# Combining equations to make one!

#### chunkylumber111

Hello this question is based on a transfer function problem and I have the following formulas and need to arrive at an equation for T/VD in terms of k0, k1, k2 and kt

where E = VD - VM, I = Ek1, T = k0I, Vt = ktT and Vm = k2Vt

My attempt:

T = k0I where I = Ek1 therefore T = k0Ek1 where E = VD - VM hence T = K0(VD - VM)K1

Also we know that T = Vt/kt where Vt = VM/k2 therefore T = VM/(k2kt) hence VM = Tk2kt and substituting VM = Tk2kt into T = K0(VD - VM)K1

we have T = K0(VD - Tk2kt)K1 expanding brackets: T = K0K1VD - K0k1Tk2kt I've tried various rearragements of this but don't seem to come up with T/VD can anyone show me where I'm going wrong?
I'm really stuck

Thanks!

#### princeps

$$\displaystyle \frac{T}{V_D}=\frac{k_0 \cdot I}{E+V_M}=\frac{k_0 \cdot I}{\frac{I}{k_1}+k_2 \cdot V_t}=\frac{k_0 \cdot I}{\frac{I}{k_1}+k_2 \cdot k_t \cdot T}=$$

$$\displaystyle =\frac{k_0 \cdot I}{\frac{I}{k_1}+k_2 \cdot k_t \cdot k_0 \cdot I}=\frac{k_0}{\frac{1}{k_1}+k_2 \cdot k_t \cdot k_0}=\frac{k_1 \cdot k_0}{1+k_1 \cdot k_2 \cdot k_t \cdot k_0}$$

#### chunkylumber111

Got it now, thanks for your help!