User:John R. Brews/Sample

In the theory of electrical circuits, Norton's theorem allows the replacement of a two-terminal portion of a linear circuit by a simplified circuit consisting of a current source, called the Norton voltage source, in parallel with an impedance, called the Norton impedance. Norton's theorem is the dual of Thévenin's theorem, which replaces a two-terminal portion of a linear circuit by a simplified circuit consisting of a voltage source in series with an impedance.

Finding the Norton components

To determine the Norton current, short circuit the two terminals and measure or calculate the short-circuit current between the terminals.

To determine the Norton impedance, set all independent sources between the two terminals to zero (leave the dependent sources active), and determine the impedance seen looking into the two terminals. For example, apply a known voltage across the port terminals and measure the current drawn, or apply a known current and determine the voltage developed. The Norton impedance is then the ratio of the voltage across the terminals to the current passing through the terminals.

In the case of a frequency dependent applied voltage or current, the impedance may be dependent upon frequency and exhibit a phase dependent upon frequency, as normally expressed using an impedance represented by a complex number.

Example

(PD) Image: John R. Brews
Bipolar current buffer.

(PD) Image: John R. Brews
Small-signal circuit to find output current.

The bipolar circuit in the figure at left represents a current buffer. The operation of this circuit is as follows. The DC current source I_E determines the current leaving the emitter in the quiescent state when no signal is applied. By appropriate selection of the DC current source it can be arranged that the transistor is configured to operate usefully, that is, it is in active mode.

A signal voltage i_S is applied to the emitter. The time-varying current drawn through the base of the transistor is very small compared to that in the emitter. The DC current source I_E does not permit any time varying current to pass through it, so at the signal frequency it behaves as an open circuit. Consequently as the signal changes the emitter current, the collector current follows suit, and the output current i_out in the load resistor R_L is a copy of the input signal i_in.

As this discussion indicates, the current gain of this circuit is very nearly unity, so amplification is not achieved. The purpose of this circuit is to isolate the signal source from the load R_L, with the object of preventing the load from affecting the performance of the source. In particular, if the source is forced to draw a large current, that may adversely impact its performance, and this circuit insures that only a small current is demanded of the source. Because the object of this circuit is not current amplification, but isolation of the driver, it is called a current buffer.

The transistor is a very nonlinear device, so the output signal current in the load resistor is not a faithful amplification of the input signal at the emitter unless the signal amplitude is small. For purposes of estimating the amplifier behavior, the assumption is made that the signal amplitudes are very small so the amplifier appears to behave as a linear circuit. Under these circumstances, the transistor can be approximated by a linear circuit composed of resistors called the hybrid pi model. The linear small-signal approximation to the circuit using the transistor hybrid pi model is shown in the figure with a few algebraic expressions derived by application of Kirchhoff's current law. This circuit is used to determine the Norton current (the short-circuit current) of the buffer.

The value of the Norton current is the output current through a short-circuit load, that is, with R_L = 0. From the small-signal circuit on the right, this current is i_O, which is found by equating the values of the emitter voltage found from the output and the input sides of the circuit:

$v_{E}=(i_{S}-i_{O})(r_{\pi }//R_{S})\ ,$

and:

$v_{E}=i_{O}R_{L}+(i_{O}-g_{m}v_{\pi })r_{O}\ .$

Equating thise expressions, the Norton current is found as:

$i_{N}=i_{O}=i_{s}\left({\frac {r_{\pi }//R_{S}}{r_{\pi }//R_{S}+{\frac {r_{O}+R_{C}//R_{L}}{1+g_{m}r_{O}}}}}\right)\approx i_{s}\left({\frac {g_{m}(r_{\pi }//R_{S})}{1+g_{m}(r_{\pi }//R_{S})}}\right)\ ,$

where the approximation holds because R_L is taken as zero for a short-circuit output condition, and because g_mr_O >> 1.