Saturday, June 1, 2013

Lecture 14 (RC, RL and RLC AC circuits)
In this lecture complex numbers are used to analyse A.C. series circuits, in particular:
• Resistance Capacitance (RC) circuits
• Resistance (Pure) Inductance (RL) circuits
• Resistance (Pure) Inductance and Capacitance (RLC) circuits
• Resistance (Real) Inductance and Capacitance (RLC) circuits




RC series A.C. circuits
diagram
The e.m.f. that is supplied to the circuit is distributed between the resistor and the capacitor. Since the same current must flow in each element, the resistor and capacitor are in series. The common current can often be taken to have the reference phase.
Complex current equation
In a series circuit, the potential differences are added up around the circuit.
(In a parallel circuit where the emf is the same across all elements, the currents are added).



diagram
On a phasor diagram this is:
diagram


The physical current and potentials are:

diagram


The applied emf is φ rad behind the current in the circuit.


Example
A 255V, 500/π Hz supply is connected in series with a 100R resistor and a 2μF capacitor. Taking the phase of the emf as a reference, find the complex and rms values of
(a) the current in the circuit, and
(b) the potential difference across each element.

First write the complex emf and how it is distributed around the circuit.
emf equations


1.37 radians is about 780. The total impedance of the circuit is seen in the relationship between emf and current. The complex and rms currents are now calculated.

current equations

The current leads the applied emf phase reference by 1.37 radians or 780.
phase graph


The potential differences across the resistor and capacitor are now calculated.
current equations
current equations


The resistor potential difference is in phase with the current and the capacitor potential difference lags the current phase by π/2 (or 900).

pd graph


RC high pass filter circuit
Since the impedance of the RC series circuit depends on frequency, as indicated above, the circuit can be used to filter out unwanted low frequencies.
diagram
The complex e.m.f. supplied is:
diagram

The complex potential across the output resistor is:
diagram


The physical potential across the output resistor is:
diagram
A graph of output versus frequency gives:
diagram
The output potential is zero for a D.C. potential, and Em for very high frequency. Low frequencies are suppressed and high frequencies are not really affected. The cut-off frequency is arbitrarily chosen as the frequency where only half the input power is output.
diagram


The half power angular frequency is the reciprocal of the time constant RC. The phase will be π/4 at the half power frequency.



RC low pass filter circuit
As above, the complex e.m.f. supplied is:
diagram

diagram
The complex potential across the output capacitor is:
diagram


The physical potential across the output capacitor is:
diagram
A graph of output potential versus frequency gives:
diagram
The output potential is Em for a D.C. potential, and zero for very high frequency. High frequencies are suppressed and low frequencies are not really affected. The cut-off frequency is also chosen as the frequency where only half the input power is output.
diagram



The half power angular frequency is again the reciprocal of the time constant RC. The phase will also be π/4 at the half power frequency.



RL series A.C. circuits
circuit diagram
The e.m.f. that is supplied to the circuit is distributed between the resistor and the capacitor. Since the resistor and capacitor are in series the common current is taken to have the reference phase.
diagram
Adding the potentials around the circuit:
diagram
On a phasor diagram this is:
diagram
The physical current and potentials are:
diagram


The applied emf is φ rad ahead of the current in the circuit.



Example
A 100V, 1000/π Hz supply is connected in series with a 30R resistor and a 20mH inductor. Take the emf as the reference phase and find:
(a) the complex impedance of the circuit
(b) the complex, real (i.e. physical) and rms currents, and
(c) the complex, real (i.e. physical) and rms potential differences across each element.



diagram

The complex impedance for the circuit is 50 Ω, and the phase angle between current and applied emf is 0.93 radians (or about 530).
diagram


diagram




The emf is the reference phase.





The real (i.e. physical) current is the imaginary part of the complex current and lags behind the applied emf with -0.93 radians (-530).
The rms current is an equivalent dc current of 2 A and has no phase.
diagram
The complex potential difference across the resistor is in phase with the current.
The rms potential difference is 60 V.
diagram
The complex potential difference across the inductor leads the emf by 0.64 radians (370).
The rms potential difference is 80 V.
diagram



RL high pass filter circuit
circuit diagram
The complex e.m.f. supplied is:
emf equation
The complex potential across the output inductor is:
diagram
The physical potential across the output inductor is:
diagram



The equations have the same physical form as the RC high pass filter, but with time constant L/R instead of RC. The output potential is Em for a very high frequency, and zero for D.C. potential. Low frequencies are suppressed and high frequencies are not really affected. The half power angular frequency is again the reciprocal of the time constant.
diagram



RL low pass filter circuit
diagram
The complex e.m.f. supplied is:
diagram


The complex potential across the output resistor is:
diagram
The physical potential across the output resistor is:
diagram
The equations have the same physical form as the RC low pass filter, but with time constant L/R instead of RC. The output potential is Em for a D.C. potential, and zero for very high frequency. High frequencies are suppressed and low frequencies are not really affected. The half power angular frequency is again the reciprocal of the time constant.
diagram




RLC series A.C. circuits
diagram
The e.m.f. that is supplied to the circuit is distributed between the resistor, the inductor, and the capacitor. Since the elements are in series the common current is taken to have the reference phase.
diagram
Adding the potentials around the circuit:
diagram
On a phasor diagram this is:
diagram



The physical current and potentials are:
diagram




Example
A 240V, 250/π Hz supply is connected in series with 60R, 180mH and 50μF. Take the emf as the reference phase and find:
(a) the complex impedance of the circuit
(b) the complex, real (i.e. physical) and rms currents, and
(c) the complex, real (i.e. physical) and rms potential differences across each element.






diagram

The complex impedance for the circuit is 78.1 Ω, and the phase angle between current and applied emf is 0.69 radians (or 39.80).

diagram
complex current




The emf is the reference phase.





