ISC0100

CYBERELECTRONICS

Fall 2018

The 3rd lecture

Martin Jaanus NRG-308

martin.jaanus@ttu.ee 56 91 31 93

Learning environment : http://isc.ttu.ee

Materials : http://isc.ttu.ee/martin

Topics

1. Voltage, current

2. DC, AC

3. Values, calculations

4. Capacitor, Inductor

Alternative current

Voltage, electric potential difference, electric pressure or

electric tension (formally denoted ∆V or ∆U, but more often

simply as V or U, for instance in the context of Ohm's or

Kirchhoff's laws) is the difference in electric potential energy

between two points per unit electric charge. The voltage

between two points is equal to the work done per unit of

charge against a static electric field to move the test charge

between two points and is measured in units of volts (a joule

per coulomb).

The volt (symbol: V) is the derived unit for electric potential, electric potential

difference (voltage), and electromotive force. The volt is named in honour of the

Italian physicist Alessandro Volta (1745–1827), who invented the voltaic pile,

possibly the first chemical battery.

.

Voltage

An electric current is a flow of electric charge. In electric

circuits this charge is often carried by moving electrons in a

wire. It can also be carried by ions in an electrolyte, or by both

ions and electrons such as in a plasma.

The SI unit for measuring an electric current is the ampere

(A), which is the flow of electric charge across a surface at the

rate of one coulomb per second.

Current

The electrical resistance of an electrical conductor is a

measure of the difficulty to pass an electric current through

that conductor. The inverse quantity is electrical conductance,

and is the ease with which an electric current passes.

Electrical resistance shares some conceptual parallels with

the notion of mechanical friction. The SI unit of electrical

resistance is the ohm (Ω), while electrical conductance is

measured in siemens (S).

An object of uniform cross section has a resistance

proportional to its resistivity and length and inversely

proportional to its cross-sectional area. All materials show

some resistance, except for superconductors, which have a

resistance of zero.

Electrical resistance and conductance

DC, Direct current Direct current (DC) is the unidirectional

flow of electric charge. It is not changing during observation.

0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,5 1 1,5 2

V

t

0

0,5

1

1,5

2

2,5

3

3,5

4

0 0,5 1 1,5 2

V

t

Direct Current

AC, Alternative current is an electric current in which the flow

of electric charge (periodically) reverses direction. Different

signal values.

-350

-250

-150

-50

50

150

250

350

0 0,5 1 1,5 2

V

t -20

-15

-10

-5

0

5

10

15

20

0 0,5 1 1,5 2

V

t

Periodic Non periodic

Alternative Current)

Alternative Current

How AC can be produced.

Image :wikipedia Image :circuitlab.com

Alternative Current - values

-350

-250

-150

-50

50

150

250

350

0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2

V

t

amplituded (A)

Max (V)

amplitude (A)

Min (V)

p-p

Perod (T)

Periodic signal, amplitude, peak to peak, period

Example:

u(t)=A*sin(2πt+φ)

Signal has several values

• Amplitude

• Peak value (max,min)

• Mean value:

• Absolute mean value

• rms,

root mean square value

𝑈𝑚𝑘 =1

𝑇න0

𝑇

|𝑢 𝑡 |𝑑𝑡

𝑈0 =1

𝑇න0

𝑇

𝑢 𝑡 𝑑𝑡

𝑈𝑚𝑘 =1

𝑇න0

𝑇

𝑢2 𝑡 𝑑𝑡

frequency f=1/T , unit hertz (Hz)

Alternative Current - RMS

𝑈𝑚𝑘 =1

𝑇න0

𝑇

𝑢2 𝑡 𝑑𝑡Effective value

Root mean square (RMS) amplitude is used especially in electrical engineering:

the RMS is defined as the square root of the mean over time of the square of the

vertical distance of the graph from the rest statei.., the RMS of the AC waveform

(with no DC component).

RMS value equals with DC value whitch has the same energy..

RMS value describes a power of signal .

The power is proportional with root of the signal..

All measurement devices (meters) are showing RMS value.But actually most of them are showing absolute mean value !!!!

If we use RMS values, the formulas are same than in DC values..

Alternative Current - measurement

How AC can measured

Scale is graduated using RMS

Value. Real difference is 1,11x

Correct result only when

Measuring sine wave

𝑈𝑚𝑘 =1

𝑇න0

𝑇

|𝑢 𝑡 |𝑑𝑡

Below 1V unlinear !

Recifier Filter Amplifier

Alternative Current - true RMS

True RMS multimeter

x2 𝑋 V

It is expencive realization in analogue electronics (toot and square root)

In digital meters, real time sampling is required.

All measurement devices (meters) are showing RMS value.

But actually most of them are showing absolute mean value !!!!

𝑈𝑚𝑘 =1

𝑇න0

𝑇

𝑢2 𝑡 𝑑𝑡

Input: 2 V sine

Alternative Current - measurement

Input: 6 V sine

Alternative Current - measurement

Input: 6 V sine ???

