g16.4427 practical mri 1 – 26 th march 2015 g16.4427 practical mri 1 review of circuits and...

G16.4427 Practical MRI 1 – 26th March 2015

G16.4427 Practical MRI 1

Review of Circuits and Electronics


Current• Current is the flow of electrical charge through an

electronic circuit– The direction of a current is opposite to the direction of

electron flow

• Current is measured in Amperes (amps)– 1 A = 1 C/s

French physicist and mathematician

20th January 1775 - 10th June 1836André-Marie Ampère


Voltage• Voltage, or electric potential difference, is the

electrical force that causes current to flow in a circuit

• Voltage is measured in Volts (V)– One volt is the difference in electric potential across a

wire when an electric current of one ampere dissipates one watt of power: 1 V = 1 W/A

Italian physicist, inventor of the battery

28th February 1745 - 5th March 1827Alessandro Volta


Resistance• The electric resistance is the opposition to the

passage of an electric current through an element

• Resistance is measured in Ohms (Ω)– One ohm is the resistance between two points of a

conductor when a constant potential difference of one volt produces a current of 1 ampere: 1 Ω = 1 V/A

German physicist

26th March 1789 - 6th July 1854Georg Simon Ohm


Ohm’s Law• Defined the relationship between voltage, current

and resistance in an electric circuit• It states that the current in a resistor varies in

direct proportion to the voltage applied and it is inversely proportional to the resistor’s value

V

I R


Kirchhoff’s Laws• Kirchhoff’s voltage law (KVL)– The algebraic sum of the voltages around any closed

path (electric circuit) equal to zero

• Kirchhoff’s current law (KCL)– The algebraic sum of the currents entering a node

equal to zero

German physicist

12th March 1824 - 17th October 1887Gustav Kirchhoff


Kirchhoff’s Voltage Law (KVL)

+

+

+

+

_

_

_

_

v1

v2

v4

v3

•

v3 + v4 – v2 – v1 = 0


Kirchhoff’s Current Law (KCL)

i1

i2 i3

i2 + i3 – i1 = 0


Problem

Use Kirchhoff's Voltage Law to calculate the magnitude and polarity of the voltage across resistor R4 in this resistor network


Problem

Use Kirchhoff's Voltage Law to calculate the magnitudes and directions of currents through all resistors in this circuit


Inductor• The energy stored in magnetic fields has effects on

voltage and current. We use the inductor component to model these effects

• An inductor is a passive element designed to store energy in the magnetic field


Physical Meaning• When the current through an inductor is a

constant, then the voltage across the inductor is zero, same as a short circuit

• No abrupt change of the current through an inductor is possible except an infinite voltage across the inductor is applied

• The inductor can be used to generate a high voltage, for example, used as an igniting element


Inductance• The ability of an inductor to store energy in a magnetic

field

• Inductance is measured in Henries (H)– If the rate of change of current in a circuit is one ampere per

second and the resulting electromotive force is one volt, then the inductance of the circuit is one henry: 1 H = 1 Vs/A

American scientist, first secretary of the

Smithsonian Institution

17th December 1797 - 13th May 1878Joseph Henry


How Inductors Are Made• An inductor is made of a coil of conducting wire

μ = μrμ0

μ0 = 4π × 10-7 (H/m)


Energy Stored in an Inductor

power

Energy stored inan inductor


Capacitor• The energy stored in electric fields has effects on

voltage and current. We use the capacitor component to model these effects

• A capacitor is a passive element designed to store energy in the electric field


Physical Meaning• A constant voltage across a capacitor creates

no current through the capacitor, the capacitor in this case is the same as an open circuit

• If the voltage is abruptly changed, then the current will have an infinite value that is practically impossible. Hence, a capacitor is impossible to have an abrupt change in its voltage except an infinite current is applied


Capacitance• The ability of a capacitor to store energy in an

electric field

• Capacitance is measured in Farad (F)– A farad is the charge in coulombs which a capacitor

will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second: 1 F = 1 As/V

British scientist, Chemist, physicist and

philosopher

22nd September 1791 - 25th August 1867Michael Faraday


How Capacitors Are Made• A capacitor consists of two conducting plates separated by an

insulator (or dielectric)

ε = εrε0

ε0 = 8.854 × 10-12 (F/m)


Energy Stored in a Capacitor

power

Energy stored inan inductor


Resonance in Electric Circuits• Any passive electric circuit will resonate if it has an

inductor and capacitor• Resonance is characterized by the input voltage and

current being in phase– The driving point impedance (or admittance) is completely real

when this condition exists

“RLC Circuit”


