Electronics in High Energy Physics Introduction to electronics in HEP Electrical Circuits (based on P.Farthoaut lecture at Cern)

Electrical Circuits Generators Thevenin / Norton representation Power Components Sinusoidal signal Laplace transform Impedance Transfer function Bode diagram RC-CR networks Quadrupole

Sources v r R + - I Voltage Generator Current Generator

Thevenin theorem (1) Rth Vth A A B B Any two-terminal network of resistors and sources is equivalent to a single resistor with a single voltage source Vth = open-circuit voltage Rth = Vth / Ishort

Thevenin theorem (2) V R1 R2 + - A R1//R2 A Voltage divider

Norton representation B A Rno Ino Vth Rth Any voltage source followed by an impedance can be represented by a current source with a resistor in parallel

Power transfer v r R + - I Power in the load R P is maximum for R = r

Sinusoidal regime

Complex notation Signal : v1(t) = V cos( t + ) v2(t) = V sin( t + ) v(t) = v1 + j v2 = V e j( t + ) = V ej ej t = S ej t Interest: S = V ej contains only phase and amplitude ej t contains time and frequency Real signal = R [ S ej t ] In case of several signals of same  only complex amplitude are significant and one can forget ej t One can separate phase and time

Complex impedance In a linear network with v(t) and i(t), the instantaneous ratio v/i is often meaningless as it changes during a period To i(t) and v(t) one can associate J ej t and S ej t S / J is now independent of the time and characterizes the linear network Z = S / J is the complex impedance of the network Z = R + j X = z ej R is the resistance, X the reactance z is the module,  is the phase z, R and X are in Ohms Examples of impedances: Resistor Z = R Capacitance (perfect) Z = -j / C; Phase = - /2 100 pF at 1MHz  1600 Ohms 100 pF at 100 MHz  16 Ohms Inductance (perfect) Z = jL; Phase = + /2 100 nH at 1 MHz  0.63 Ohms 100 nH at 100 MHz  63 Ohms

Power in sinusoidal regime i = IM cos  t in an impedance Z = R + j X = z ej v = z IM cos( t + ) = R IM cos t - X IM sin t p = v i = R IM2 cos2t – X IM2 cost sin t = R IM2 /2 (1+cos2t ) - X IM2 /2 sin2 t p = P (1+cos2 t ) - Pq sin2 t = pa + pq pa is the active power (Watts); pa = P (1+ cos2t) Mean value > 0; R IM2 /2 pq is the reactive power (volt-ampere); pq = Pq sin2t Mean value = 0 Pq = X IM2 /2 In an inductance X = L ; Pq > 0 : the inductance absorbs some reactive energy In a capacitance X = -1/C; Pq < 0 : the capacitance gives some reactive energy

Real capacitance A perfect capacitance does not absorb any active power it exchanges reactive power with the source Pq = - IM2 /2C In reality it does absorb an active power P Loss coefficient tg  = |P/Pq| Equivalent circuit Resistor in series or in parallel tg  = RsCs tg  = 1/RpCp Cs Rp Rs Cp

Real inductance Similarly a quality coefficient is defined Q = Pq/P Equivalent circuit Resistor in series or in parallel Q = Ls/Rs Q = Rp/Lp Ls Rp Rs Lp

Laplace Transform (1) v = f(i) integro-differential relations In sinusoidal regime, one can use the complex notation and the complex impedance V = Z I Laplace transform allows to extend it to any kind of signals Two important functions Heaviside (t) = 0 for t < 0 = 1 for t  0 Dirac impulsion (t) = ’(t) = 0 for t  0

Laplace Transform (2) Examples Linearity Derivation, Integration Translation

Laplace Transform (3) Change of time scale Derivation, Integration of the Laplace transform Initial and final value

Impedances i(t) Z v(t) I(p) Z(p) V(p) Network v(t), I(t) Generalisation V(p) = Z(p) I(p)

Transfer Functions I1 I2 Transfer Function V2 V1 Input V1, I1; Output V2, I2 Voltage gain V2(p) / V1(p) Current gain I1(p) / I2(p) Transadmittance I2(p) / V1(p) Transimpedance V2(p) / I1(p) Transfer function Out(p) = F(p) In(p) Convolution in time domain:

Bode diagram (1) Replacing p with j in F(p), one obtains the imaginary form of the function transfer F(j) = |F| ej() Logarithmic unit: Decibel In decibel the module |F| will be The phase of each separate functions add Functions to be studied

Bode diagram (2) a F(p) = p + a ; |F1|db= 20 log | j  + a|  [rad/s] 20 dB per decade 6 dB per octave a 3 dB error F(p) = p + a ; |F1|db= 20 log | j  + a| Bode diagram = asymptotic diagram  < a, |F1| approximated with A = 20 log(a)  > a, |F1| approximated with A = 20 log() 6 dB per octave (20 log2) or 20 dB per decade (20 log10) Maximum error when  = a 20 log| j a + a| - 20 log(a) = 20 log (21/2) = 3 dB

