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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 = 8.2
Vo
Vi

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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
p
u p ∗ u
H10P2:
IMPEDANCES
For each of the following circuits compute the impedance.

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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 ω
w ω
ω → 0 ω → ∞
=ZA
ω → 0 →ZA
ω → ∞ →ZA
=ZB
ω → 0 →ZB
ω → ∞ →ZB
=ZC
ω → 0 →ZC

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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 R2
ω w ω

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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.
Vo
Vi
ω = 0 ω → ∞
Vf
If
ω = 0 ω → ∞
R1 R2
L C
R1 >R1 R2
Cmatch
Lmatch
= 1000.0ΩR1 = 50.0ΩR2
f = 990.0
=Cmatch

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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