The real (i.e. physical) current is the imaginary part of the complex current and lags behind the applied emf with -0.69 radians (-39.80).
The rms current is an equivalent dc current of 3 A and has no phase.
complex potential difference across the resistor
The complex potential difference across the resistor is in phase with the current.
The rms potential difference is 180 V.
complex potential difference across the inductor
The complex potential difference across the inductor leads the emf by 0.88 radians (50.20).
The rms potential difference is 270 V.
complex potential difference across the capacitor
The complex potential difference across the capacitor lags the emf with -2.27 radians (-129.80). (A negative angle is measured clockwise from the positive "x" axis).
The rms potential difference is 120 V.
diagram



Impure or Practical Inductors in A.C. series circuits
diagram
In general, an inductor will have resistance because it is made of normally resistive wire. The potential difference across the inductor includes both elements because they cannot be physically separated.
Adding the potentials around the circuit:
diagram
On a phasor diagram this is:
diagram
The physical current and potentials are:
diagram





Example
A source of alternating current provides an r.m.s. potential difference of 195V at 1000 rad.s-1. A resistor (30R), a real inductor (20R, 200m), and a capacitor (12μ5) are connected in series with the supply. Find
Take the emf as the reference phase and find:
(a) the complex impedances of the circuit elements and the total circuit,
(b) the complex, real (i.e. physical) and rms currents, and
(c) the complex, real (i.e. physical) and rms potential differences across each element.
diagram
The complex impedance for the circuit is 130 Ω, and the phase angle between current and applied emf is 1.18 radians (or 67.40).
diagram

diagram



The emf is the reference phase.





The real (i.e. physical) current is the imaginary part of the complex current and lags behind the applied emf with -1.18 radians (-67.40).
The rms current is an equivalent dc current of 1.5 A and has no phase.
diagram
The complex potential difference across the resistor is in phase with the current.
The rms potential difference is 45 V.
diagram
The complex potential difference across the inductor leads the emf by 0.29 radians (16.90).
The rms potential difference is 301 V.
diagram
The complex potential difference across the capacitor lags the emf with -2.75 radians (-1570). (A negative angle is measured clockwise from the positive "x" axis).
The rms potential difference is 120 V.
diagram



Average Power dissipated in RLC series A.C. circuits
Power is not dissipated in inductance and capacitance; it is only dissipated in resistance.
Average Power is calculated with rms quanties.
In general for a A.C. circuit with an applied e.m.f., E, and any series combination of the three circuit elements, Resistance, Inductance and Capacitance, there will be a total resistance, R, and a resultant reactance, X.

This will produce
a Real (Resistive) Power, P,
a Reactive Power, jQ, and
a Complex Power, diagram, which is the sum of them.

For a resultant Inductive Reactance, there will be the kind of diagram shown on the right and the following relationships (in r.m.s. terms) can be derived it.
diagram
diagram

The Apparent Power is the size of diagram, (i.e. EI ), but the Real Power (P) dissipated is less than this when the current and applied potential are not in phase. The factor cosφ is called the power factor.
Only the Real Power is given the Unit of Watt.
Apparent and Complex power are given the Unit VA.
Reactive Power (Q) is given the unit VAR (Volt Amp Reactive).
P, Q and S are related by a Pythagorean relationship.
When the applied potential is designated as the phase reference, then the diagram will be rotated clockwise by φ and φ will be negative for a resultant Inductive Reactance.



Example
A 45 V rms, 1000 rad.s-1 supply is connected in series with a 50R6 resistor and a practical inductor which has 40m inductance and 30R resistance combined. Take the applied potential as the reference phase and angles in radian. Find
(a) the complex impedances of the inductor and the total circuit
(b) the complex current in the circuit,
(c) the potential differences across the resistor and the inductor
(d) the apparent and real total power dissipated, and
(e) the real power dissipated by the resistor and inductor.



power example



Summarising:
Complex potentials and currents hold both magnitude and phase information.
Resistor/Capacitor and Resistor/Inductor circuits can form filters to block high or low frequency signals.
Average Power is calculated with rms quanties.
Apparent Power is the product of applied emf and current.
Real Power is the product of applied emf, current and cos(the phase angle between emf and current) (in Watt). cos(phase angle between emf and current) is called the power factor.
Apparent and Complex power are given the unit VA (Volt Amp).
Reactive Power is given the unit VAR (Volt Amp Reactive).

Acknowledgement: These notes are based in part on "Alternating Current Circuit Theory" by G.J.Russell and K.Mann NSWUP 1969.

1 comment:

  1. Thanks for looking at Peter Eylands website www.insula.com.au - see more free physics pages there.

    ReplyDelete