Bode diagram (3) a |F2|db= - 20 log | j  + a|  [rad/s] - 20 dB per decade -6 dB per octave a 3 dB error |F2|db= - 20 log | j  + a| Bode diagram = asymptotic diagram  < a, |F2| approximated with A = - 20 log(a)  > a, |F2| approximated with A = - 20 log() - 6 dB per octave (20 log2) or - 20 dB per decade (20 log10) Maximum error when  = a 20 log| j a + a| - 20 log(a) = 20 log (21/2) = 3 dB

Bode diagram (4) Low pass filters As before but: |F|dB  [rad/s] -20 dB per decade Low pass filters -40 dB per decade As before but: Slope 6*n dB per octave (20*n dB per decade) Error at =a is 3*n dB

Bode diagram (5) Phase of F1(j ) = (j  + a) Asymptotic diagram tg  = /a Asymptotic diagram  = 0 when  < a  = /4 when  = a  = /2 when  > a

Bode diagram (6) Phase of F2(j ) = 1/(j  + a) Asymptotic diagram tg  =- /a Asymptotic diagram  = 0 when  < a  = - /4 when  = a  = - /2 when  > a

Bode diagram (7) |F3|dB = 20 log|b2 - 2 + 2aj| Asymptotic diagram 40 dB per decade b |F3|dB = 20 log|b2 - 2 + 2aj| Asymptotic diagram  --> 0 A = 40 log b  --> ∞ A’ = 20 log 2 = 40 log  A = A’ for  = b Error depends on a and b p2 + 2a p + b2 = b2[(p/b)2 + 2(a/b)(p/b) + 1] Z = a/b U = /b

Bode diagram (8) |F4|dB = - 20 log|b2 - 2 + 2aj| Asymptotic diagram -40 dB per decade b |F4|dB = - 20 log|b2 - 2 + 2aj| Asymptotic diagram  --> 0 A = - 40 log b  --> ∞ A’ = - 20 log 2 = - 40 log  A = A’ for  = b Error depends on a and b Z = a/b U = /b

Bode diagram (9) Z = 0.1 Z = 1 Phase of F3(j) = (b2 - 2 + 2aj) and F4(j) = 1/(b2 - 2 + 2aj) tg  = 2a/ (b2 - 2) Asymptotic diagram  = 0 when  < b  = ± /2 when  = b  = ±  when  > b

RC-CR networks (1) V1 V2 R C Integrator; RC = time constant Dirac response Heaviside response t/RC V C R V1 V2 Integrator; RC = time constant

RC-CR networks (2) C R V1 V2 Low pass filter c = 1/RC

RC-CR networks (3) V1 V2 C R Derivator; RC = time constant Dirac response Heaviside response RC = 1 t/RC V C R V1 V2 Derivator; RC = time constant

RC-CR networks (4) R2 C1 C2 V1 R1 V2 Dirac response Heaviside response RiCi = RC t/RC V V1 C1 R1 V2 R2 C2

RC-CR networks (5) V1 C1 R1 V2 R2 C2 Band pass filter

Time or frequency analysis (1) A signal x(t) has a spectral representation |X(f)|; X(f) = Fourier transform of x(t) The transfer function of a circuit has also a Fourier transform F(f) The transformation of a signal when applied to this circuit can be looked at in time or frequency domain x(t) X(f) y(t) = x(t) * f(t) Y(f) = X(f) F(f) f(t) F(f)

Time or frequency analysis (2) The 2 types of analysis are useful Simple example: Pulse signal (100 ns width) (1) What happens when going through a R-C network? Time analysis (2) How can we avoid to distort it? Frequency analysis

Time or frequency analysis (3) Time analysis C R X(t) Y(t)

Time or frequency analysis (4) Contains all frequencies Most of the signal within 10 MHz To avoid huge distortion the minimum bandwidth is 10-20 MHz Used to define the optimum filter to increase signal-to-noise ratio

Quadrupole I1 I2 V1 V2 Passive Active Parameters Network of R, C and L Internal linked sources Parameters V1, V2, I1, I2 Matrix representation

Parameters Impedances Admittances Hybrids

Input and output impedances Input impedance: as seen when output loaded Zin = Z11 - (Z12 Z21 / (Z22 + Zu)) Zin = h11 - (h12 h21 / (h22 + 1/Zu)) Output impedance: as seen from output when input loaded with the output impedance of the previous stage Zout = Z22 - (Z12 Z21 / (Z11 + Zg)) 1/Zout = h22 - (h12 h21 / (h11+ Zg))