H8P1

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 H8P1: Charging an Inductor

Consider the following circuit:

Let

R=220.0Ω

and

L=0.1

H. Suppose time is measured in milliseconds. Starting at time

t=−5

ms and continuing, we know that

V(t)=V

0

+Λδ(t)

V, where

V

0

=4.0

V and

Λ=0.007

Webers.

Also, assume that we measured

i

L

(−1)=−0.01

A.

What is the current, in Amperes, through the inductor at

t=0

−, a time infinitesimally before the impulse is delivered? Hint: The system has not yet reached steady state. 0.015 What is the current, in Amperes, through the inductor at

t=0

+, a time infinitesimally after the impulse is delivered? 0.085 What is the current, in Amperes, through the inductor at

t=1

ms, after the impulse is delivered? 0.025

 H8P2: Physiological Model

The electrical circuit language is a powerful language for modeling other kinds of systems, such as mechanical, chemical, thermal, and fluidic systems. Here is an example from physiology: drug delivery.

Suppose we want to understand what happens when a medical doctor injects a drug into an person. For example, the doctor may need to know how the concentration of an antibiotic will vary at a target site with time. To be effective, the concentration must be high enough for long enough to kill the invading bacteria, but not so high as to damage healthy tissues.

We can model the amount of the drug by charge and the concentration of the drug by voltage. Thus, we get a very simple model:

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I

models the rate of injection of the drug into the body.

v

C models the concentration of the drug in the body.

1/R

models the rate of elimination of the drug.

C

models the size of the body.

Of course, this model is very crude, but it can be elaborated. With more circuitry we could model the concentrations in the blood, the elimination in the kidneys, the concentrations in various organs, the degradation of the drug in the liver, the elimination of the degraded products by the kidneys, etc. So this can be very complicated, for another time!

If the time constants for these processes are on the order of hours, and the injection takes tens of seconds, it is apparent that the exact profile of the rate of injection does not matter. If the doctor takes 5 seconds or 10 seconds to inject the same amount of antibiotic, the profile of

concentration in the blood should be about the same for times on the order of hours. Thus, an injection can be modeled as an impulse. (This is just a mental experiment: most antibiotics cannot be injected this way, they can be given intravenously, over a period of an hour, thus making a pulse rather than an impulse.)

So, let's do some numbers. Consider the consequences of injecting

Q=400.0

mg of cipro into a

C=70

kg person. Assume

The person is entirely made of water, which weighs 1kg/L.

By the end of the injection the antibiotic is uniformly distributed in all of the body tissues. What is the initial concentration after the injection

v

C

(0

+

)

, of the antibiotic in the body, in mg/L? 400/70

The half-life of the antibiotic in the person is about

t

h

=4.0

Hours. Thus, the concentration will drop to half of its initial value in that time. What is the resistance, in seconds/liter, in our model circuit, that will produce this rate of loss of concentration? 296.78

If the doctor wants to keep the concentration between 1/4 and 2/3 of the initial concentration, what is longest time, in hours, that may be allowed to elapse before a new injection is required?

8

What dose, in milligrams, will be required at that time to get the concentration back to 2/3 of the initial concentration? 166.67

Note that now the doctor can repeat this calculation to determine how often he must give a this new dose to maintain the concentrations in the range he believes will be effective.

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 H8P3: Memory

--This problem's grader was previously accepting incorrect answers. It has been fixed. If you resubmit, you will use the newer, more correct grader to check your response --

One possible implementation of one bit of a digital memory might be something like the circuit below. The memory element here is the gate-to-source capacitance of transistor

Q

2. Since the capacitance is leaky, this can hold a value only for a time determined by the time constant of the capacitance and the leakage resistance. Thus, such a memory has to be "refreshed" periodically, by reading out its value and rewriting that value.

We assume an "ancient"

1μ

m technology satisfying the static discipline:

V

S

=5.0

V,

V

OH

=3.5

V,

V

IH

=3.0

V,

V

IL

=0.9

V,

V

OL

=0.5

V The gate-source capacitance

C

GS

=3.5

fF and

V

T

=1.0

V.

