Electronics Lecture 8 A circuit analysis usg phasors 8. Introduction The preious lecture discussed the transient response of an circuit to a step oltage by switchg a battery. This lecture will estigate the frequency response of the same circuit to an A oltage. Our knowledge of the impedance of a capacitor makes it relatiely easy to analyse this circuit by treatg it as a oltage diider. The results are best plotted on a logarithmic scale and the db units we troduced lecture 3 will proe useful. The results will show that the frequency doma the circuit can behae as a low or high-pass filter (dependg on configuration) which rejects high or low frequency put signals. Practical uses of such circuits will be discussed, particular the transmission of telephone and radio signals a little later. In the followg two lectures the method of circuit analysis usg complex quantities presented here will be extended to clude ductors and eentually to resonant L circuits. 8. A generalised rule for impedances The concept of impedance was troduced lecture 7. Impedance has the units of Ω (can you see why?) but for capacitors and ductors there is a phase difference between the oltage and current at the termals of these components, unlike resistors where Z Z Figure 8. these are always phase. Figure 8. shows two impedances series. These may be resistors, capacitors or ductors. We assume for the moment that the total impedance is gien by: Z total Z +Z. This is clearly correct for two resistors as we showed lecture but what ab capacitors? Substitutg Z /j and Z /j gies Z total /j total /j +/j ; remog the constant j term leads directly to the rule for capacitors series shown the last lecture. Extendg this to the general form for impedances parallel /Z total /Z + /Z gies the correct result for resistors, and for capacitors gies total +. We could extend this rule to coer combations of resistors, capacitors (and later) ductors. The results will, general, yield complex numbers but we already know how to rationalise these to obta alues of the modulus and argument which are real and measurable. 8.3 Analysis of an circuit The circuit shown figure 8. is known as a low pass filter. We will analyse this circuit the frequency doma, that is, determe its behaiour oer a wide range of frequency. Proided the frequency of the oltage Figure 8. source is not too high (say the GHz range) the resistor will behae as an ideal lear component with a constant alue of. Section 7.6 of the preious lecture showed that the impedance of the capacitor Z /j and the presence of the denomator tells us that the capacitor impedance is low at high frequency and large at low frequency. onsequently we anticipate that for low frequencies where Z >> most of the put oltage will be dropped across the capacitor whilst at high frequencies Z << only a small fraction of will be dropped across the capacitor. Therefore we analyse the circuit by monitorg the oltage across the capacitor, labelled the diagram. Note that we hae not said anythg yet ab phase
changes between and. We will fd that the analysis usg the complex impedance of the capacitor leads naturally to the changes of phase. Applyg KVL - - We saw lecture how this led to the oltage diider rule when considerg two resistors. Here, we use the same method but with two impedances to get: Z (8.) + Z We will analyse many circuits like this the remag lectures and it is common to use the ratio /. This is also known as the oltage transfer ratio or more generally, the ga, A. It may seem odd to use the term ga sce we know that will always be smaller than makg the ga less than unity but you will see when we discuss amplifiers later the course that the / ratio can be used quite generally. Note also that A is a phasor operator which acts on the phasors representg and, i.e. V AV. Substitutg for Z : / j + / j + j Where the presence of the j term dicates a phase change between and. We can rationalise (8.) by multiplyg the numerator and denomator with the complex conjugate of the numerator ( - j) to gie: (8.) j (8.3) + + which gies the rectangular (a + jb) form of A. If we were performg an experiment the laboratory we would measure and (or their ratio) and any phase change between them usg an oscilloscope. In dog so we would be measurg the modulus and angle of (8.3). These are gien by: ( + / ( a + b ) and / φ b/ a ) tan (8.4) 8.4 Frequency dependence of A and φ - Bode plots The modulus of / or ga is plotted figure 8.3 oer a wide frequency range for alues kω and 3 nf. As we predicted, the ga is close to unity for low frequencies droppg to ab. by 5 rad/s. This circuit is known, for obious reasons, as a low pass filter and is used many applications to limit the bandwidth of signals as will be discussed later. Neertheless the form of the graph is rather unterestg. If stead we plot the data on logarithmic axes as shown figure 8.4 where the ordate is gien db units the. Figure 8.3 n modulus of V/Vi.8.6.4 kω 3 nf V V ( + ). 4 6 8 (rad/s)
