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 signal U m - maximum value (magnitude) of signal  - initial phase (phase corresponding to t=0)  t+  - phase angle at time t f=1/T - frequency in Hz T - period of sinusoidal signal  - angular frequency measured in radians per second

RMS value of signal For sinusoidal signal voltage current

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

General solution of circuit The general solution of the circuit in time domain is composed of two components: x(t)=x s (t)+x t (t) Steady state component – part x s (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 x t (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)=x s (t)+x t (t)

Solution in steady state Symbolic represenation of voltage excitation Symbolic represenation of current response Symbolic equation of circuit

Solution in steady state (cont.) After performing the appropriate manipulations we get The complex RMS notations of current and voltage The complex RMS equation of the circuit

Complex represenation of the RLC elements Resistor Inductor Capacitor

Complex impedances Reactance of inductor Reactance of capacitor Impedance of inductor Impedance of capacitor

Final solution of RLC circuit Complex algebraic equation of RLC circuit Complex current Magnitude RMS value of current Phase of current

Kirchhoff’s laws for complex representation KCL KVL Ohm’s complex law Y=1/Z - complex admittance

Symbolic method - summary Conversion: time-complex representation of sources 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 = 5e j0 = 5 Z L = jωL = j5 Z C = -j/(ωC) = -j10 Admittance and impedance of the circuit

Solution (cont.) Voltage and currents

Solution (cont.) Time representation of the signals

Phasor diagram for resistor Equation

Phasor diagram for inductor Equation

Phasor diagram for capacitor Equation

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.

Example Draw the phasor diagram for the circuit RLC circuit structure

Construction of phasor diagram