8.4.1. SERIES CONNECTION OF R AND C, OR R AND L
If a series connection of a resistor and a capacitor, or of a
resistor and an inductor is connected across an ac-voltage they stand for two
different impedances. Leaving aside that the capacitor and the inductor have
always a small ohmic resistance, we can simplify the situation by looking at
them at first as solely capacitive or inductive reactances Xc or X1.
Recollecting our knowledge about phase relations at L and C, we
find that the voltages appearing in the circuits shown in fig. 69 must have
special relations. As we know:
fig. 69
In a series connection of resistances the curent in both
components is equal.
Intending to draw a phasor diagram we start therefore with the
phasor of the current Itot. We know in both circuits the voltage at the resistor
Vr must be exactly in phase with that current.
While the voltage at the capacitor must be lagging for 90
degrees in relation to the current and the voltage at the inductor must be
leading for 90 degrees. As we know too: phasors are added geometrically.
fig. 70
Therefore the overall voltage Vtot will be found by shifting the
start of Vr up to the end of Vc or V1 and by drawing a line from the noughtpoint
up to the end of Vc or V1 we get the overall voltage Vtot necessary to let the
current Itot flow through the circuit.
The voltages found at those components are depending on Ohm's
Law, therefore Vr = I x R, Vc = I x Xc, and V1 = I x X1
These formulas demonstrate too: the relation between the
voltages is equal to the relation between the reactances. In order to get an
imagination of the behaviour of one of those circuits we can therefore draw
instead of the voltage-triangle a triangle made up from the resistance, the
reactance and showing the overall impedance.
fig. 71
This triangle shows very clearly: the impedance of the circuit
can be calculated by using the old formula of PHYTHAGORAS. This combination
introduced here can be used for so called PASSES.
fig. 72
If we observe how the OUTPUT VOLTAGE is changing while the
frequency of the INPUT VOLTAGE is increased over a certain range we observe that
the output voltage is:
- either changing from low to high values
(HIGH-PASS)
- or from high values to low values (LOW-PASS)
The combination of R and C or R and L offers four different
possibilities depending on where the components are positioned.
fig. 73
fig. 75a
fig. 75b
As we can see from the graphs showing the outputvoltage is not
suddenly cut off totally at a special frequency, but the outputvoltage is fading
out over a wide range of frequency.
We can calculate the output-voltages at various frequencies at
each PASS-combination by the following formulas:
“HIGH” - Passes |
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“LOW” - Passes |
_{} |
_{} |
Nevertheless for technicians it is necessary to compare
different passes in relation to their ability to pass or to cut off the input
signal.
Therefore there was defined a certain “LIMITTING
FREQUENCY” which is considered as the frequency from which on the
output-voltage is defined as “cut-off”. This limitting frequency is
reached if the output-voltage is equal or lower than 70.7% of the input-signal.
This limiting frequency can be calculated by the following formulas:
limit frequency with RC - combinations |
limit frequency with RL - combinations |
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CHECK YOURSELF.
1. What does the term PASS mean 7
2. What is the difference between a HIGH - and a LOWPASS.
3. An RL Highpass should have a limitting frequency of 120 Hz.
You have a coil with 150 mH. What is the resistance necessary for this pass.
4. What is the limitting frequency of a Low pass which is
consisting of a resistance R=120 Ohms and a capacitor of 2 mikroFarad?
5. The limitting frequency of an amplifier should be 28 Hz. The
coupling-capacitor. The coupling-capacitor has a capacity of 4.7 nF. Which
resistance must have the
resistor?