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# Module 10.2

# Practical Parallel Circuits

**After studying this section, you should be able to:**- • Describe the action of practical LCR parallel circuits with the use of phasor diagrams .

### Fig. 10.2.1a

### Fig 10.2.1a Looking at the inductive (LR) branch of the parallel circuit.

Fig 10.2.1a shows a practical LCR parallel circuit, where R is the internal resistance of the inductor L, plus any additional resistance in the inductive arm of the circuit. Before considering the whole circuit, the inductive branch will be examined as though it was a separate LR series circuit, and the arm containing C will be temporarily ignored. An understanding of what happens in L and R will be the foundation for a better understanding of the whole circuit.

### Fig 10.2.1b Phasors for the L and R

### Fig. 10.2.1b

Fig 10.2.1b shows a phasor diagram for the LR branch of the circuit in Fig 10.2.1a, drawn as it would be for an LR series circuit. The branch of the circuit containing C is being ignored. The reference phasor is (I_{S}) and because the same current (I_{S}) passes through both R and L, the phasors for I_{L} and V_{R} will be in the same phase. V_{S} is the phasor sum of V_{L} and V_{R}.

In a parallel circuit it will be the supply voltage V_{S} that is common to all components and so will be used as the reference phasor in Fig 10.2.2.

### Fig 10.2.2 Phasors for the LR branch of a parallel LCR circuit

### Fig. 10.2.2

Fig 10.2.2 shows Fig 10.2.1b modified for a parallel circuit. The complete diagram is rotated so that the phasor for V_{S} is horizontal and used as the reference phasor. This is because, when describing PARALLEL circuits, it is the supply voltage (V_{S}) that is common to all components.

The phasors for I_{L} and V_{R} are in phase with each other, and V_{L} leads I_{L} by 90°. However the phase angle θ between V_{S} and I_{L} (and I_{S}) will vary with frequency. This is because the value of X_{L} and therefore V_{L} will increase as frequency increases. Because V_{L} changes in length, and V_{S} is fixed, angle θ will change, which will have an effect on the phasor diagrams for the complete LCR circuit.

### Fig 10.2.3a Phasors for the LR branch of a parallel LCR circuit at HIGH frequency.

### Fig. 10.2.3a

Fig 10.2.3a represents the condition when the frequency of the supply is high, so X_{L} and therefore V_{L} will be large. V_{S} is the phasor sum of V_{R} and V_{L}.

It follows then, that the phase angle θ is some value between 0° and 90° with I_{L} lagging on V_{S}. In the ideal circuit I_{L} always lags on V_{S} by 90°, so the effect of adding some resistance will be to reduce the angle of lag (θ). At higher frequencies however V_{L} and θ increase and the circuit becomes more like a pure inductor.

It is important to note that the value of X_{L} depends on both the frequency and the value of inductance. The value of R will also depend on the design of the inductor and so V_{L} and θ will depend on both the frequency of V_{S} and on component values.

### Fig 10.2.3b Phasors for the LR branch of a parallel LCR circuit at LOW frequency.

### Fig. 10.2.3b

Fig 10.2.3b shows the effect of reducing the frequency of V_{S} to a low value. X_{L} will now be smaller, and so will V_{L}.

V_{S} is still the phasor sum of V_{R} and V_{L}, due to the reduction in X_{L}, I_{L} will increase and most of the supply voltage will be developed across R, increasing V_{R}. With V_{L} reduced in amplitude and V_{R} increased, angle θ is very small making I_{S} and V_{S} nearly in phase, making the circuit much more resistive than inductive.

This means that in a practical circuit, where the inductor must possess some resistance, the angle θ by which I_{L} lags V_{S} is not the 90° difference that would be expected of a pure inductor, but will be somewhere between 0° and 90°, depending on the frequency of the supply. At frequencies where X_{L} is much greater than R the circuit is predominantly inductive but at comparatively low frequencies where the normally small value of R may become comparable or even greater than X_{L} the circuit becomes more predominantly resistive.

### Fig 10.2.4a The complete LCR parallel circuit.

### Fig. 10.2.4a

Returning to the whole LCR circuit, three phasors, I_{C}, I_{L} and the reference phasor V_{S} are used to show the operation of the complete parallel circuit shown in Fig 10.2.4a.

Current phasors for L and C are used because V_{L} (combined with its internal resistance R_{L}) and V_{C} will be the same as they are connected in parallel across the supply. It is the currents through L and through C that will differ. The phasor for I_{C} leads V_{S} (which is also the voltage across C and L) by 90° and I_{L} lags V_{S} by somewhere between 0°and 90°, depending on component values and supply frequency.

### Fig 10.2.4b Phasors for the complete LCR parallel circuit.

### Fig. 10.2.4b

In Fig 10.2.4b a fourth phasor I_{S} (the supply current) will be the phasor sum of I_{C} and I_{L}, which in this diagram is larger than I_{C}. The two current phasors I_{C} and I_{L} are not in exact anti phase so the phasor for I_{S} is lagging that for V_{S}. Therefore the circuit is inductive.