylindrical coil with many turns of conducting wire.

voltage across the inductor is directly proportional to

the time rate of change of the current.

Inductance is the property whereby an inductor

Properties of an inductor exhibits opposition to the change of current flowing

through it, measured in henrys (H).

Inductance depends on its physical dimension and

When the voltage across a construction.

capacitor is not changing with v L di

time (eg. DC voltage), the dt

current through the capacitor

is zero. L constant of proportionality

A capacitor is open circuit to L inductance (Unit Henry)

DC

However, if a battery (DC

voltage) is connected across a

capacitor, the capacitor

charges. Topic 5

The voltage on the capacitor Energy Storage Elements

must be continuous.

The voltage on a capacitor

cannot change abruptly. (Inductor)

Voltage-current relationship for an inductor

Basically,

v L di

dt

vdt Ldi

[email protected]1vdt

L

where Parallel and Series Inductor

i(t0 ) total current for t t0

and i() 0. i L1 L2 L3 LN

The idea i() 0 is reasonable, because V1 V2 V3 VN

there must be a time in the past when there was

1 no current in the inductor. V

L

di vdt

1 Energy _ stored

L

i vdt p vi L di i Based on KVL,

dt

i 1 t v(t)dt i(t0 ) di

L t0 w pdt t pdt t Li di dt V V1 V2 V3 ... VN ; V L dt

dtt0 t0

V L1 di L2 di L3 di ... LN di

t L t 1 1 dt dt dt dt

2 i2 t0 2 Li2 (t) 2 Li2 (t0 )

L idi N di di

V k1 Lk dt Leq dt

t0

1 Li2

2 Leq L1 L2 L3 ... LN

Important Properties i i1 i2 i3 iN

Voltage across an inductor is zero when the current is V L1 L2 L3 LN

constant.

Inductor acts like a short-circuit to DC

The current through an inductor cannot change Based on KCL,

instantaneously. ii1 i2 i3 ... 1 Vdt i(t0 )

The ideal capacitor does not dissipate energy. The iN;i L

inductor takes power from the circuit when it storing i1 t 1 t

energy and delivers power to the circuit when L1 t0 Vdt i1(t0 ) L2 t0 Vdt i2(t0)

returning previously stored energy.

1 t 1 t iN(t0)

L3 t0 Vdt i3(t0) ... LN t0 Vdt

i 1 t Vdt i(t0 )

Leq t0

1 1 1 1 ... 1

Leq L1 L2 L3 LN

12afandia@ut hm. edu. my

First Order Circuit

Definition Source-Free RC Circuit

RC and RL circuits are circuits that combined resistors with Since the capacitor is fully charged, the v(0) V0

capacitors and resistors with inductors respectively. initial voltage and energy are given as

w(0) 1 CV0

The analysis techniques used in RC and RL circuits are the Applying KCL at top node of the 2

same as in resistive circuits but they involve differential circuit gives

equations. iC iR 0

By definition, i = Cdv/dt and iR = v/R,

Both RC and RL circuits produce first order differential thus C dv v 0

equations, thus these circuits are known as first-order dt R

circuits. Rearranging the equations and

integrating both sides ln v t ln A

There are two ways of supplying energy to these circuits: RC

Initial condition of storage elements in the circuit itself known

as source-free circuit or natural response.

Applying independent sources to the circuit known as step

response.

Combining two type of first order circuits and two ways of

exciting the circuit give four different types of first order circuit

variations.

Source-Free RL Circuit Putting variable v on right hand side t

gives

v(t) Ae RC

Topic 6 From initial condition, A = v(0) = V0 t

v(t) V0 e RC

First-Order Circuits t

Natural response for RC circuit v(t) V0e RC

Note:

From the equation obtained, it is obvious that the voltage

In RL circuit, inductor current is taken as the i(0) I0 response of RC circuit is reduced exponentially from its

response. Its current and energy are given as initial value.

1 Since there is no electrical source involved, therefore the

w(0) 2 LI 0 response is called the natural response.

In other words, natural response of an RC circuit solely

depends on the energy stored in the capacitor.

Applying KVL around the loop of the circuit

gives vL vR 0 τ RC v(t) V0e t

L di iR 0 τ

[email protected] definition, vL = Ldi/dt and vR = iR,

dt It is clear that the smaller the time constant, the faster

the response, i.e. the faster the voltage decays to reach

Rearranging the equations and integrating ln i Rt ln A the steady state response.

both sides L The response during process of decaying from its initial

value is noted as transient response.

The natural response of RC circuit in the following

Rt figure.

