Circuit Analysis and Power Electronics

Question 1

Decibel described in terms of Vout and Vin

Answer 1

Question 2

What does the plot of the magnitude response as a function of frequency look like?

Answer 2

Question 3

What dictates the linearity of a circuit?

Answer 3

• An input signal v₁ gives an output signal v_1out = Av₁
• an input signal v₂ gives an output signal v_2out = A_v2
• an input signal (v₁ + v₂) gives an output signal v_1out + v_2out = A(v₁ + v₂)
• So, for a sinusoidal input voltage of frequency f, any output of the circuit is also sinusoidal with frequency f. Note that the output need not be in phase with the input.

Question 4

Active vs Passive circuits

Answer 4

Active circuits include devices that draw power from a power supply.

Passive circuits don’t draw extra power.

Question 5

What is the best representation to add and subtract impedances?

Answer 5

cartesian coordinates

Question 6

what is the best representation to multiply and divide?

Answer 6

Polar coordinates

Question 7

Finding Polar from Cartesian

Answer 7

Question 8

What is the instantaneous power of a resistor? (what does plot look like )

Answer 8

For the resistor the +ve instantaneous power p(t) corresponds to energy being transferred to the resistor – the average power P =

averaged over a cycle is non- zero as shown on the previous slide. This net transfer of energy is that dissipated in the resistor per cycle, i.e. lost as heat into the surroundings.

Question 9

What does the instantaneous power of an inductor look like? (relationship between current and voltage)

Answer 9

Voltage leads current by pi/2.

Average power is zero.

Question 10

What does the instantaneous power of a capacitor look like?

Answer 10

Voltage lags Current by pi/2

Average power is zero.

Question 11

Impedance and Admittance of R L C components.

Answer 11

Question 12

To calculate the average power dissipated in a resistor over a cycle what equation would you use.

Answer 12

Question 13

Mathematical expression for mean value value of a waveform v(t) over an interval T

Answer 13

Question 14

How would you calculate the RMS of a waveform?

Answer 14

To calculate the root-mean-squared:

First, square the function.

Then, calculate the mean of the square.

Lastly, take the square root.

Question 15

For sinusoidal signals, what is a quick way of determining the RMS?

Can this be used for other sorts of waveforms?

Answer 15

Amplitude divided by the square root of 2 gives RMS.

This cannot be used for other signals. Important that you only use this on sinusoidal signals! Other waveforms will need to have their RMS through the longer procedure.

Question 16

Calculate RMS for this signal

Answer 16

Question 17

What is the power factor?

Answer 17

cos(ϕ) is the power factor.

If loads in a circuit are purely resistive, cos(ϕ)=1.

If cos(ϕ) = 0 (i.e. ϕ = ±90), then purely reactive loads.

Question 18

For a driven LCR circuit, how does the phase angle change across a range of frequencies?

Why is this?

Answer 18

The phase is negative for frequencies lower than the resonance, this is because the voltage across the inductor becomes small (for lower frequencies) and the capacitor dominates. Since the current leads the voltage in a capacitor, the resultant phase is negative.

It is the opposite for positive phase angle… The inductor dominates at higher frequencies and with an inductor the voltage leads the current.

Question 19

The equation relating R_in and R_out of a transformer.

What generalisation can you make for the load?

Answer 19

Can be generalized for any complex load ZL. The impedance seen on the primary side is that of the load multiplied by the square of the turns ratio*.

*Note carefully here that the turns ratio is usually defined as Np:Ns not Ns:Np

Question 20

What is something that all cables have at high frequencies?

Answer 20

At high freq all cables have characteristic RF impedance Z₀. It is independent of the cable length and is real(resistive).

Again, it is only seen by high-frequency RF signals and it isn’t the resistance of the wire nor is it the impedance you measure at lower frequencies.

Question 21

Mutual inductance: what does M depend on?

Answer 21

M will not depend on I₁ (which will cancel) but only on geometric factors such as the dimensions, separation, relative orientation, number of turns (of both coils) and the relative permeability (µ_r) of the medium between the coils.

Question 22

How do you calculate the normalized coupling parameter?

Answer 22

Question 23

Why do the coils in a transformer have a k ~0.99?

Answer 23

Coils on a transformer have a k ~0.99 because it is wound a closed ferromagnetic core with high relative permeability mu_r > 100. Acts as a magnetic circuit, all the flux generated by the first coil is concentrated inside the core with only a very small fraction leaking out.

Question 24

What are the major applications of transformers?

Answer 24

1. To step up (increase) or step down (decrease) the amplitude of ac voltages, e.g. in power transmission or in home electronics
2. To match the impedance of electrical components to maximize power transfer for lower frequency signals (or minimize signal reflection for rf signals).
3. To isolate circuits from one another for safety or other issues.
4. To couple resonant circuits together - e.g. radio detection

Question 25

Why the N_s

Turns of secondary

Answer 25

The reason is that in faradays law the emf is related to the change of flux-linkage rather than just “flux”

Question 26

emf related to the turns ratio

Answer 26

Question 27

How can you change the sign of the secondary coil voltage output with respect to the primary?

Answer 27

Changing out the coils are wound affects the sign.

Question 28

What does the dot notation represent in terms of transformers?

Answer 28

The polarity of the voltage is the same at the dots.

With a load, if the primary current is going in at the primary dot, then it is coming out at the secondary dot.

Question 29

What is the net effect of adding a load to the secondary outputs of a transformer?

Answer 29

Primary current increases.

When we add the load the increase in primary current means that more energy is now being drawn from the source and this is the energy that is being dissipated in the load.

Question 30

When you compare this to the voltage turns ratio, what does it tell you?

Answer 30

It tells you that if the transformer steps up the voltage to the load, the current must step down by the same ratio (vice-versa). The transformer cannot supply more power than is supplied by the source.