Series Resonance

• The input impedance is given by:

• The magnitude of the circuit current is:


Resonant Frequency

• Resonance occurs when the impedance is real:

• We define the Q (quality factor) of the circuit as:

• Q is the peak energy stored in the circuit divided by the average energy dissipated per cycle at resonance– Low Q circuits are damped and lossy– High Q circuits are underdamped

“Resonant Frequency”


Parallel Resonance

• The relation between the current and the voltage is:

• Same equations as series resonance with the substitutions: – R 1/R, L C, C L:


Problem

Determine the resonant frequency of the RLC circuit above


Transmission Lines• Fundamental component of any RF system– Allow signal propagation and power transfer

between scanner RF components• All lines have a characteristic impedance (V/I)– RF design for MRI almost always use Z0 = 50 Ω

• Input and output of transmission lines have a phase difference corresponding to the time it takes wave to go from one end to the other

• Length is usually given with respect to λ


Geometry


Transmission Line Reflections

• A wave generated by an RF source is traveling down a transmission line

• The termination impedance (Zload) may be a resistor, RF coil, preamplifier or another transmission line

• In general there will be reflected and transmitted waves at the load

Z0


Circuit Model

• Small sections of the line can be approximated with a series inductor and a shunt capacitors

• The transmission line is approximated as a series of these basic elements


Reflection and Transmission

Forward wave (forward power):

Reflected wave (reflected power):

Transmitted wave (transmitted power):

reflection coefficient

transmission coefficient

S11

S21


High Γ in High Power Applications• Decreases the power transfer to load

consequently causing loss of expensive RF power

• Increases line loss: 3 dB power loss can increase to > 9 dB with a severely mismatched load

• Causes standing waves and increased voltage or current at specific locations along the transmission line


Useful Facts to Remember• If Zload = Z0 then there is no reflected wave• If the length of the line is λ/4 (or odd multiples)– Short one end, open other end– Can be considered a resonant structure with high

current at shorted end and high voltage at open end• If the length of the line is λ/2 (or multiples)– Same impedance at both ends– With open at both ends this can also considered a

resonant structure with high current at center and high voltage at ends


Impedance Transformation• Given the importance of reflections, it is

generally desirable to match a given device to the characteristic impedance of the cable

• Can use broadband or narrowband matching circuits– Most MRI systems operate over a limited

bandwidth, so narrowband matching works fine for passive devices such as RF coils

• There are many circuits that can be used for impedance transformation


The Smith Chart• A useful tool for analyzing transmission lines

reflection coefficient at the load

Input impedance of a line of length d, with Z0 and Zload

Smith Chart is the polar plot of Γ with circles of constant r and x overlaid

reflection coefficient at distance d from the load

On the Smith Chart we can convert Γ to Z (or the reverse) by graphic inspection


Smith Chart InterpretationCircles correspond to constant r• the centers are always on the horizontal axis (i.e. real part of the reflection coefficient)

Partial circles correspond to constant x

The intersection of an r circle and an x circle specifies the normalized impedance

The distance between such point and the center of chart is the reflection coefficient (real + imaginary). Any point on the line can be on the circle with such radius


Smith Chart Interpretation

The termination is perfectly matched (i.e. reflection coefficient is zero) for a point at the center of the Smith Chart (i.e. r = 1, x = j0 and radius of the circle = 0)

Question: which point correspond to the termination being an open circuit?

Answer: The right most point on the x-axis, which corresponds to infinite z and Γ = 1. The left most point corresponds to z = 0 and reflection coefficient Γ = –1.


Smith Chart Example 1Locate these normalized impedances on the simplified Smith Chart:

• z = 1 + j0

• z = 0.5 - j0.5

• z = 0 + j0

• z = 0 - j1

• z = 1 + j2

• z = ∞


Smith Chart Example 2Graphically find the admittance corresponding to the impedance:

• z = 0.5 + j0.5

In fact: y = 1/(0.5 + j0.5) = 1 – j1

1. Locate the impedance2. Draw a circle centered at

the center of the Smith Chart and passing through the impedance

3. Plot a straight line through the impedance and the center of the Smith Chart

4. The intersection of the line with the circle yields the value for the admittance


Any questions?


See you next lecture!

g16.4427 practical mri 1 – 26 th march 2015 g16.4427 practical mri 1 review of circuits and...

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