CIRCUITS and SYSTEMS – part I

Prof. dr hab. Stanisław Osowski

Electrical Engineering (B.Sc.) Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie

Lecture 2

Analysis of circuits in steady state at sinusoidal excitation

Sinusoidal signal

u(t) - instantaneous value of signalUm - maximum value (magnitude) of signal

- initial phase (phase corresponding to t=0)

t+ - phase angle at time tf=1/T - frequency in HzT - period of sinusoidal signal

- angular frequency measured in radians per second

)sin()( tUtu m

RMS value of signal

2

)sin()( mm

UUtUtu

Tot

t

dttfT

F0

2 )(1

For sinusoidal signal

• voltage

• current

2

)sin()( mm

IItIti

Steady state of the circuit

Steady state of the circuit is the state in which the character of the circuit response is the same as the excitation. It means that at sinusidal excitation the response is also sinusidal of the same frequency.

For the need of steady state analysis we introduce the so symbolic method of complex numbers. This method converts all differential and integral equations into algebraic equations of complex character.

Symbolic method for RLC circuit

The RLC circuit under analysis

The circuit equation in time domain

dt

diLidt

CRitUm

1)sin(

General solution of circuit

The general solution of the circuit in time domain is composed of two components: x(t)=xs(t)+xt(t)

•Steady state component – part xs(t) of general solution for which the signal has the same character as excitation (at sinusidal excitation the response is also sinusidal of the same frequency). This state is theoretically achieved after intinite time (in practice this time is finite).•Transient component - part xt(t) of general solution for which the signal may take different form from excitation (for example at DC excitation it may be sinusoidal or exponential). The general solution is just the sum of these two parts

x(t)=xs(t)+xt(t)

Solution in steady state

tjjmm eeUtUtUtu )( )sin()(

tjjmm eeItItIti )( )sin()(

Symbolic represenation of voltage excitation

Symbolic represenation of current response

dttICdt

tdILtRItU )(

1)()()(

Symbolic equation of circuit

Solution in steady state (cont.)

After performing the appropriate manipulations we get

iii jmjmjmjm eI

Cje

ILje

IRe

U

2

1

222

2

,2

ijmjm eI

IeU

U

The complex RMS notations of current and voltage

ICj

LIjRIU

1

The complex RMS equation of the circuit

Complex represenation of the RLC elements

Resistor

Inductor

Capacitor

R Z R RIU R

LjLIjU L L Z

Cj

CjI

CjUC

11 Z

1C

Complex impedances• Reactance of inductor

LL jXZ

• Reactance of capacitor

LX L

CXC

1

• Impedance of inductor

• Impedance of capacitor

CC jXZ

Final solution of RLC circuit

• Complex algebraic equation of RLC circuit

ZIIZIZRIU CL

• Complex current

R

CLjj

eCLR

U

CLjR

eU

Z

UI

)/(1arctg

22 ))/(1(/(1

• Magnitude RMS value of current

• Phase of current

22 ))/(1( CLR

U

Z

UI

R

CLi

)/(1arctg

Kirchhoff’s laws for complex representation

• KCL

• KVL

• Ohm’s complex law

Y=1/Z - complex admittance

k

kI 0

k

kU 0

YUIZIU

Symbolic method - summary

• Conversion: time-complex representation of sources

i

u

jmim

jmum

eI

tIti

eU

tUtu

2 )sin()(

2 )sin()(

• Complex represenattion of RLC elements

• Kirchhoff’s laws for complex values

• Solution of complex equations -> complex currents & voltages.

Example

Determine the currents in in steady state of the circuit at the following values of parameters: R=10Ω, C=0,0001F, L=5mH, i(t)=7.07sin(1000t) A.

Circuit structure

Solution

Complex symbolic values of parameters:

ω = 1000

I = 5ej0 = 5

ZL = jωL = j5

ZC = -j/(ωC) = -j10

Admittance and impedance of the circuitoj

CL

eY

ZjZZR

Y 45

2

101 1,01,0

111

Solution (cont.)

Voltage and currents

o

o

o

o

j

CC

j

LL

jR

j

eZ

UI

eZ

UI

eR

UI

eZIU

135

45

45

45

2

5

2

102

52

50

Solution (cont.)

Time representation of the signals

)1351000sin(5)(

)451000sin(10)(

)451000sin(5)(

)451000sin(50)(

oC

oL

oR

o

tti

tti

tti

ttu

Phasor diagram for resistor

Equation

ojRRR eRIRIU 0

Phasor diagram for inductor

Equation

ojLLL eLILIjU 90

Phasor diagram for capacitor

Equation

ojCCC eI

CI

CjU 9011

Phasor diagram for RLC circuit• The construction starts from the farest branch from the source. For

series connected elements of this branch start from current; for parallel connected elements start from voltage. Next we draw alternatingly the currents and voltages for the succeeding branches, approaching in this way the source.

• The relation of the input voltage towards the input current determines the reactive character of the circuit. – If the input voltage leads its current the character is inductive.– If (opposite) the input voltage lags its current the character of the circuit is

capacitive. – When the voltage is in phase with current – the circuit is of resistive character.

ExampleDraw the phasor diagram for the circuit

RLC circuit structure

Construction of phasor diagram