form of the plot proides us at a glance with the important aspects of the circuit, namely that the ga is (or -4-3dB Figure 8.4 Ga (db) -9-4 -9 3 4 5 6-4 -9 db) up to a alue slightly greater than 4 rad/s aboe which it drops rapidly. Followg on from our time doma analysis of a capacitor we troduced the time constant τ ; we do somethg similar here and assign /. At (8.4) becomes. 77 (8.5) / + ecallg from section 3.4 that G( db) log we see that the ga 3dB at as dicated on the graph. The graph can be approximately fitted by two asymptotes which meet at. For this reason is also called the 3dB pot or the corner frequency. Below the ga is unity whilst aboe it decreases with a constant slope. Sce this roll off occurs at moderately high frequencies where > we can approximate (8.5) by ga so for eery doublg of the frequency the ga reduces by. A reduction by a factor of means a reduction of 6dB and the slope has a alue of -6dB per octae. Similar reasong gies the alue - db per decade.. Figure 8.5 φ -π/4 -π/ 3 4 5 6 Equation (8.4) shows that the phase change between and is also dependent on frequency and figure 8.5 shows a plot of φ agast log. Summarisg the three regions of the graph: Up to ~. and are phase aboe ~ lags by π/ for termediate alues there is an almost lear shift and at, lags by π/4 Amplitude and phase graphs are known collectiely as Bode plots. Their great adantage is that only the alue of need be calculated and the form of the plot is easy to draw. 3
Some further sight can be obtaed by lookg back at figure 8. and applyg KVL: +. Sce the components form a series circuit the stantaneous current through and flowg to or of the capacitor termals must be the same. Let this current be i I cos t, then I cos t. For the capacitor we know that i d /dt so / I dt which is / s t. These functions are shown figure 8.6 for the case of small Amplitude 5 4 3 - - -3-4 -5 I cos t / s t Angle (radians) Figure 8.6. Under this condition Z >> and most of the put oltage is dropped across the capacitor. The put oltage is the addition of these two functions and is dicated by the lighter, grey le. We can see immediately that the amplitude of ~ as expected and the phase difference is small. It is easy to enisage the situation when is large; then Z << and the amplitude of ~ and these waes are almost phase and π/ of phase with. The same formation can be represented on an Argand diagram shown figure 8.xx. As before the current is gien by ii cos t. The oltage across the I/ directly to (8.4) φ I resistor i I cos t and we show the phasor representg (V I ) lyg along the real axis accordg to our conention. The oltage across the capacitor ( ) lags the current by π/ and is represented by the phasor V IX s t which lies along the negatie imagary axis as shown. To fd the phasor representg we add the two phasors usg the parallelogram rule as shown by the dot/dashed le. The amplitude of is (I / +I ) / while the phase difference between the put and put oltages is gien by tan φ -. In phasor form this is V I (/ + ) / arctan -, which leads 8.5 The high pass filter If we terchange the capacitor and resistor as shown figure 8.5 it is easy to show that: Figure 8.5 A (8.6) + / j j / ationalisg this complex expression gies ( + / / ) (8.7) At low frequencies where << the ga~/, is small and creases with creasg At high frequencies the ga ~. 4
This behaiour is the opposite to that found for the low pass filter and not surprisgly this circuit is called a high pass filter. The phase shift is gien by φtan - /. At low frequency (< /) leads by π/; at high frequencies (> ) and are phase. At leads by π/4. For a simple Jaa ersion of the low and high pass filters see: https://www.st-andrews.ac.uk/~www_pa/scots_guide/experiment/lowpass/lpf.html 8.6 ascaded low pass filters If two low pass filters are cascaded the put oltage is not simply the product of two transfer functions. If you try to analyse such a circuit usg phasors and an Argand diagram you will soon realise the shortcomgs of that technique. This is because the second filter acts as a load for the first. Although slightly more oled than anythg presented until now it is possible to show that the ga is gien by: ( ) + j3 When /, and the amplitude of the ga is/3 and the phase is -π/ j3 Howeer, we shall later that a simple operational amplifier circuit called the unity ga buffer acts as an impedance matcher so that loadg of the first filter by the second does not occur. For such an arrangement employg different alues of and correspondg to different alues of ( and ): ( + j / )( + j / where and are the corner frequencies of the two circuits. The figure shows the resultg Bode plot and also ) Ga db (rad/s) 3 4 5 6 dicates a useful property the log-log plots, namely that they can simply be added. Aboe the roll-off is twice that of the sgle circuit i.e. - db per octae. 5