RESISTANCES AND CAPACITANCES 



by the scalar IR, the product is a directed quantity {Figure 3.15). Similarly 

 when sin Mt is multiphed by the scalar —IjcoC the product is another directed 

 quantity (downward, because capacitive reactance is by convention negative) 

 {Figure 3.16). Their sum is obtained by vector addition {Figure 3.17). The 



Yf^ of length//? 



IR 



Figure 3.15 



I/cuc 



V(,, of length 



Figure 3.16 



Figure 3.17 



angle between y^ and y^^ is the phase angle ^, which is now seen to be equal 

 to 



tan 



-1 ^/^ 



= tan-i 



1 



IR mCR 



and the length of t;^^, which gives us [y^^l, is now seen by Pythagoras to be 



\{IRf + 



\o)c) 



2U/2 



i\r^ + 



2U/2 



Thus the complete description of y^^ is 



modulus 



phase angle 

 Y 



'AB 



{ i 1 \2U/2 i 1 



/ W + 1^1 j cos(w? + ^) where ^ - tan-i ^j^^ 



Impedance — The impedance of the circuit is the ratio of terminal voltage 

 to current, is measured in ohms, and is symbohzed by Z. It expresses with 

 circuits containing resistances and capacitances the amount of opposition 

 to the flow of current, in a manner analogous to the notions of resistance 

 and reactance in circuits containing, respectively, resistors and capacitors 

 only. The presence of the phase difference between voltage and current 

 makes impedance a 'complex number', containing a modulus part and an 

 angle part. If, for the present, we restrict ourselves to the modulus, then 



\vj^j,\=I\Z\ {cf.V=IR) 

 and on comparing this with the expression for y^^ above 



or, remembering that capacitive reactance, X^ = lIcoC 



\z\ = {R' + {Xc)'yi^ 



The network we have been discussing is an extremely simple one, yet the 

 analysis has yielded a square root term, and square root terms are notoriously 

 tiresome in calculations. It is not hard to see that in the analysis of more 

 elaborate resistance and capacitance networks, the equations are liable to 

 become very unwieldy. There is an analytical technique which we shall use 



32 



