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Transition between real sinusoidal signals (time domain) and complex variable signals (frequency domain) General case, when voltages and/currents also have a phase shift: Amplitude PhaseTime- dependent (oscillating ) component Complex amplitude (includes phase) Time- dependent (oscillating) component

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I-Vs in time-domain and on the complex plane Capacitor Inductor Resistor Time domain (real variables) Complex variables Assuming v(t) and i(t) are the sinusoidal signals with the angular frequency Complex amplitudes complex amplitude

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Impedances Capacitor Inductor Resistor Complex amplitude I-Vs Impedances Complex amplitude I-V relationships using impedances

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Impedance, resistance, reactance Capacitor Inductor Resistor Impedance is in general, a complex number. Therefore, in general, the impedance contains real and imaginary parts: Z = R + jX; where R is resistance; X is reactance no imaginary part: reactance = 0 no real part: resistance = 0

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Impedance of series connections Impedance is in general, a complex number. Therefore, in general, the impedance contains real and imaginary parts: Z = R + jX; where R is resistance; X is reactance R-C in series Using complex variables, a series R-C connection can be considered as a single impedance with the real part (resistance) = R and imaginary part (reactance) = X C = -1/( C) R-L in series Using complex variables, a series R-L connection can be considered as a single impedance with the real part (resistance) = R and imaginary part (reactance) = X L = L Using complex variables, a series L-C connection can be considered as a single impedance with the real part (resistance) = 0 and imaginary part (reactance) = X LC = L -1/( C) L-C in series RC RL LC

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v(t)= 9*cos( t) V C 3 pF R 50 Ohm RC circuit example Find the current amplitude in the circuit at the signal frequency f = 1 GHz 1. Complex source voltage amplitude, V S = 9 V 2. Angular frequency, = 2 f = Total circuit impedance: The reactance X = -1/(2 9*3e-12) = Ohm; Z T = 50 – j50.35 Ohm 4. Complex current amplitude in the circuit: I = V S / Z T 5. Complex current amplitude can be found if we present the complex number describing I in the Eulers form: I = I M e j I M = ; = 0.79 rad I = i

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Complex numbers There are two commonly accepted notations for the j or i Some examples 1 + 4i is a complex number with a real part =1, imaginary part= j2 is a complex number with a real part =2, imaginary part= i is a complex number with a real part = 0, imaginary part = -4 5is a complex number with a real part = 5, imaginary part =0

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Phasors and complex numbers VMVM Phasor representing complex amplitude (V M L is a combination of two parameters: the amplitude V M (the length of the arrow) and the phase angle. This pair of two parameters fully describes the AC waveform of a given angular frequency x y R x0x0 y0y0 A pair of two parameters: the radius R and the angle fully describes the position of the point (red dot) on the x – y plane The position of the point can also be described by a pair of two numbers: x 0 and y 0 The pair (R,j) or (x 0, y 0 ) is called a complex number

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Complex numbers Complex numbers provide a convenient technique to describe the phasors related to AC voltages and currents. The real number, the square of which is equal to (-1) does not exist. Therefore, j is not a real number. Complex number is the sum of two parts: Real part, Re(N), and Imaginary part, Im(N) multiplied by a so- called imaginary unit j (sometimes also denoted as i). Imaginary unit is a special number such that j 2 = -1; Introduction of the imaginary unit is needed to provide simple and convenient rules to operate the pairs of two number (complex numbers) as a single number. N = x + jy, x = Re(N); y = Im(N);

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Complex numbers There are two equivalent ways to define the complex number: by Real and Imaginary parts (x and y coordinates) or by the radius and the angle of the arrow poining to the number. This radius is also called a modulus, or an absolute value of the complex number. x Complex number N represented by the position of the dot on x-y plane x1x1 jy 1 jy |R|

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Complex numbers The angle that the vector makes with the axis x is called an argument or phase angle x Complex number N represented by the position of the dot on x-y plane x1x1 jy 1 jy x1 = |R| cos ( ); y1 = |R| sin ( ) |R|

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Complex numbers To summarize, any complex number can be described in two ways: x Complex number N x1x1 jy 1 jy N = |R| L (Phasor, or Polar form ) N = x 1 + j y 1 (Algebraic, or Cartesian form)

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The algebra of Complex numbers: all the algebraic operations, addition, subtraction, multiplication and division follow the algebra rules for real numbers; whenever there is a term i 2 or (j 2 ) replace it with (-1) z1 = a + bi and z2 = c + di; addition: z1 + z2 =a + bi + c + di =(a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i

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The algebra of Complex numbers: all the algebraic operations, addition, subtraction, multiplication and division follow the algebra rules for real numbers; whenever there is a term i 2 or (j 2 ) replace it with (-1) z1 = a + bi and z2 = c + di; multiplication: z 1 z 2 = (a + bi)(c + di) = ac + adi + bci + bdi 2 ; i 2 = -1, therefore: z 1 z 2 = (a + bi)(c + di) = ac + adi + bci - bd z 1 z 2 =ac - bd + (ad + bc)i

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The algebra of Complex numbers: all the algebraic operations, addition, subtraction, multiplication and division follow the algebra rules for real numbers; whenever there is a term i 2 or (j 2 ) replace it with (-1) z1 = a + bi and z2 = c + di; division

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The algebra of Complex numbers: Polar –> Cartesian form conversion x Complex number N x1x1 jy 1 jy |N| (Phasor, or Polar form ) N = x 1 + j y 1 (Algebraic, or Cartesian form) Re (N) = x = |N|*cos( ); Im (N) = y = |N|*sin( );

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The algebra of Complex numbers: Cartesian -> Polar form conversion x Complex number N x1x1 jy 1 jy |N| (Phasor, or Polar form ) N = x 1 + j y 1 (Algebraic, or Cartesian form) tan( = y/x = Im(N)/Re(N), or: = arctan (y/x) = arctan [Im(N)/Re(N)]

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v(t)= 9*cos( t) V C 3 pF R 50 Ohm RC circuit example for MATLAB simulations Find the voltage amplitude across the resistor R in the frequency range f = 10 MHz – 10 GHz See the MATLAB code on the next slide

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MATLAB codeComments clc Vm=9; C=3e-12;R=50; f=1e7:1e7:1e10; om=6.28*f; Z = R+(1./(i*om*C)); Vr = Vm./Z*R; Vrm = abs(Vr); AlphaR = angle(Vr); figure(1) plot(f, Vrm) xlabel('Frequency, Hz') ylabel('Voltage amplitude, V') figure(2) plot(f, AlphaR) xlabel('Frequency, Hz') ylabel('Voltage phase, rad') Clear matlab screen Define the variables … Define the frequency array Angular frequency Total impedance; note the. Complex voltage amplitude across resistance Voltage amplitude across R Resistor voltage phase shift (with respect to the source voltage) Plotting the frequency dependences: Figure (1) – resistor voltage amplitude Figure (2) – resistor voltage phase

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Figure 1Figure 2

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