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Impedance in an AC circuit. The stress resonance

Consider the circuit in which the series connected resistor, an inductor and a capacitor (Fig. 4). The voltage at the terminals of the circuit generated by an external source, like the previous examples, the changes harmonically with an amplitude

In general, current in the circuit and the voltage does not vary in the same phase, so

external voltage:

Fig. 4. Fig. 5.

In accordance with a voltage , и can be written as:

- in phase with the current;

- ahead the current strength on phase:

- lags behind from current strength on phase.

Substituting the values ​​of in after trigonometric transformations may be an expression for the impedance Z in the AC circuit and the phase difference . However, a simple and easy to manage to do that using the vector diagrams.

In fig.5. along the currents axis directed vector of current strength amplitude Since in the entire circuit amplitude of the current is the same, then the amplitude of the voltages at the sites postpone relative this vector: vector - in one phase with a current; vector - ahead current strength in phase by , vector - with a lag of current strength in phase by . Summing up three vectors we find graphically the values ​​and . Using the Pythagorean theorem, we have

Substituting the expressions of these amplitudes and taking into account Ohm's law, we find , where Z-total resistivity of the AC is called impedance.

Therefore:

We obtain

Difference of inductive and capacitive reactance is called reactive resistivity. We write Ohm's law as applied to the amplitudes of voltage and current in the circuit (see.fig.5)

From Figure 5. we also find an expression for in terms of known quantities:

If the inductive and capacitive resistance of the circuit when connected in series would be the same , then [see (5)] Z = R. Such case of forced electrical oscillations is called the stress.

Since then . Hence we find the resonant circular frequency:

Under this condition, the impedance circuit Z has the smallest value equal to R (when data R, L and C), for the resonance of voltage in the circuit shown in Fig. 6.

If then and , current strength lags on phase of the applied voltage. When we have and Current strength is ahead on the phase of voltages.

Figure 6. represented the graph of resonance curve, showing the behavior of the current when the voltage near the resonance frequency is changed. The higher the resistance of the body, the lower will be the wave. During filling of given site with blood, its resistance decreases, and this causes an increase in conductivity, i.e. growth recorded the current. Impedance depends on the resistive and capacitive resistances. At high frequency of current capacitive reactance value close to zero, and therefore the impedance of tissue depends mainly on the active resistance of the blood supply and including systolic wave.

Problems:

1.In a chain of alternating current with a frequency of 400 Hz is enabled the coil with inductance of 0.1 GN. At which current capacity in this chain will be resonance?

2.Into the network with voltage of 36 V and a frequency of 1000 Hz sequentially included 4 Om resistance, inductance and capacitance 2 mH 8 uF. To determine the current in the chain, the terminal voltage active resistance, inductance, capacitance, and the phase difference between current and voltage.

3.In the AC line voltage U = 110 V and a frequency of γ = 100 Hz sequentially included capacitor C = 5 * 10 F and an inductor L = 0.2 Gn and ohmic resistance R = 4 Om.

Determine 1) effective force current in the circuit 2) current frequency at which resonance occurs voltages (resonant frequency) 3) By the resonant current and voltage across the inductance and capacitance for resonance

4. Find the amplitude value of the current in a circuit containing a capacitor 1 mF. Tension in the chain 250 V, and the resistance of 2.5 ohms. The capacitor and resistor are connected in series. Frequency of 50 Hz.

5. The phase shift between current and voltage during the passage of the alternating current frequency of 25 Hz through the muscle was 35 degrees. What is the capacitance in the equivalent circuit of series-connected resistor and a capacitor if the resistance k = 0,5?

6 Strike slip of phases between current and voltage during the passage of the alternating current frequency of 30 Hz through the muscle is 65 degrees. What is the resistance of the resistor in the equivalent circuit of series-connected capacitor and resistor, if the capacitor 3.6 mF?

6.The finiteness which is under the electrodes has 1 Om resistance and capacitance of 0.2 mF. Determine the phase shift angle between current and voltage for the frequency of 50 Hz, connected in series.

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