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

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. 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).

On a phasor diagram this is:

The physical current and potentials are:

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.

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.

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

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

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

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.

The complex e.m.f. supplied is: The complex potential across the output resistor is:

The physical potential across the output resistor is:

A graph of output versus frequency gives:

The output potential is zero for a D.C. potential, and E_m 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.

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:

The complex potential across the output capacitor is:

The physical potential across the output capacitor is:

A graph of output potential versus frequency gives:

The output potential is E_m 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.

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

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.

Adding the potentials around the circuit:

On a phasor diagram this is:

The physical current and potentials are:

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.

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).
	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.
The complex potential difference across the resistor is in phase with the current. The rms potential difference is 60 V.	The complex potential difference across the inductor leads the emf by 0.64 radians (370). The rms potential difference is 80 V.

RL high pass filter circuit

The complex e.m.f. supplied is:

The complex potential across the output inductor is:

The physical potential across the output inductor is:

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 E_m 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.

RL low pass filter circuit

The complex e.m.f. supplied is:

The complex potential across the output resistor is:

The physical potential across the output resistor is:

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 E_m 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.

RLC series A.C. circuits

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.

Adding the potentials around the circuit:

On a phasor diagram this is:

The physical current and potentials are:

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.

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).
	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.
	The complex potential difference across the resistor is in phase with the current. The rms potential difference is 180 V.
	The complex potential difference across the inductor leads the emf by 0.88 radians (50.20). The rms potential difference is 270 V.
	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.

Impure or Practical Inductors in A.C. series circuits

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:

On a phasor diagram this is:

The physical current and potentials are:

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.

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).
	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.
	The complex potential difference across the resistor is in phase with the current. The rms potential difference is 45 V.
	The complex potential difference across the inductor leads the emf by 0.29 radians (16.90). The rms potential difference is 301 V.
	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.

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, , 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.

The Apparent Power is the size of , (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.

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.

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