H8P1

 H8P1: Charging an Inductor Consider the following circuit:

Let R=220.0Ω and L=0.1H. Suppose time is measured in milliseconds. Starting at time

t=−5ms and continuing, we know that V(t)=V0+Λδ(t)V, where V0=4.0V and Λ=0.007Webers. Also, assume that we measured iL(−1)=−0.01A. What is the current, in Amperes, through the inductor at t=0−, a time infinitesimally before the 0.015

impulse is delivered? Hint: The system has not yet reached steady state. What is the current, in Amperes, through the inductor at t=0+, a time infinitesimally after the impulse is delivered?

0.085

impulse is delivered?

What is the current, in Amperes, through the inductor at t=1ms, after the 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:

I models the rate of injection of the drug into the body. vC 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=400.0mg of cipro into a

C=70kg 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 vC(0+), of the antibiotic in the body, in 400/70

mg/L? The half-life of the antibiotic in the person is about th=4.0Hours. Thus, the concentration will drop to half of its initial value in that time. What is the resistance, in seconds/liter, in our model 296.78

circuit, that will produce this rate of loss of concentration? 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 166.67

initial concentration? 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.

 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 Q2. 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" 1μm technology satisfying the static discipline:

VS=5.0V, VOH=3.5V, VIH=3.0V, VIL=0.9V, VOL=0.5V The gate-source capacitance CGS=3.5fF and VT=1.0V.

RON=1800.0Ω, ROFF=95.0MΩ and RPU=10.0kΩ. The specifications for this logic family say that we have to refresh this memory every 64ms. So, if we charge the gate capacitance to VOH it will take more than 64ms to decay to VIH. If it takes

0.1s to decay that far, what must be the parasitic resistance, in TerraOhms from the gate to the 185.34 source of Q2? Note: (1 TΩ=1012Ω). 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 vDS is positive.) For Q1 and Q2 this is not an issue, but for Q3 sometimes we are charging the gate of Q2, 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 DIN is low Q1 turns off and its drain goes high. What is the maximum value of voltage on the drain of Q1, in Volts?

5

Now, suppose the drain of Q1 is high, as above, and the store line is held at the same voltage as

the drain of Q1. What is the maximum voltage, in Volts, that the gate of Q2 can be charged to? Note, this value must be larger than VOH to satisfy the static discipline.

3.8

The drain of Q1 is still high and the gate of Q2 is at VOL. A STORE pulse comes in and turns on Q3. How long must the STORE pulse be on, in picoseconds, to charge the gate capacitance of Q2 to VOH?

45

Note: In fact, the SR model is not very accurate for Q3 in this circuit

because Q3 is in saturation, but use the SR model anyway, for simplicity. Now, suppose DIN is high, the drain of Q1 is low, and the gate of Q2 is at VOH. A STORE pulse comes in and turns on Q3. How long must the STORE pulse be on, in picoseconds, to discharge the gate capacitance of Q2 to VIL?

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.

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 component value, remembering that the other component is a 1kΩ resistor.

Voltage and current plots for Circuit 1

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

R

Component value for Device #1:

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

1000

C

Component value for Device #2:

60p

Voltage and current plots for Circuit 2

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

L

Component value for Device #1:

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

80u

R

Component value for Device #2:

1000

Voltage and current plots for Circuit 3

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

C

Component value for Device #1:

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

160p

R

Component value for Device #2:

1000

Voltage and current plots for Circuit 4

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

R

Component value for Device #1:

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

1000

L

Component value for Device #2:

40u

Voltage and current plots for Circuit 5

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

R

Component value for Device #1:

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

3300

R

Component value for Device #2:

1000