Week10

19/05/12 21:26Courseware – MITx 6.002x

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H10P1:

MAGNITUDE

AND ANGLE

Here are four first-order circuits:

The parameters k , nF, and mH.

For each of these circuits there is a magnitude and phase of the voltage-transfer ratio

among the following two graphs. We want you to choose, for each circuit the appropriate

magnitude and angle graph.

R = 8.2 Ω C = 0.61 L = 8.2VoVi



In the spaces provided please enter your choices. For example, if you chose magnitude and

angle for circuit A we want you to enter their product .

Circuit A:

Circuit B:

Circuit C:

Circuit D:

Check

pu p ∗ u

H10P2:

IMPEDANCES

For each of the following circuits compute the impedance.



In each space write an algebraic expression for the impedance in terms of , , , and . (As

usual, use for in your expressions.) In each case we also ask, "How does the impedance

behave as and as ?" If the answer is zero, enter a "0", if the answer is a constant

enter the algebraic expression for the constant, and if the answer is infinity, enter the symbol

"inf".

The impedance of circuit A,

As

As

The impedance of circuit B,

As

As

The impedance of circuit C,

As

Z R C L ωw ωω → 0 ω → ∞

=ZA

ω → 0 →ZA

ω → ∞ →ZA

=ZB

ω → 0 →ZB

ω → ∞ →ZB

=ZC

ω → 0 →ZC



As

The impedance of circuit D,

As

As

Check

ω → ∞ →ZC

=ZD

ω → 0 →ZD

ω → ∞ →ZD

H10P3:

AN L

NETWORK

The inductor and capacitor in the diagram below are part of the output-coupling network of a radio

transmitter. The rest of the transmitter (the source of radio-frequency energy) is represented as a

Thevenin source, and the antenna load is represented by a resistor.

In this problem we will examine some of the characteristics of this circuit. In the spaces provided

below you will write algebraic expressions in terms of the part parameters , , , , and the

angular frequency . (As usual, use for in your expressions.)

L C R1 R2ω w ω



One thing we want to know is the voltage-transfer ratio (the ratio of the complex amplitude of the

output voltage to the complex amplitude of the input voltage) of this network, as a function of the

operating frequency. Now that we know about impedances this is just like solving a resistive ladder!

In the space provided below write an algebraic expression for this ratio.

Look carefully at what you just computed. What is it for ? What happens as ? You

should always examine system functions this way.

Another important value is the driving-point impedance that the final amplifier "sees" looking at the

antenna through the coupling network. This is the ratio of the complex amplitude of the voltage

across the input port to the complex amplitude of the current into that port. In this circuit it is . In

the space provided below write an algebraic expression for this impedance. (Hint: The algebra is

often easier if you invert parallel impedances to make admittances. They then just add.)

Again, look carefully at what you just computed. What is it for ? What happens as ?

Remember that we found that in resistive circuits the load that absorbs the maximum power from a

Thevenin source is the one where the load resistance is the same as the source resistance. Here we

have an antenna that we want to transfer power to, but both the amplifier and the load have given

resistances and . Since capacitors and inductors do not eat power, they just store the energy

temporarily, perhaps if we choose the inductance and capcitance wisely we can couple the amplifier

to the antenna very well.

It is possible to find values of and that make the driving-point impedance you just computed

exactly , if . This will "match the antenna to the amplifier".

In the space provided below write an algebraic expression for the capacitance that allows this

match:

In the space provided below write an algebraic expression for the inductance that allows this

match:

Now let's look at some real numbers. For a big transmitting amplifier the output resistance may be

. A typical antenna has a radiation resistance of . Consider an AM

broadcast transmitter at kHz. In the spaces provided below, write the numerical values of

the capacitance (in picoFarads) and inductance (in microHenrys) for match.

VoVi

ω = 0 ω → ∞

VfIf

ω = 0 ω → ∞

R1 R2

L CR1 >R1 R2

Cmatch

Lmatch

= 1000.0ΩR1 = 50.0ΩR2f = 990.0

=Cmatch



By the way, AM broadcast transmitters can be very large: up to 50kW. The parts used for such power

levels are impressive. For example, an inductor may be made of large gauge silver-coated copper

tubing. There is a nice picture here .

Check

=Lmatch

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