1. 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

2. 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

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

4. 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

5. 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

6. 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) = Ohm;
ZT = 50 – j50.35 Ohm
4. Complex current amplitude in the circuit: I = VS / ZT
I = i
5. Complex current amplitude can be found if we present the complex number describing I in the Euler’s form: I = IM ejF IM = ; F = 0.79 rad

7. 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

8. Phasors and complex numbersf 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

9. 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.

10. 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

11. 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)

12. 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 )

13. 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

14. 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

15. 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

16. 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);

17. 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)]

18. 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

19. 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

20. Figure 1
Figure 2