Putting variable i on right hand side gives i(t) Ae L

From initial condition, A = i(0) = I0 Rt

i(t) I0e L

Natural response for RL circuit Rt

i(t) I0e L Current and power dissipated in resistor

Note: iR (t) v(t) V0 e t

The current response of RL circuit is reduced R R τ

exponentially from its initial value. p V02 e 2t

Since the energy is only coming from inductor, τ

thus the response is known as natural

response.

τ L i(t) t R

R τ

I0e

The smaller the time constant, the faster the Summary of Source-Free RC Circuit

inductor current decays from its initial value.

Determine the initial voltage, V0, of the capacitor.

The inductor current decays much faster during Find the time constant of the circuit , τ = RC.

the transient response before it reaches the Calculate the capacitor voltage to complete the RC natural

steady state response.

response.

The step response of inductor current is Use natural response of capacitor voltage to find expression

illustrated in the following figure.

of capacitor current iC, resistor voltage vR and resistor

current iR.

Current and power dissipated in resistor

t Summary of Source-Free RL Circuit

τ

vR (t) iR I Re

0 Determine the initial inductor current, I0.

Calculate the time constant of the circuit, τ

2t = L/R

τ Finalise the RL natural response by finding the inductor current

p I 2 Re

0

response.

From natural response of inductor current, determine the inductor

voltage vL, resistor voltage vR and resistor current iR.

13 [email protected]

First Order Circuit

Step Response RC Circuit

Assume that the initial capacitor voltage just before v0 v0 V0

and after the voltage source is applied

C dv v VSut 0

Applying KCL in the circuit produces

dt R

Rearranging the equation yields dv dt

v VS RC

Integrating both sides and putting initial ln v VS 1

conditions V0 VS RC

Simplify the equation by taking v to the right hand vt VS V0 VS e t τ t 0

side and substitutes RC term with τ

Uncharged capacitor Charged capacitor A complete step response of an RC circuit is vt V0 , t 0

given as t0

VS V0 VS e t τ ,

complete step response If the capacitor is assumed not charged vt 0, 1 - e tτ , t0

initially, therefore V0=0 VS t0

Complete response = natural response + forced response

v = vn + vf

where vn V0e t τ vt VS 1- e t τ ut

vf VS1 e t τ

It also can be written as

Note: The current through the uncharged capacitor is it VS e t τ ut

Natural response, vn, is actually the given as

R

response due to capacitor itself it will Step Response RL Circuit

continue to decay and eventually

dies.

Forced response, vf, is the response

produced due to independent voltage

source.

Natural response normally exists

along with the transient response

[email protected] theforcedresponseexists

Topic 6

First-Order Circuits

along with steady-state response. i 0 i0 I0

Complete response = transient response + steady-state response Assume that the initial inductor current just di VSu t

before and after the current source is applied dt

v = vt + vss Applying KVL around the loop produces

vt V0 - VS e t τ Rearranging the equation yields L iR 0

where vss VS

di Rdt

i - VS L

step response form in general R

vt v v0 ve t τ i VS R

ln R

where v is the final capacitor voltage Integrating both sides and putting initial VS L

v 0 is the initial capacitor voltage conditions R

I0

τ is the time constant

Simplify the equation by taking i to the right i(t) VS I0 VS e t t0

hand side and substitutes R/L term with τ R R

Summary of Step Response RC Circuit If the capacitor is assumed not charged

initially, therefore I0 = 0

Initial capacitor voltage, V0, if the 0, t 0

capacitor has been charged

it VS tτ

Final capacitor voltage, VS R 1 - e , t0

Time constant, τ

Summary of Step Response RL Circuit It can also be written as it VS 1 - e tτ ut

R

Initial inductor current, I0, if the inductor The voltage across the uncharged inductor is

has been charged given as vt VS e t τ ut

Final inductor current, IS general form for step response of RL circuit

Time constant, τ

The equation of step response can be it i i0 i e t τ

used to find the instantaneous current at

any time.

where i is the final inductor current Uncharged inductor Charged inductor

i0 is the initial inductor current

τ is the time constant

14afandia@ut hm. edu. my

Second Order Circuit

Source Free Series RLC Circuit

Initial and Final Values With initial capacitor voltage V0 and initial inductor current I0.