R

ON

=1800.0Ω

,

R

OFF

=95.0

M

Ω

and

R

PU

=10.0

k

Ω

.

The specifications for this logic family say that we have to refresh this memory every 64ms. So, if we charge the gate capacitance to

V

OH it will take more than 64ms to decay to

V

IH. If it takes

0.1

s to decay that far, what must be the parasitic resistance, in TerraOhms from the gate to the source of

Q

2? Note: (1 T

Ω=10

12

Ω

). 185.34

On a chip a MOSFET is physically symmetrical: the source and the drain are interchangeable. For the purposes of the formulae that describe an n-channel MOSFET the source is just the electrode with the lowest potential and the drain is the one with the highest potential. (That is, the voltage from the drain to the source

v

DS is positive.) For

Q

1 and

Q

2 this is not an issue, but for

Q

3 sometimes we are charging the gate of

Q

2, to store a "0", and sometimes we are

discharging it, to store a "1". (Note that there is an inversion on input and output.)

When

D

IN is low

Q

1 turns off and its drain goes high. What is the maximum value of voltage on the drain of

Q

1, in Volts? 5

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the drain of

Q

1. What is the maximum voltage, in Volts, that the gate of

Q

2 can be charged to? Note, this value must be larger than

V

OH to satisfy the static discipline. 3.8

The drain of

Q

1 is still high and the gate of

Q

2 is at

V

OL. A STORE pulse comes in and turns on

Q

3. How long must the STORE pulse be on, in picoseconds, to charge the gate capacitance of

Q

2 to

V

OH? 45 Note: In fact, the SR model is not very accurate for

Q

3 in this circuit because

Q

3 is in saturation, but use the SR model anyway, for simplicity.

Now, suppose

D

IN is high, the drain of

Q

1 is low, and the gate of

Q

2 is at

V

OH. A STORE pulse comes in and turns on

Q

3. How long must the STORE pulse be on, in picoseconds, to discharge the gate capacitance of

Q

2 to

V

IL? 34.83

Lab 8: First-order Transients

In this lab we'll be exploring how linear circuits respond to changes in their inputs. You may find it useful to review the first three sections of Chapter 10 in the text.

Figure 1 below shows the circuit we'll be using to explore the response of two linear devices hooked in series with the input driven by a voltage source that has a step from

0V

to

1V

at

t=0

.

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Figure 1. Circuit for testing step response

The circuit includes probes for measuring the voltage across Device #2 and the current entering the "+" terminal of Device #2. We'll be asking you determine the type and value for each of the two devices based on the voltage and current plots shown below.

For each of the voltage and current plot pairs shown below, give the component type (R, C, L) and value in the appropriate units (ohm, farads, henries) for each device. One of the two devices will always be a

1kΩ

resistor; it could be in either position. The other device is a resistor (R), capacitor (C) or inductor (L) of unknown value.

You may find it helpful to run experiments on your own copy of the test circuit in the circuit sandbox, which can be found in the Overview section of the website. Remember to

click CHECK at the bottom of that page in order to save your schematic between visits. If you include both the current and voltage probes, both curves will appear in the same plot and may overlay each other. To get separate plots like below, you'll need to insert the probes one at a time.

The step response for series connections of RC and RL circuits is discussed in Chapter 10. For series CR and LR circuits, you'll need to complete (or find!) a similar derivation in order to have the formulas you'll need to compute the unknown component values. Looking at the shape of the voltage and current curves is enough to determine the configuration of the series circuit given that one component is known to be a resistor. To determine the value of the other component, use the measurements of

(t,v)

or

(t,i)

shown on the plots to solve the formula for the

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Voltage and current plots for Circuit 1

Type of Device #1 (R, C, L): R

Component value for Device #1: 1000

Type of Device #2 (R, C, L): C

Component value for Device #2: 60p

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Type of Device #1 (R, C, L): L

Component value for Device #1: 80u

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Type of Device #1 (R, C, L): C

Component value for Device #1: 160p

Type of Device #2 (R, C, L): L

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