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 Time-dependent (“oscillating”) component Phase Complex amplitude (includes phase) Time-dependent (“oscillating”) component

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

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

Impedance, resistance, reactance 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 Resistor no imaginary part: reactance = 0 Capacitor no real part: resistance = 0 no real part: resistance = 0 Inductor

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 R C 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) = XC = -1/(wC) R L 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) = XL = wL R-L in series L C 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) = XLC = wL -1/(wC) L-C in series

RC circuit example Find the current amplitude in the circuit at the signal frequency f = 1 GHz 1. Complex source voltage amplitude, VS= 9 V 2. Angular frequency, w = 2pf =2p*1E9 3. Total circuit impedance: v(t)= 9*cos(wt) V C 3 pF R 50 Ohm The reactance X = -1/(2p*1E9*3e-12) = - 53.05 Ohm; ZT = 50 – j50.35 Ohm 4. Complex current amplitude in the circuit: I = VS / ZT I = 0.0894 + 0.0900i 5. Complex current amplitude can be found if we present the complex number describing I in the Euler’s form: I = IM ejF IM = 0.1268; F = 0.79 rad

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= 4 2 - j2 is a complex number with a real part =2, imaginary part= -2. -4i is a complex number with a real part = 0, imaginary part = -4 5 is a complex number with a real part = 5, imaginary part =0

Phasors and complex numbers f VM x y f R x0 y0 A pair of two parameters: the radius R and the angle f 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: x0 and y0 The pair (R,j) or (x0, y0) is called a complex number Phasor representing complex amplitude (VM L f ) is a combination of two parameters: the amplitude VM (the length of the arrow) and the phase angle f . This pair of two parameters fully describes the AC waveform of a given angular frequency w

N = x + jy, x = Re(N); y = Im(N); Complex numbers Complex numbers provide a convenient technique to describe the phasors related to AC voltages and currents. N = x + jy, x = Re(N); y = Im(N); 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 j2 = -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. The real number, the square of which is equal to (-1) does not exist. Therefore, j is not a real number.

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. jy |R| Complex number N represented by the position of the dot on x-y plane jy1 x x1

Complex number N represented by the position of the dot on x-y plane Complex numbers The angle that the vector makes with the axis x is called an argument or phase angle |R| jy Complex number N represented by the position of the dot on x-y plane jy1 f x x1 x1 = |R| cos (f); y1 = |R| sin (f)

To summarize, any complex number can be described in two ways: Complex numbers To summarize, any complex number can be described in two ways: x Complex number N x1 jy1 jy f N = x1 + j y1 (Algebraic, or Cartesian form) N = |R| L f (Phasor, or Polar form )

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 i2 or (j2) 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

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 i2 or (j2) replace it with (-1) z1 = a + bi and z2 = c + di; multiplication: z1z2 = (a + bi)(c + di) = ac + adi + bci + bdi2; i2 = -1, therefore: z1z2 = (a + bi)(c + di) = ac + adi + bci - bd z1z2=ac - bd + (ad + bc)i

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 i2 or (j2) replace it with (-1) z1 = a + bi and z2 = c + di; division

The algebra of Complex numbers: Polar –> Cartesian form conversion Complex number N x1 jy1 jy f N = x1 + j y1 (Algebraic, or Cartesian form) |N| f (Phasor, or Polar form ) Re (N) = x = |N|*cos(f); Im (N) = y = |N|*sin(f);

The algebra of Complex numbers: Cartesian -> Polar form conversion Complex number N x1 jy1 jy f N = x1 + j y1 (Algebraic, or Cartesian form) |N| f (Phasor, or Polar form ) tan(f) = y/x = Im(N)/Re(N), or: f = arctan (y/x) = arctan [Im(N)/Re(N)]

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

MATLAB code Comments 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) 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

Figure 1 Figure 2