Question 31

What is the equation that relates R_in to R_L of a transformer?

Answer 31

*Note*

Here the turns ratio is flipped compared to the voltage/turns ratio equation.

So remember that for this equation the primary is the numerator and secondary denominator.

Question 32

What does decreasing the secondary coils relative to the primary do?

Answer 32

It increases the R_Ltrans (R_Ltrans is equivalent R_in)

Question 33

At high frequencies what are all cables characterized by?

Answer 33

By characteristic impedance Z₀

Which relates the voltage drop dV and the current I along an infinitesimal length dl.

Note: the characteristic impedance is only something that is seen by high frequency rf signals.

Question 34

How does the autotransformer use less copper than a regular transformer?

And what is one of its disadvantages?

Answer 34

It prevents you from having to use thicker wires when having to transfer larger current due to a step up.

Question 35

What are the 4 types of losses?

Answer 35

1. Copper losses (ohmic losses) - currents flowing in the electrical resistance of the wire making up the windings of the transformer result in losses. Since windings are usually made out of copper, they are called “copper losses”.
   
   1. Copper losses in primary = i_p²(rms) x R_p
   2. Copper losses in secondary = i_s² (rms) x Rs
2. Imperfect Coupling - imperfect magnetic coupling. These losses can be minimized by winding the transformer on a ferromagnetic core.
3. Eddy currents - When the flux in the transformer core changes as a function of time, a current will be induced in the core itself if it is made of a conducting material and hence there will be losses in the resistance of the core.
   
   1. The usual way of overcoming eddy currents is to make the core out of thin sheets of ferromagnetic material insulated from each other. The laminations are oriented to break up the eddy current flow and minimize the dissipation.
4. Hysteresis - magnetism in ferromagnetic materials is attributable to the orientation of tiny magnetic domains, whose magnetic fields align with an externally applied magnetic field.
   
   1. Magnetic domains have a certain inertia to this change, energy is required to continually cause them to reverse, commonly known as hysteresis loss.
   2. The term hysteresis comes from a loop formed when you plot the magnetization of a ferromagnetic material versus the applied field. The resulting area enclosed by the loop is the work that needs to be done.

Sum of the eddy current and hysteresis losses is collectively known as the core loss.

Question 36

How do we represent the effect of leakage flux?

Answer 36

L_p and L_s are the total inductances of primary and secondary respectively.

L_pm refers to the mutual flux

L_pl refers to the leakage flux

Same for secondary.

Question 37

What are the high/ low-frequency behaviors of real transformers?

Answer 37

At very high frequencies capacitative effects become important.

At high frequencies the ‘capacitors’ tend to acts as shorts. At low frequencies the primary inductance L_pm tends to short the load.

Question 38

What does the frequency response of a real transformer look like?

Answer 38

Remember, that the impedance of an inductor is jw L and the impedance of a capacitor is 1/jwC.

Question 39

What are the stages in the analysis of real transformers?

Answer 39

Question 40

How do you determine the emf of a loop rotating in a magnetic field?

Answer 40

Question 41

Whats the difference between phase and line voltage?

Answer 41

Phase voltage is between any phase (live wire) and neutral

Line voltage is between any two phases (two live wires)

Question 42

What generator is this?

Answer 42

Four wire ‘Star’ connected generator

Has a neutral wire.

So it has a phase and line voltage.

Question 43

What generator is this ?

What ‘voltages’ does it have

Answer 43

Three-wire delta configuration

Name comes from triangular shape.

The delta doesn’t have neutral wire.

So only line voltages.

Question 44

How do you find the voltage across two components that are out of phase with each other?

What about currents?

Answer 44

Determined using phasor diagram

They need to be determined using a phasor diagram.

Question 45

Why use three-phase electricity?

Answer 45

1. Can deliver more kVa (power) than single-phase or DC using the same total amount of conductor (copper).
   
   1. For a given power requirement can use much less material (thinner wire) ⇒ cheaper because less material is used and less maintenance costs. Lighter lines are easier to install, supporting structures not as big.
2. Lower transmission losses over long distances compared to single phase.
3. 3-phase loads like 3-phase induction motors have several advantages:
   
   1. doesn’t need slip rings or require heavy insulated coils to turn.
   2. runs more smoothly than single phase - single phase falls to 0 every cycle, for 3 phase it is constant.
   3. They have self-starting properties
4. Flexibility/Ease of supply and distribution. You can either use the large line voltage between 2 phases in the star arrangement or each phase on its own e.g. to houses in a street.

Question 46

What is the relationship between phase (load resistance) and line voltage for star configuration?

And current ?

Answer 46

V_ph * √(3) = V_L

or

V_ph = V_L/ √(3)

The current through each load resistance is the same as the line current

Question 47

What is the current and voltage relationship between the load resistance and the line voltage for a balanced delta load?

Answer 47

The voltage across each load resistance is the same as the line voltage.

The current through each connecting line is √3 times the load current.

Question 48

What is the relationship between load currents in a balanced delta load configuration?

How do you find the line current?

Answer 48

they are 120 degrees out of phase

line current is found to be the vector difference of the two load currents

Question 49

Explain why a 3 phase star configuration has less energy loss than single phase.

Answer 49

Comes down to the fact that the current is spread between three wires compared to 1 wire for one phase.

From P = I²R you can see that multiple I²R_T by 3 will lead to a lower transmission loss than (3I)²R_T .

Question 50

How do you draw the equivalent circuit for a real transformer?

Answer 50

Question 51

Kirchoffs 1st law

Answer 51

At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

Question 52

kirchoffs 2nd law

Answer 52

The algebraic sum of all the voltages around any closed loop in a circuit is zero