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Theoretical foundations of electrical engineering

# Calculation of the frequency characteristics of the given two-port terminal device

Problem situation
«1. Calculate for a given circuit:
а) complex function of the input resistance ZВХ(jw), its frequency response ZВХ(jω) and phase-frequency φz(jω) characteristics;
в) complex function of the voltage transfer ratio KU(jω), its frequency response KU(ω) and phase-frequency φk(ω) characteristics.
2. Construct diagrams at given circuit elements Zвх(ω), φz(ω), KU(ω), φk(ω) (in linear and logarithmic scale on the frequency axis).
3. Construct frequency locus (diagrams of the frequency response characteristics) ZВХ(jω), KU(jω)).
4. Define characteristic frequencies.
5. Explain qualitatively steps of the received characteristics.»

Given
R1=R2=R3=10k ohms;
C1=1 μF.

## Solution

Input resistance of the circuit:

Module of the input resistance:

Input resistance phase:

Voltage transfer function in the most compact form and in the form with separate real and imaginary part:

Module of the transfer function:

Argument of the transfer function:

Module diagram of the input resistance depending on the frequency (in linear scale)::

Module diagram of the input resistance depending on the frequency (in logarithmic scale – limit construction is expanded to 10 kHz):

Argument diagram of the input resistance depending on the frequency (in linear scale):

Argument diagram of the input resistance depending on the frequency (in logarithmic scale – limit construction is expanded to 10 kHz):

Module diagram of the voltage transfer function depending on the frequency (in linear scale):

Module diagram of the voltage transfer function depending on the frequency (in logarithmic scale – limit construction is expanded to 10 kHz):

Argument diagram of the voltage transfer function depending on the frequency (in linear scale):

Argument diagram of the voltage transfer function depending on the frequency (in logarithmic scale – limit construction is expanded to 10 kHz):

Frequency locus of the input resistance function:

Frequency locus of the voltage transfer function:

Determination of characteristic frequencies
In this scheme there is only one reactive element, therefore, the resonance cannot occur. The only characteristic frequency is frequency of the maximum phase shift of the output voltage.

Qualitative explanation of the received characteristics
Input resistance
At DC (zero frequency) the capacitor is equivalent to the circuit break, current does not pass through it, and therefore, resistance is infinite.
At infinite frequency the capacitor has no time to charge/discharge and therefore, is equivalent to a wire. Therefore, the input resistance consists of the resistance of three-series connected resistors. Total 10k ohms +10k ohms + 10k ohms=30k ohms.
When the resistance is determined by the capacitor (low frequencies)- the shift phase will be -90о, because

When the resistance is determined by resistors (high frequencies) – there is no shift phase because the current through active resistance is codirectional with the voltage drop on it.
All qualitative suggestions are shown in diagrams.
Voltage transfer function
The transfer function of the circuit only with passive elements cannot be more than one.
At low frequency the condenser is equal to the circuit break, therefore, input and output voltages coincide – transfer coefficient is equal to one.
At infinite frequency – the condenser is not taken into account and we see that output voltage is taken from one of the three series resistors. Therefore, transfer function should be equal 1/3. Phase characteristic. We assume that the test voltage source has zero phase shift then the phase of the transfer function is a phase of the output voltage.
Phase characteristic. We assume that the test voltage source has zero phase shift then the phase of the transfer function is a phase of the output voltage.
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