Keskväärtus on 0

Alternative Current - measurement

Input: 6 V triangle

Alternative Current - measurement

Input: 6 V rectangular

Alternative Current - measurement

Alternative Current - calculations

Calculation (look also waveform.pdf)

A

t

Part of signal mean value RMS value

A

t

A

t

B

𝐴 ∗ 𝑡 𝐴2 ∗ 𝑡

𝐴 + 𝐵

2∗ 𝑡

𝐴2 + 𝐴𝐵 + 𝐵2

3∗ 𝑡

𝐴2

π∗ 𝑡

𝐴2

2∗ 𝑡

Mean value

Xk=𝑋1+𝑋2+𝑋…𝑡1+𝑡2+𝑡…

RMS value

Xrms=𝑋1+𝑋2+𝑋…𝑡1+𝑡2+𝑡…

Alternative Current - calculations

Example :

RMS

V=25∗02+25∗

42

2+25∗

(−4)2

2+25∗02

100=

02+42

2+(−4)2

2+02

4= 2 𝑉

AVG

V=25∗0+25∗4∗

2

π+25∗ −4 ∗

2

π+25∗0

100=

0+4∗2

π+ −4 ∗

2

π+0

4= 0 V

Alternative Current - calculations

Example :

RMS

V=25∗

42+4∗ −1 + −1 2

3+25∗ −1 2+25∗

−1 2+ −1 ∗0+ 0 2

3+25∗

02+0∗4+42

3

100=1.66 𝑉

AVG

V=25∗

4+ −1

2+25∗ −1 +25∗

−1 +0

2+25∗

0+4

2

100= 0.5 V

Alternative Current - calculations

Example :

RMS V=3262

2+

1832

2=264.3𝑉

Periods are different !

Capacitor

Capacitor prysical quantity - capacitance C, unit farad (F)

Energy is tored into electrical field 𝐸 =𝐶𝑉2

2

Integrates current , voltage misses current a quater period.

Conductance reactive 𝑌𝑐 = 𝑗ω𝐶 𝑍𝑐 =1

𝑗ω𝐶ω = 2π𝑓 ω−angular frequency(rad/s)

In case of series or parallel connection , use conducance formulas !!!

Concuctance is proportional with frequency.

Capacitor•Capacitors:

• state variable: voltage

• Fundamental circuit equation:

• The capacitance gives an indication of electric inertia. Compare the

above equation with Newton’s

• Capacitors will tend to hold its voltage fixed.

• For a finite current with an infinite capacitance, the voltage must be

constant. Hence, capacitors tend to behave like voltage sources

(the larger the capacitance, the closer they resemble a voltage source)

• A capacitor’s energy is

CC

dvi C

dt

dvF m

dt

21

2CW Cv

Inductor

Physical quantity – inductivity L, unit henry (H)

Energy is stored in magnetic field 𝐸 =𝐿𝐼2

2

Integrate voltage , current misses a quater period .

Resistance is reactive ZL= 𝑗ω𝐿 𝑌𝐿 =1

𝑗ω𝐿ω = 2π𝑓 ω− angular freq.(rad/s)

In parallel or series connection use resistance formulas.

Resistance is proportional with frequency.

Inductor• Inductors

• state variable: current

• Fundamental circuit equation:

• The inductance gives an indication of electric inertia. Inductors will

tend to hold its current fixed.

• Any attempt to change the current in an inductor will be answered with

an opposing voltage by the inductor. If the current tends to drop, the

voltage generated will tend to act as an electromotive force. If the

current tends to increase, the voltage across the inductor will drop, like

a resistance.

• For a finite voltage with an infinite inductance, the current must be

constant. Hence, inductors tend to behave like current sources (the

larger the inductance, the closer they resemble a current source)

• An inductor’s energy is

LL

div L

dt

21

2LW Li

Inductors and capacitors..

• …are present even if we do not want it !!.

• Also the resistance

• Parasite (invisible) (components)

• In the rise of frequency , will apply parasite L and C

If you are designig electronics you must accept it !!!

You buy this:

But you get this:

Since capacitors behave like constant voltage sources you shall never connect a

switch in parallel with a capacitor. Any attempt to violate this load will lead to high

currents. Likewise, you shall never connect a switch in series with an inductor. Any

attempt to violate this rule will lead to high voltages.

Inductors and capacitors

500kV Substation arc.

Screenshot from video https://www.youtube.com/watch?v=hIkNY5xjy5k

R, L, C..

Low-pass filter

High-pass filter

𝐹𝐶=1

2𝜋𝑅𝐶

𝐹𝐶=1

2𝜋𝑅𝐶

RC- time constant

R, L, C..

Images:wikipedia

Resonance

Resonance is a condition in an RLC circuit in which the capacitive and reactive

reactance are equal in magnitude, thereby resulting in a purely resistive impedance.

Resonance circuits are useful for constructing filters and used in many application.

Series resonance

Series resonance

Parallel resonance

At resonance, the impedance consists only

conductance G.

The value of current will be minimum since the total

admittance is minimum.

The voltage and current are in phase.

RC transfer calulation

L

+

–R V

+

R=600Ω

L=300mH

=2krad/s

Find transfer ratio in dB

1. Find the transfer function.

Using Ohm’s law

I=V/Z , where Z total resistance of

inductor and resistor Z=R+jωL. Voltage, applied to resistor

equals

Vr=I*R . We get

Vr/V=R/(R+jωL) or K=R

R+jωL. Since we are interested in amplitude,

We need length of the vector

Kv=20log (R

R2+(ωL)2

)

RC transfer calulation 1

L

+

–R V

+

R=600Ω

L=300mH

=2krad/s

Kv=20log (R

R2+(ωL)2

)

Kv=20log (600

6002+6002

) =20log(0.707)=-3dB

RC transfer calulation 1

L

+

–C V

+

C=1nF

L=400mH

=49krad/s

Kv=20log 1

1-490002*400*10-3*1*10-9=20log(1/0.0396)=20log(25)=27.9dB

Transfer function:

K=1/j ωC

jωL+1/jωC=

=1

j (ωL-1/ωC) * jC=

1

1- 2LC

j*j=-1 1/j=-j

RC transfer calulation 2