Thus, at initial condition (t = 0):

v0, i0, dv0, di0, v, i v0 V0 i0 I0

dt dt Ri L di 1 t

Applying KVL around the loop in the circuit

idt 0

dt C

where v denotes the capacitor

voltage and i denotes the the Differentiate each term to eliminate the d2i R di i 0

inductor current. integral and rearrange them dt 2 L dt LC

Note:

Another important point to

remember is that both variables v To solve the second order equation, there must be initial Ri0 L di0 V0 0

and i must be defined according values of i and di/dt or initial values of some v and i.

to passive sign convention. The initial value of i is I0 and to find initial value of di/dt is dt

Capacitor voltage is always di0 1 RI0 V0

continuous, the same goes to L

inductor current First order solution is i Aest dt

v 0 v0 and i 0 i 0 As2est AR sest A est 0

Characteristic equation L LC

Aest s2 R s 1

L LC

[email protected]

Expressing the roots

s2 R s 1 0 s1 R R 2 1 s1 α α2 ω02 s2 α α2 ω02

L LC 2L 2L LC where α R ,

1

R 2 1 2L ω0 LC

2L LC

s2 R

2L

Second-Order Circuits s1 and s2 are called natural frequencies since they are associated

with natural response,

0 is the resonant frequency and

is the damping factor.

Solution of natural response for series RLC i1 A1es1t i2 A2es2t

If > 0, the solution is overdamped, i t A1es1t A2es2t

If = 0, the solution is critically damped, and

If < 0, the solution is underdamped.

Overdamped case Critically damped case Underdamped case

This occurs when a > w0, or when C > 4L/R2. This occurs when = 0, or when C = 4L/R2 It happens when < 0, or when C < 4L/R2

Both roots s1 and s2 are negative and real. The natural response solution: The natural response solution:

The response is it A1t A2 eαt it eαt B1cosdt B2sindt

i t A1es1t A2es2t where d is damped natural frequency

15 [email protected]

Second Order Circuit

Source Free Parallel RLC Circuit

With initial voltage is given as V0 and initial inductor current I0.

Applying KCL at the top node yields

v 1 t vtdt C dv 0

R L dt

Differentiate each element w.r.t. t gives

d2v 1 dv 1 v 0

dt 2 RC dt LC

Characteristic equation

s2 1 s 1 0

RC LC

Expressing the roots

1 1 2 1 s1 α α2 ω02 s2 α α2 ω02

2RC 2RC LC where α R ,

2L

[email protected] 1 1 2 1 ω0 1

2RC 2RC LC LC

s2

s1 and s2 are called natural frequencies since they are associated

with natural response,

0 is the resonant frequency and

is the damping factor.

Topic 7

Second-Order Circuits

Overdamped case Critically damped case Underdamped case

This occurs when a > w0, or when C > 4L/R2. This occurs when = 0, or when C = 4L/R2 It happens when < 0, or when C < 4L/R2

Both roots s1 and s2 are negative and real. The natural response solution: The natural response solution:

The response is

vt A1t A2 eαt vt eαt A1cosdt A2sindt

v t A1es1t A2es2t

where d is damped natural frequency

16afandia@ut hm. edu. my

Second Order Circuit

Step Response Series RLC Circuit Applying KVL around the loop in the circuit

L di iR v VS

dt

It is known that i C dv , therefore d2v R dv v VS

dt dt 2 L dt LC LC

Solution for step response vt vt vss

The solution of transient response is the same that are

obtained in natural response, which are

v t A 1e s1t A 2 e s2t Overdamped case

v t A 1t A 2 e αt Critically damped case

vt eαt A1cosd t A 2sind t Underdamped case

Steady state response is the same as the final value of v(t)

vss t v VS

Complete solution for step response for series RLC

v t VS A1es1t A es2t Overdamped case

2

Topic 7 v t VS A1t A2 eαt Critically damped case

[email protected] Circuits

v t VS eαt A1cosd t A2sind t Underdamped case

Note:

The values for A1 and A2 are obtained from

the initial conditions v(0) and dv(0)/dt.

Step Response Parallel RLC Circuit Applying KCL at the top node for t > 0 yields

Complete solution for step response C dv i v IS

for series RLC circuit dt R

i t IS A1es1t A2es2t Overdamped case It is known that v L di , therefore d2i R di i IS

i t IS A1t A2 eαt Critically damped case dt dt 2 L dt LC LC

i t IS eαt A1cosdt A2sindt Underdamped case

The form of the equation obtained is the same as

17 [email protected] natural response parallel RLC but with variable i.

Solution for step response i t it iss

The solution of transient response

i t A1es1t A2es2t Overdamped case

i t A1t A2 eαt Critically damped case

i t eαt A1cosdt A2sindt Underdamped case

Steady state response is the same as the final value of i(t)

iss t i IS

Basic Circuit Analysis – Quick Reference Notes

Disember 2018